List of space groups

{{Short description|None}} There are 230 [[space group#Table of space groups in 3 dimensions|space groups]] in three dimensions, given by a number index, and a full name in [[Hermann–Mauguin notation]], and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a [[point group]] of the unit cell.

==Symbols== In [[Hermann–Mauguin notation]], space groups are named by a symbol combining the [[point group]] identifier with the uppercase letters describing the [[Bravais lattice#In 3 dimensions|lattice type]]. Translations within the lattice in the form of [[screw axis|screw axes]] and [[glide planes]] are also noted, giving a complete crystallographic space group.

These are the [[Bravais lattice#In 3 dimensions|Bravais lattices in three dimensions]]: *'''P''' primitive *'''I''' body-centered (from the German ''Innenzentriert'') *'''F''' face-centered (from the German ''Flächenzentriert'') *'''S''' base-centered (from the German ''Seitenflächenzentriert''), or specifically: **'''A''' centered on A faces only **'''B''' centered on B faces only **'''C''' centered on C faces only *'''R''' rhombohedral

A reflection plane '''m''' within the point groups can be replaced by a [[glide plane]], labeled as '''a''', '''b''', or '''c''' depending on which axis the glide is along. There is also the '''n''' glide, which is a glide along the half of a diagonal of '''a''' face, and the '''d''' glide, which is along a quarter of either a face or space diagonal of the unit cell. The '''d''' glide is often called the diamond glide plane as it features in the [[diamond]] structure. * <math>a</math>, <math>b</math>, or <math>c</math>: glide translation along half the lattice vector of this face * <math>n</math>: glide translation along half the diagonal of this face * <math>d</math>: glide planes with translation along a quarter of a face diagonal * <math>e</math>: two glides with the same glide plane and translation along two (different) half-lattice vectors.{{refn|group=note|name=e|The symbol <math>e</math> was introduced by the [[International Union of Crystallography|IUCR]] in 1992. Prior to this, the space groups Aem2 (No. 39), Aea2 (No. 41), Cmce (No. 64), Cmme (No. 67), and Ccce (No. 68) were known as Abm2 (No. 39), Aba2 (No. 41), Cmca (No. 64), Cmma (No. 67), and Ccca (No. 68) respectively. Historical literature may refer to the old names, but their meaning is unchanged.<ref>{{cite journal |last1=de Wolff |first1=P. M. |last2=Billiet |first2=Y. |last3=Donnay |first3=J. D. H. |last4=Fischer |first4=W. |last5=Galiulin |first5=R. B. |last6=Glazer |first6=A. M. |last7=Hahn |first7=T. |last8=Senechal |first8=M. |last9=Shoemaker |first9=D. P. |last10=Wondratschek |first10=H. |last11=Wilson |first11=A. J. C. |last12=Abrahams |first12=S. C. |title=Symbols for symmetry elements and symmetry operations. Final report of the IUCr Ad-Hoc Committee on the Nomenclature of Symmetry |journal=Acta Crystallographica Section A |volume=48 |issue=5 |date=1992-09-01 |issn=0108-7673 |pages=727–732 |doi=10.1107/s0108767392003428 |doi-access=free|bibcode=1992AcCrA..48..727D }}</ref>}}

A gyration point can be replaced by a [[screw axis]] denoted by a number, ''n'', where the angle of rotation is <math>\color{Black}\tfrac{360^\circ}{n}</math>. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (twofold) rotation followed by a translation of {{sfrac|1|2}} of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of {{sfrac|1|3}} of the lattice vector. The possible screw axes are: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅.

Wherever there is both a rotation or screw axis ''n'' and a mirror or glide plane ''m'' along the same crystallographic direction, they are represented as a fraction <math display="inline">\frac{n}{m}</math> or ''n/m''. For example, 4₁/a means that the crystallographic axis in question contains both a 4₁ screw axis as well as a glide plane along '''a'''.

In '''Schoenflies notation''', the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form <math>\Gamma_x^y</math> which specifies the Bravais lattice. Here <math>x \in \{t, m, o, q, rh, h, c\}</math> is the lattice system, and <math>y \in \{\empty, b, v, f\}</math> is the centering type.<ref>{{cite book |last1=Bradley |first1=C. J. |last2=Cracknell |first2=A. P. |title=The mathematical theory of symmetry in solids: representation theory for point groups and space groups |publisher=Clarendon Press |location=Oxford New York |year=2010 |isbn=978-0-19-958258-7 |oclc=859155300 |pages=127–134}}</ref>

In '''Fedorov symbol''', the type of space group is denoted as ''s'' (''symmorphic'' ), ''h'' (''hemisymmorphic''), or ''a'' (''asymmorphic''). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.

===Symmorphic=== The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Example for point group 4/mmm (<math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math>): the symmorphic space groups are P4/mmm (<math>P\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math>, ''36s'') and I4/mmm (<math>I\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math>, ''37s'').

===Hemisymmorphic=== The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Example for point group 4/mmm (<math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math>): hemisymmorphic space groups contain the axial combination 422, but at least one mirror plane ''m'' will be substituted with glide plane, for example P4/mcc (<math>P\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{c}</math>, ''35h''), P4/nbm (<math>P\tfrac{4}{n}\tfrac{2}{b}\tfrac{2}{m}</math>, ''36h''), P4/nnc (<math>P\tfrac{4}{n}\tfrac{2}{n}\tfrac{2}{c}</math>, ''37h''), and I4/mcm (<math>I\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{m}</math>, ''38h'').

===Asymmorphic=== The remaining 103 space groups are asymmorphic. Example for point group 4/mmm (<math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math>): P4/mbm (<math>P\tfrac{4}{m}\tfrac{2_1}{b}\tfrac{2}{m}</math>, ''54a''), P4₂/mmc (<math>P\tfrac{4_2}{m}\tfrac{2}{m}\tfrac{2}{c}</math>, ''60a''), I4₁/acd (<math>I\tfrac{4_1}{a}\tfrac{2}{c}\tfrac{2}{d}</math>, ''58a'') - none of these groups contains the axial combination 422.

==List of triclinic== {| class="wikitable floatright" style="text-align:center;" |+ Triclinic Bravais lattice |- | [[File:Triclinic.svg|80px]] |}

{| class=wikitable |+ [[Triclinic crystal system]] ! Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] |- align=center |1||1||<math>1</math>||P1|| P 1|| <math>\Gamma_tC_1^1</math> || ''1s''||<math>(a/b/c)\cdot 1</math> || <math>(\circ)</math> |- align=center |2||{{overline|1}}||<math>\times</math>||P{{overline|1}}|| P {{overline|1}}|| <math>\Gamma_tC_i^1</math> || ''2s''||<math>(a/b/c)\cdot \tilde 2</math> || <math>(2222)</math> |}

==List of monoclinic== {| class="wikitable floatright" |+ Monoclinic Bravais lattice |- ! Simple (P) ! Base (S) |- | [[File:Monoclinic.svg|80px]] | [[File:Base-centered monoclinic.svg|80px]] |} {{sticky header}} {| class="wikitable sticky-header" |+ [[Monoclinic crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! colspan=2|Full name(s) ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] (primary) ! [[Fibrifold notation|Fibrifold]] (secondary) |- align=center |{{anchor|3-5}}3||rowspan=3|2||rowspan=3|<math>22</math>||P2|| P 1 2 1||P 1 1 2 || <math>\Gamma_mC_2^1</math> || ''3s'' || <math>(b:(c/a)):2</math> || <math>(2_02_02_02_0)</math> || <math>({*}_0{*}_0)</math> |- align=center |4||P2₁||P 1 2₁ 1||P 1 1 2₁ || <math>\Gamma_mC_2^2</math> || ''1a'' || <math>(b:(c/a)):2_1</math> || <math>(2_12_12_12_1)</math> || <math>(\bar{\times}\bar{\times})</math> |- align=center |5||C2|| C 1 2 1||B 1 1 2 || <math>\Gamma_m^bC_2^3</math> || ''4s'' || <math>\left ( \tfrac{a+b}{2}/b:(c/a)\right ) :2</math> || <math>(2_02_02_12_1)</math> || <math>({*}_1{*}_1)</math>, <math>({*}\bar{\times})</math> |- align=center |{{anchor|6-9}}6||rowspan=4|m||rowspan=4|<math>*</math>||Pm|| P 1 m 1||P 1 1 m || <math>\Gamma_mC_s^1</math> || ''5s'' || <math>(b:(c/a))\cdot m</math> || <math>[\circ_0]</math> || <math>({*}{\cdot}{*}{\cdot})</math> |- align=center |7||Pc|| P 1 c 1||P 1 1 b || <math>\Gamma_mC_s^2</math> || ''1h'' || <math>(b:(c/a))\cdot \tilde c</math> || <math>(\bar\circ_0)</math> || <math>({*}{:}{*}{:})</math>, <math>({\times}{\times}_0)</math> |- align=center |8||Cm|| C 1 m 1||B 1 1 m || <math>\Gamma_m^bC_s^3</math> || ''6s'' || <math>\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m</math> || <math>[\circ_1]</math> || <math>({*}{\cdot}{*}{:})</math>, <math>({*}{\cdot}{\times})</math> |- align=center |9||Cc|| C 1 c 1||B 1 1 b || <math>\Gamma_m^bC_s^4</math> || ''2h'' || <math>\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c</math> || <math>(\bar\circ_1)</math> || <math>({*}{:}{\times})</math>, <math>({\times}{\times}_1)</math> |- align=center |{{anchor|10-15}}10||rowspan=6|2/m||rowspan=6|<math>2*</math>||P2/m||P 1 2/m 1||P 1 1 2/m || <math>\Gamma_mC_{2h}^1</math> || ''7s'' || <math>(b:(c/a))\cdot m:2</math> || <math>[2_02_02_02_0]</math> || <math>(*2{\cdot}22{\cdot}2)</math> |- align=center |11||P2₁/m||P 1 2₁/m 1||P 1 1 2₁/m || <math>\Gamma_mC_{2h}^2</math> || ''2a'' || <math>(b:(c/a))\cdot m:2_1</math> || <math>[2_12_12_12_1]</math> || <math>(22{*}{\cdot})</math> |- align=center |12||C2/m||C 1 2/m 1||B 1 1 2/m || <math>\Gamma_m^bC_{2h}^3</math> || ''8s'' || <math>\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m:2</math> || <math>[2_02_02_12_1]</math> || <math>(*2{\cdot}22{:}2)</math>, <math>(2\bar{*}2{\cdot}2)</math> |- align=center |13||P2/c||P 1 2/c 1||P 1 1 2/b || <math>\Gamma_mC_{2h}^4</math> || ''3h'' || <math>(b:(c/a))\cdot \tilde c:2</math> || <math>(2_02_022)</math> || <math>(*2{:}22{:}2)</math>, <math>(22{*}_0)</math> |- align=center |14||P2₁/c||P 1 2₁/c 1||P 1 1 2₁/b || <math>\Gamma_mC_{2h}^5</math> || ''3a'' || <math>(b:(c/a))\cdot \tilde c:2_1</math> || <math>(2_12_122)</math> || <math>(22{*}{:})</math>, <math>(22{\times})</math> |- align=center |15||C2/c||C 1 2/c 1||B 1 1 2/b || <math>\Gamma_m^bC_{2h}^6</math> || ''4h'' || <math>\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c:2</math> || <math>(2_02_122)</math> || <math>(2\bar{*}2{:}2)</math>, <math>(22{*}_1)</math> |}

==List of orthorhombic== {| class=wikitable style="text-align:center;" |+ Orthorhombic Bravais lattice |- ! Simple (P) ! Body (I) ! Face (F) ! Base (S) |- | [[File:Orthorhombic.svg|80px]] | [[File:Orthorhombic-body-centered.svg|80px]] | [[File:Orthorhombic-face-centered.svg|80px]] | [[File:Orthorhombic-base-centered.svg|80px]] |} {{sticky header}} {| class="wikitable sticky-header" |+ [[Orthorhombic crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] (primary) ! [[Fibrifold notation|Fibrifold]] (secondary) |- align=center |{{anchor|16-24}}16||rowspan=9|222||rowspan=9|<math>222</math>||P222||P 2 2 2|| <math>\Gamma_oD_2^1</math> || ''9s'' || <math>(c:a:b):2:2</math> || <math>(*2_02_02_02_0)</math> || |- align=center |17||P222₁||P 2 2 2₁|| <math>\Gamma_oD_2^2</math> || ''4a'' || <math>(c:a:b):2_1:2</math> || <math>(*2_12_12_12_1)</math> || <math>(2_02_0{*})</math> |- align=center |18||P2₁2₁2||P 2₁ 2₁ 2|| <math>\Gamma_oD_2^3</math> || ''7a'' ||<math>(c:a:b):2</math> [[File:Circled_colon.png|16px]] <math>2_1</math> || <math>(2_02_0\bar{\times})</math> || <math>(2_12_1{*})</math> |- align=center |19||P2₁2₁2₁||P 2₁ 2₁ 2₁|| <math>\Gamma_oD_2^4</math> || ''8a'' || <math>(c:a:b):2_1</math> [[File:Circled_colon.png|16px]] <math>2_1</math> || <math>(2_12_1\bar{\times})</math> || |- align=center |20||C222₁||C 2 2 2₁|| <math>\Gamma_o^bD_2^5</math> || ''5a'' || <math>\left ( \tfrac{a+b}{2}:c:a:b\right ) :2_1:2</math> || <math>(2_1{*}2_12_1)</math> || <math>(2_02_1{*})</math> |- align=center |21||C222||C 2 2 2|| <math>\Gamma_o^bD_2^6</math> || ''10s'' || <math>\left ( \tfrac{a+b}{2}:c:a:b\right ) :2:2</math> || <math>(2_0{*}2_02_0)</math> || <math>(*2_02_02_12_1)</math> |- align=center |22||F222||F 2 2 2|| <math>\Gamma_o^fD_2^7</math> || ''12s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :2:2</math> || <math>(*2_02_12_02_1)</math> || |- align=center |23||I222||I 2 2 2|| <math>\Gamma_o^vD_2^8</math> || ''11s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:b\right ) :2:2</math> || <math>(2_1{*}2_02_0)</math> || |- align=center |24||I2₁2₁2₁||I 2₁ 2₁ 2₁|| <math>\Gamma_o^vD_2^9</math> || ''6a'' ||<math>\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :2:2_1</math> || <math>(2_0{*}2_12_1)</math> || |- align=center |{{anchor|25-35}}25||rowspan=22|mm2||rowspan=22|<math>*22</math>||Pmm2||P m m 2|| <math>\Gamma_oC_{2v}^1</math> || ''13s'' || <math>(c:a:b):m \cdot 2</math> || <math>(*{\cdot}2{\cdot}2{\cdot}2{\cdot}2)</math> || <math>[{*}_0{\cdot}{*}_0{\cdot}]</math> |- align=center |26||Pmc<!-- Not a PMCID -->2₁||P m c 2₁|| <math>\Gamma_oC_{2v}^2</math> || ''9a'' || <math>(c:a:b): \tilde c \cdot 2_1</math> || <math>(*{\cdot}2{:}2{\cdot}2{:}2)</math> || <math>(\bar{*}{\cdot}\bar{*}{\cdot})</math>, <math>[{\times_0}{\times_0}]</math> |- align=center |27||Pcc2||P c c 2 || <math>\Gamma_oC_{2v}^3</math> || ''5h'' || <math>(c:a:b): \tilde c \cdot 2</math> || <math>(*{:}2{:}2{:}2{:}2)</math> || <math>(\bar{*}_0\bar{*}_0)</math> |- align=center |28||Pma2||P m a 2 || <math>\Gamma_oC_{2v}^4</math> || ''6h'' || <math>(c:a:b): \tilde a \cdot 2</math> || <math>(2_02_0{*}{\cdot})</math> || <math>[{*}_0{:}{*}_0{:}]</math>, <math>(*{\cdot}{*}_0)</math> |- align=center |29||Pca2₁||P c a 2₁ || <math>\Gamma_oC_{2v}^5</math> || ''11a'' || <math>(c:a:b): \tilde a \cdot 2_1</math> || <math>(2_12_1{*}{:})</math> || <math>(\bar{*}{:}\bar{*}{:})</math> |- align=center |30||Pnc2||P n c 2 || <math>\Gamma_oC_{2v}^6</math> || ''7h'' || <math>(c:a:b): \tilde c \odot 2</math> || <math>(2_02_0{*}{:})</math> || <math>(\bar{*}_1\bar{*}_1)</math>, <math>({*}_0{\times}_0)</math> |- align=center |31||Pmn2₁||P m n 2₁ || <math>\Gamma_oC_{2v}^7</math> || ''10a'' || <math>(c:a:b): \widetilde{ac} \cdot 2_1</math> || <math>(2_12_1{*}{\cdot})</math> || <math>(*{\cdot}\bar{\times})</math>, <math>[{\times}_0{\times}_1]</math> |- align=center |32||Pba2||P b a 2 || <math>\Gamma_oC_{2v}^8</math> || ''9h'' || <math>(c:a:b): \tilde a \odot 2</math> || <math>(2_02_0{\times}_0)</math> || <math>(*{:}{*}_0)</math> |- align=center |33||Pna2₁||P n a 2₁ || <math>\Gamma_oC_{2v}^9</math> || ''12a'' || <math>(c:a:b): \tilde a \odot 2_1</math> || <math>(2_12_1{\times})</math> || <math>(*{:}{\times})</math>, <math>({\times}{\times}_1)</math> |- align=center |34||Pnn2||P n n 2 || <math>\Gamma_oC_{2v}^{10}</math> || ''8h'' || <math>(c:a:b): \widetilde{ac} \odot 2</math> || <math>(2_02_0{\times}_1)</math> || <math>(*_0{\times}_1)</math> |- align=center |35||Cmm2||C m m 2|| <math>\Gamma_o^bC_{2v}^{11}</math> || ''14s'' || <math>\left ( \tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2</math> || <math>(2_0{*}{\cdot}2{\cdot}2)</math> || <math>[*_0{\cdot}{*}_0{:}]</math> |- align=center |{{anchor|36-46}}36||Cmc2₁||C m c 2₁ || <math>\Gamma_o^bC_{2v}^{12}</math> || ''13a'' || <math>\left ( \tfrac{a+b}{2}:c:a:b\right ) :\tilde c \cdot 2_1</math> || <math>(2_1{*}{\cdot}2{:}2)</math> || <math>(\bar{*}{\cdot}\bar{*}{:})</math>, <math>[{\times}_1{\times}_1]</math> |- align=center |37||Ccc2||C c c 2 || <math>\Gamma_o^bC_{2v}^{13}</math> || ''10h'' || <math>\left ( \tfrac{a+b}{2}:c:a:b\right ) : \tilde c \cdot 2</math> || <math>(2_0{*}{:}2{:}2)</math> || <math>(\bar{*}_0\bar{*}_1)</math> |- align=center |38||Amm2||A m m 2 || <math>\Gamma_o^bC_{2v}^{14}</math> || ''15s'' || <math>\left ( \tfrac{b+c}{2}/c:a:b\right ):m \cdot 2</math> || <math>(*{\cdot}2{\cdot}2{\cdot}2{:}2)</math> || <math>[{*}_1{\cdot}{*}_1{\cdot}]</math>, <math>[*{\cdot}{\times}_0]</math> |- align=center |39||Aem2||A b m 2 || <math>\Gamma_o^bC_{2v}^{15}</math> || ''11h'' || <math>\left ( \tfrac{b+c}{2}/c:a:b\right ) :m \cdot 2_1</math> || <math>(*{\cdot}2{:}2{:}2{:}2)</math> || <math>[{*}_1{:}{*}_1{:}]</math>, <math>(\bar{*}{\cdot}\bar{*}_0)</math> |- align=center |40||Ama2||A m a 2 || <math>\Gamma_o^bC_{2v}^{16}</math> || ''12h'' || <math>\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2</math> || <math>(2_02_1{*}{\cdot})</math> || <math>(*{\cdot}{*}_1)</math>, <math>[*{:}{\times}_1]</math> |- align=center |41||Aea2||A b a 2 || <math>\Gamma_o^bC_{2v}^{17}</math> || ''13h'' || <math>\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2_1</math> || <math>(2_02_1{*}{:})</math> || <math>(*{:}{*}_1)</math>, <math>(\bar{*}{:}\bar{*}_1)</math> |- align=center |42||Fmm2||F m m 2 || <math>\Gamma_o^fC_{2v}^{18}</math> || ''17s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2</math> || <math>(*{\cdot}2{\cdot}2{:}2{:}2)</math> || <math>[{*}_1{\cdot}{*}_1{:}]</math> |- align=center |43||Fdd2||F d d 2 || <math>\Gamma_o^fC_{2v}^{19}</math> || ''16h'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b \right ) : \tfrac{1}{2} \widetilde{ac} \odot 2</math> || <math>(2_02_1{\times})</math> || <math>({*}_1{\times})</math> |- align=center |44||Imm2||I m m 2 || <math>\Gamma_o^vC_{2v}^{20}</math> || ''16s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :m \cdot 2</math> || <math>(2_1{*}{\cdot}2{\cdot}2)</math> || <math>[*{\cdot}{\times}_1]</math> |- align=center |45||Iba2||I b a 2 || <math>\Gamma_o^vC_{2v}^{21}</math> || ''15h'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde c \cdot 2</math> || <math>(2_1{*}{:}2{:}2)</math> || <math>(\bar{*}{:}\bar{*}_0)</math> |- align=center |46||Ima2||I m a 2 || <math>\Gamma_o^vC_{2v}^{22}</math> || ''14h'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde a \cdot 2</math> || <math>(2_0{*}{\cdot}2{:}2)</math> || <math>(\bar{*}{\cdot}\bar{*}_1)</math>, <math>[*{:}{\times}_0]</math> |- align=center |{{anchor|47-55}}47||rowspan=28|2/m 2/m 2/m (mmm)||rowspan=28|<math>*222</math>||Pmmm||P 2/m 2/m 2/m || <math>\Gamma_oD_{2h}^1</math> || ''18s'' || <math>\left ( c:a:b \right ) \cdot m:2 \cdot m</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]</math> || |- align=center |48||Pnnn||P 2/n 2/n 2/n || <math>\Gamma_oD_{2h}^2</math> || ''19h'' || <math>\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \widetilde{ac}</math> || <math>(2\bar{*}_12_02_0)</math>|| |- align=center |49||Pccm||P 2/c 2/c 2/m || <math>\Gamma_oD_{2h}^3</math> || ''17h'' || <math>\left ( c:a:b \right ) \cdot m:2 \cdot \tilde c</math> || <math>[*{:}2{:}2{:}2{:}2]</math> || <math>(*2_02_02{\cdot}2)</math> |- align=center |50||Pban||P 2/b 2/a 2/n || <math>\Gamma_oD_{2h}^4</math> || ''18h'' || <math>\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \tilde a</math> || <math>(2\bar{*}_02_02_0)</math> || <math>(*2_02_02{:}2)</math> |- align=center |51||Pmma||P 2₁/m 2/m 2/a || <math>\Gamma_oD_{2h}^5</math> || ''14a'' || <math>\left ( c:a:b \right ) \cdot \tilde a :2 \cdot m</math> || <math>[2_02_0{*}{\cdot}]</math> || <math>[*{\cdot}2{:}2{\cdot}2{:}2]</math>, <math>[*2{\cdot}2{\cdot}2{\cdot}2]</math> |- align=center |52||Pnna||P 2/n 2₁/n 2/a || <math>\Gamma_oD_{2h}^6</math> || ''17a'' || <math>\left ( c:a:b \right ) \cdot \tilde a:2 \odot \widetilde{ac}</math> || <math>(2_02\bar{*}_1)</math> || <math>(2_0{*}2{:}2)</math>, <math>(2\bar{*}2_12_1)</math> |- align=center |53||Pmna||P 2/m 2/n 2₁/a || <math>\Gamma_oD_{2h}^7</math> || ''15a'' || <math>\left ( c:a:b \right ) \cdot \tilde a:2_1 \cdot \widetilde{ac}</math> || <math>[2_02_0{*}{:}]</math> || <math>(*2_12_12{\cdot}2)</math>, <math>(2_0{*}2{\cdot}2)</math> |- align=center |54||Pcca||P 2₁/c 2/c 2/a || <math>\Gamma_oD_{2h}^8</math> || ''16a'' || <math>\left ( c:a:b \right ) \cdot \tilde a:2 \cdot \tilde c</math> || <math>(2_02\bar{*}_0)</math> || <math>(*2{:}2{:}2{:}2)</math>, <math>(*2_12_12{:}2)</math> |- align=center |55||Pbam||P 2₁/b 2₁/a 2/m || <math>\Gamma_oD_{2h}^9</math> || ''22a'' || <math>\left ( c:a:b \right ) \cdot m:2 \odot \tilde a</math> || <math>[2_02_0{\times}_0]</math> || <math>(*2{\cdot}2{:}2{\cdot}2)</math> |- align=center |{{anchor|56-64}}56||Pccn||P 2₁/c 2₁/c 2/n || <math>\Gamma_oD_{2h}^{10}</math> || ''27a'' || <math>\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot \tilde c</math> || <math>(2\bar{*}{:}2{:}2)</math> || <math>(2_12\bar{*}_0)</math> |- align=center |57||Pbcm||P 2/b 2₁/c 2₁/m || <math>\Gamma_oD_{2h}^{11}</math> || ''23a'' || <math>\left ( c:a:b \right ) \cdot m:2_1 \odot \tilde c</math> || <math>(2_02\bar{*}{\cdot})</math> || <math>(*2{:}2{\cdot}2{:}2)</math>, <math>[2_12_1{*}{:}]</math> |- align=center |58||Pnnm||P 2₁/n 2₁/n 2/m || <math>\Gamma_oD_{2h}^{12}</math> || ''25a'' || <math>\left ( c:a:b \right ) \cdot m:2 \odot \widetilde{ac}</math> || <math>[2_02_0{\times}_1]</math> || <math>(2_1{*}2{\cdot}2)</math> |- align=center |59||Pmmn||P 2₁/m 2₁/m 2/n || <math>\Gamma_oD_{2h}^{13}</math> || ''24a'' || <math>\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot m</math> || <math>(2\bar{*}{\cdot}2{\cdot}2)</math> || <math>[2_12_1{*}{\cdot}]</math> |- align=center |60||Pbcn||P 2₁/b 2/c 2₁/n || <math>\Gamma_oD_{2h}^{14}</math> || ''26a'' || <math>\left ( c:a:b \right ) \cdot \widetilde{ab}:2_1 \odot \tilde c</math> || <math>(2_02\bar{*}{:})</math> || <math>(2_1{*}2{:}2)</math>, <math>(2_12\bar{*}_1)</math> |- align=center |61||Pbca||P 2₁/b 2₁/c 2₁/a || <math>\Gamma_oD_{2h}^{15}</math> || ''29a'' || <math>\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot \tilde c</math> || <math>(2_12\bar{*}{:})</math> || |- align=center |62||Pnma||P 2₁/n 2₁/m 2₁/a || <math>\Gamma_oD_{2h}^{16}</math> || ''28a'' || <math>\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot m</math> || <math>(2_12\bar{*}{\cdot})</math> || <math>(2\bar{*}{\cdot}2{:}2)</math>, <math>[2_12_1{\times}]</math> |- align=center |63||Cmcm||C 2/m 2/c 2₁/m || <math>\Gamma_o^bD_{2h}^{17}</math> || ''18a'' ||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2_1 \cdot \tilde c</math> || <math>[2_02_1{*}{\cdot}]</math> || <math>(*2{\cdot}2{\cdot}2{:}2)</math>, <math>[2_1{*}{\cdot}2{:}2]</math> |- align=center |64||Cmce||C 2/m 2/c 2₁/a || <math>\Gamma_o^bD_{2h}^{18}</math> || ''19a''||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2_1 \cdot \tilde c</math> || <math>[2_02_1{*}{:}]</math> || <math>(*2{\cdot}2{:}2{:}2)</math>, <math>(*2_12{\cdot}2{:}2)</math> |- align=center |{{anchor|65-74}}65||Cmmm||C 2/m 2/m 2/m || <math>\Gamma_o^bD_{2h}^{19}</math> || ''19s''||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m</math> || <math>[2_0{*}{\cdot}2{\cdot}2]</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{:}2]</math> |- align=center |66||Cccm||C 2/c 2/c 2/m || <math>\Gamma_o^bD_{2h}^{20}</math> || ''20h''||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot \tilde c</math> || <math>[2_0{*}{:}2{:}2]</math> || <math>(*2_02_12{\cdot}2)</math> |- align=center |67||Cmme||C 2/m 2/m 2/e || <math>\Gamma_o^bD_{2h}^{21}</math> || ''21h''||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot m</math> || <math>(*2_02{\cdot}2{\cdot}2)</math> || <math>[*{\cdot}2{:}2{:}2{:}2]</math> |- align=center |68||Ccce||C 2/c 2/c 2/e || <math>\Gamma_o^bD_{2h}^{22}</math> || ''22h''||<math>\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c</math> || <math>(*2_02{:}2{:}2)</math> || <math>(*2_02_12{:}2)</math> |- align=center |69||Fmmm||F 2/m 2/m 2/m || <math>\Gamma_o^fD_{2h}^{23}</math> || ''21s''|| <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m</math> || <math>[*{\cdot}2{\cdot}2{:}2{:}2]</math> || |- align=center |70||Fddd||F 2/d 2/d 2/d || <math>\Gamma_o^fD_{2h}^{24}</math> || ''24h''|| <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot \tfrac{1}{2}\widetilde{ab}:2 \odot \tfrac{1}{2}\widetilde{ac}</math> || <math>(2\bar{*}2_02_1)</math> || |- align=center |71||Immm||I 2/m 2/m 2/m || <math>\Gamma_o^vD_{2h}^{25}</math> || ''20s''|| <math>\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot m</math> || <math>[2_1{*}{\cdot}2{\cdot}2]</math> || |- align=center |72||Ibam||I 2/b 2/a 2/m || <math>\Gamma_o^vD_{2h}^{26}</math> || ''23h''|| <math>\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot \tilde c</math> || <math>[2_1{*}{:}2{:}2]</math> || <math>(*2_02{\cdot}2{:}2)</math> |- align=center |73||Ibca||I 2/b 2/c 2/a || <math>\Gamma_o^vD_{2h}^{27}</math> || ''21a''|| <math>\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c</math> || <math>(*2_12{:}2{:}2)</math> || |- align=center |74||Imma||I 2/m 2/m 2/a || <math>\Gamma_o^vD_{2h}^{28}</math> || ''20a''|| <math>\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot m</math> || <math>(*2_12{\cdot}2{\cdot}2)</math> || <math>[2_0{*}{\cdot}2{:}2]</math> |}

==List of tetragonal== {| class="wikitable floatright" |+ Tetragonal Bravais lattice |- ! Simple (P) ! Body (I) |- | [[File:Tetragonal.svg|80px]] | [[File:Tetragonal-body-centered.svg|80px]] |} {{sticky header}} {| class="wikitable sticky-header" |+ [[Tetragonal crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] |- align=center |{{anchor|75-80}}75||rowspan=6|4||rowspan=6|<math>44</math>||P4||P 4 || <math>\Gamma_qC_4^1</math> || ''22s'' || <math>(c:a:a):4</math> || <math>(4_04_02_0)</math> |- align=center |76||P4₁||P 4₁ || <math>\Gamma_qC_4^2</math> || ''30a'' || <math>(c:a:a) :4_1</math> || <math>(4_14_12_1)</math> |- align=center |77||P4₂||P 4₂ || <math>\Gamma_qC_4^3</math> || ''33a'' || <math>(c:a:a) :4_2</math> || <math>(4_24_22_0)</math> |- align=center |78||P4₃||P 4₃ || <math>\Gamma_qC_4^4</math> || ''31a'' || <math>(c:a:a) :4_3</math> || <math>(4_14_12_1)</math> |- align=center |79||I4||I 4 || <math>\Gamma_q^vC_4^5</math> || ''23s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4</math> || <math>(4_24_02_1)</math> |- align=center |80||I4₁||I 4₁ || <math>\Gamma_q^vC_4^6</math> || ''32a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1</math> || <math>(4_34_12_0)</math> |- align=center |{{anchor|81-82}}81||rowspan=2|{{overline|4}}||rowspan=2|<math>2\times</math>||P{{overline|4}}||P {{overline|4}} || <math>\Gamma_qS_4^1</math> || ''26s'' || <math>(c:a:a):\tilde 4</math> || <math>(442_0)</math> |- align=center |82||I{{overline|4}}||I {{overline|4}} || <math>\Gamma_q^vS_4^2</math> || ''27s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4</math> || <math>(442_1)</math> |- align=center |{{anchor|83-88}}83||rowspan=6|4/m||rowspan=6|<math>4*</math>||P4/m||P 4/m|| <math>\Gamma_qC_{4h}^1</math> || ''28s'' || <math>(c:a:a)\cdot m:4</math> || <math>[4_04_02_0]</math> |- align=center |84||P4₂/m||P 4₂/m|| <math>\Gamma_qC_{4h}^2</math> || ''41a'' || <math>(c:a:a)\cdot m:4_2</math> || <math>[4_24_22_0]</math> |- align=center |85||P4/n||P 4/n|| <math>\Gamma_qC_{4h}^3</math> || ''29h'' || <math>(c:a:a)\cdot \widetilde{ab}:4</math> || <math>(44_02)</math> |- align=center |86||P4₂/n||P 4₂/n|| <math>\Gamma_qC_{4h}^4</math> || ''42a'' || <math>(c:a:a)\cdot \widetilde{ab}:4_2</math> || <math>(44_22)</math> |- align=center |87||I4/m||I 4/m|| <math>\Gamma_q^vC_{4h}^5</math> || ''29s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4</math> || <math>[4_24_02_1]</math> |- align=center |88||I4₁/a||I 4₁/a|| <math>\Gamma_q^vC_{4h}^6</math> || ''40a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1</math> || <math>(44_12)</math> |- align=center |{{anchor|89-98}}89||rowspan=10|422||rowspan=10|<math>224</math>||P422||P 4 2 2 || <math>\Gamma_qD_4^1</math> || ''30s'' || <math>(c:a:a):4:2</math> || <math>(*4_04_02_0)</math> |- align=center |90||P42₁2||P42₁2 || <math>\Gamma_qD_4^2</math> || ''43a'' || <math>(c:a:a):4</math> [[File:circled_colon.png|16px]] <math>2_1</math> || <math>(4_0{*}2_0)</math> |- align=center |91||P4₁22||P 4₁ 2 2 || <math>\Gamma_qD_4^3</math> || ''44a'' || <math>(c:a:a):4_1:2</math> || <math>(*4_14_12_1)</math> |- align=center |92||P4₁2₁2||P 4₁ 2₁ 2 || <math>\Gamma_qD_4^4</math> || ''48a'' || <math>(c:a:a):4_1</math> [[File:circled_colon.png|16px]] <math>2_1</math> || <math>(4_1{*}2_1)</math> |- align=center |93||P4₂22||P 4₂ 2 2 || <math>\Gamma_qD_4^5</math> || ''47a'' || <math>(c:a:a):4_2:2</math> || <math>(*4_24_22_0)</math> |- align=center |94||P4₂2₁2||P 4₂ 2₁ 2 || <math>\Gamma_qD_4^6</math> || ''50a'' || <math>(c:a:a):4_2</math> [[File:circled_colon.png|16px]] <math>2_1</math> || <math>(4_2{*}2_0)</math> |- align=center |95||P4₃22||P 4₃ 2 2 || <math>\Gamma_qD_4^7</math> || ''45a'' || <math>(c:a:a):4_3:2</math> || <math>(*4_14_12_1)</math> |- align=center |96||P4₃2₁2||P 4₃ 2₁ 2 || <math>\Gamma_qD_4^8</math> || ''49a'' || <math>(c:a:a):4_3</math> [[File:circled_colon.png|16px]] <math>2_1</math> || <math>(4_1{*}2_1)</math> |- align=center |97||I422||I 4 2 2 || <math>\Gamma_q^vD_4^9</math> || ''31s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2</math> || <math>(*4_24_02_1)</math> |- align=center |98||I4₁22||I 4₁ 2 2 || <math>\Gamma_q^vD_4^{10}</math> || ''46a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2_1</math> || <math>(*4_34_12_0)</math> |- align=center |{{anchor|99-110}}99||rowspan=12|4mm||rowspan=12|<math>*44</math>||P4mm||P 4 m m || <math>\Gamma_qC_{4v}^1</math> || ''24s'' || <math>(c:a:a):4\cdot m</math> || <math>(*{\cdot}4{\cdot}4{\cdot}2)</math> |- align=center |100||P4bm|| P 4 b m || <math>\Gamma_qC_{4v}^2</math> || ''26h'' || <math>(c:a:a):4\odot \tilde a</math> || <math>(4_0{*}{\cdot}2)</math> |- align=center |101||P4₂cm|| P 4₂ c m || <math>\Gamma_qC_{4v}^3</math> || ''37a'' || <math>(c:a:a):4_2\cdot \tilde c</math> || <math>(*{:}4{\cdot}4{:}2)</math> |- align=center |102||P4₂nm|| P 4₂ n m || <math>\Gamma_qC_{4v}^4</math> || ''38a'' || <math>(c:a:a):4_2\odot \widetilde{ac}</math> || <math>(4_2{*}{\cdot}2)</math> |- align=center |103||P4cc|| P 4 c c || <math>\Gamma_qC_{4v}^5</math> || ''25h'' || <math>(c:a:a):4\cdot \tilde c</math> || <math>(*{:}4{:}4{:}2)</math> |- align=center |104||P4nc|| P 4 n c || <math>\Gamma_qC_{4v}^6</math> || ''27h'' || <math>(c:a:a):4\odot \widetilde{ac}</math> || <math>(4_0{*}{:}2)</math> |- align=center |105||P4₂mc|| P 4₂ m c || <math>\Gamma_qC_{4v}^7</math> || ''36a'' || <math>(c:a:a):4_2\cdot m</math> || <math>(*{\cdot}4{:}4{\cdot}2)</math> |- align=center |106||P4₂bc|| P 4₂ b c || <math>\Gamma_qC_{4v}^8</math> || ''39a'' || <math>(c:a:a):4\odot \tilde a</math> || <math>(4_2{*}{:}2)</math> |- align=center |107||I4mm|| I 4 m m || <math>\Gamma_q^vC_{4v}^9</math> || ''25s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot m</math> || <math>(*{\cdot}4{\cdot}4{:}2)</math> |- align=center |108||I4cm|| I 4 c m || <math>\Gamma_q^vC_{4v}^{10}</math> || ''28h'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot \tilde c</math> || <math>(*{\cdot}4{:}4{:}2)</math> |- align=center |109||I4₁md|| I 4₁ m d || <math>\Gamma_q^vC_{4v}^{11}</math> || ''34a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot m</math> || <math>(4_1{*}{\cdot}2)</math> |- align=center |110||I4₁cd|| I 4₁ c d || <math>\Gamma_q^vC_{4v}^{12}</math> || ''35a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot \tilde c</math> || <math>(4_1{*}{:}2)</math> |- align=center |{{anchor|111-122}}111||rowspan=12|{{overline|4}}2m||rowspan=12|<math>2{*}2</math>||P{{overline|4}}2m|| P {{overline|4}} 2 m || <math>\Gamma_qD_{2d}^1</math> || ''32s'' || <math>(c:a:a):\tilde 4 :2</math> || <math>(*4{\cdot}42_0)</math> |- align=center |112||P{{overline|4}}2c|| P {{overline|4}} 2 c || <math>\Gamma_qD_{2d}^2</math> || ''30h'' || <math>(c:a:a):\tilde 4 </math> [[File:circled_colon.png|16px]] <math>2</math> || <math>(*4{:}42_0)</math> |- align=center |113||P{{overline|4}}2₁m|| P {{overline|4}} 2₁ m || <math>\Gamma_qD_{2d}^3</math> || ''52a'' || <math>(c:a:a):\tilde 4 \cdot \widetilde{ab}</math> || <math>(4\bar{*}{\cdot}2)</math> |- align=center |114||P{{overline|4}}2₁c|| P {{overline|4}} 2₁ c || <math>\Gamma_qD_{2d}^4</math> || ''53a'' || <math>(c:a:a):\tilde 4 \cdot \widetilde{abc}</math> || <math>(4\bar{*}{:}2)</math> |- align=center |115||P{{overline|4}}m2|| P {{overline|4}} m 2 || <math>\Gamma_qD_{2d}^5</math> || ''33s'' || <math>(c:a:a):\tilde 4 \cdot m</math> || <math>(*{\cdot}44{\cdot}2)</math> |- align=center |116||P{{overline|4}}c2|| P {{overline|4}} c 2 || <math>\Gamma_qD_{2d}^6</math> || ''31h'' || <math>(c:a:a):\tilde 4 \cdot \tilde c</math> || <math>(*{:}44{:}2)</math> |- align=center |117||P{{overline|4}}b2|| P {{overline|4}} b 2 || <math>\Gamma_qD_{2d}^7</math> || ''32h'' || <math>(c:a:a):\tilde 4 \odot \tilde a</math> || <math>(4\bar{*}_02_0)</math> |- align=center |118||P{{overline|4}}n2|| P {{overline|4}} n 2 || <math>\Gamma_qD_{2d}^8</math> || ''33h'' || <math>(c:a:a):\tilde 4 \cdot \widetilde{ac}</math> || <math>(4\bar{*}_12_0)</math> |- align=center |119||I{{overline|4}}m2|| I {{overline|4}} m 2 || <math>\Gamma_q^vD_{2d}^9</math> || ''35s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot m</math> || <math>(*4{\cdot}42_1)</math> |- align=center |120||I{{overline|4}}c2|| I {{overline|4}} c 2 || <math>\Gamma_q^vD_{2d}^{10}</math> || ''34h'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot \tilde c</math> || <math>(*4{:}42_1)</math> |- align=center |121||I{{overline|4}}2m|| I {{overline|4}} 2 m || <math>\Gamma_q^vD_{2d}^{11}</math> || ''34s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 :2</math> || <math>(*{\cdot}44{:}2)</math> |- align=center |122||I{{overline|4}}2d|| I {{overline|4}} 2 d || <math>\Gamma_q^vD_{2d}^{12}</math> || ''51a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \odot \tfrac{1}{2}\widetilde{abc}</math> || <math>(4\bar{*}2_1)</math> |- align=center |{{anchor|123-132}}123||rowspan=20|4/m 2/m 2/m (4/mmm) ||rowspan=20|<math>*224</math>||P4/mmm|| P 4/m 2/m 2/m || <math>\Gamma_qD_{4h}^1</math> || ''36s'' || <math>(c:a:a)\cdot m:4\cdot m</math> || <math>[*{\cdot}4{\cdot}4{\cdot}2]</math> |- align=center |124||P4/mcc|| P 4/m 2/c 2/c || <math>\Gamma_qD_{4h}^2</math> || ''35h'' || <math>(c:a:a)\cdot m:4\cdot \tilde c</math> || <math>[*{:}4{:}4{:}2]</math> |- align=center |125||P4/nbm|| P 4/n 2/b 2/m|| <math>\Gamma_qD_{4h}^3</math> || ''36h'' || <math>(c:a:a)\cdot \widetilde{ab}:4\odot \tilde a</math> || <math>(*4_04{\cdot}2)</math> |- align=center |126||P4/nnc|| P 4/n 2/n 2/c || <math>\Gamma_qD_{4h}^4</math> || ''37h'' || <math>(c:a:a)\cdot \widetilde{ab}:4\odot \widetilde{ac}</math> || <math>(*4_04{:}2)</math> |- align=center |127||P4/mbm|| P 4/m 2₁/b 2/m || <math>\Gamma_qD_{4h}^5</math> || ''54a'' || <math>(c:a:a)\cdot m:4\odot \tilde a</math> || <math>[4_0{*}{\cdot}2]</math> |- align=center |128||P4/mnc|| P 4/m 2₁/n 2/c || <math>\Gamma_qD_{4h}^6</math> || ''56a'' || <math>(c:a:a)\cdot m:4\odot \widetilde{ac}</math> || <math>[4_0{*}{:}2]</math> |- align=center |129||P4/nmm|| P 4/n 2₁/m 2/m || <math>\Gamma_qD_{4h}^7</math> || ''55a'' || <math>(c:a:a)\cdot \widetilde{ab}:4\cdot m</math> || <math>(*4{\cdot}4{\cdot}2)</math> |- align=center |130||P4/ncc|| P 4/n 2₁/c 2/c || <math>\Gamma_qD_{4h}^8</math> || ''57a'' || <math>(c:a:a)\cdot \widetilde{ab}:4\cdot \tilde c</math> || <math>(*4{:}4{:}2)</math> |- align=center |131||P4₂/mmc|| P 4₂/m 2/m 2/c || <math>\Gamma_qD_{4h}^9</math> || ''60a'' || <math>(c:a:a)\cdot m:4_2\cdot m</math> || <math>[*{\cdot}4{:}4{\cdot}2]</math> |- align=center |132||P4₂/mcm|| P 4₂/m 2/c 2/m || <math>\Gamma_qD_{4h}^{10}</math> || ''61a'' || <math>(c:a:a)\cdot m:4_2\cdot \tilde c</math> || <math>[*{:}4{\cdot}4{:}2]</math> |- align=center |{{anchor|133-142}}133||P4₂/nbc|| P 4₂/n 2/b 2/c || <math>\Gamma_qD_{4h}^{11}</math> || ''63a'' || <math>(c:a:a)\cdot \widetilde{ab}:4_2\odot \tilde a</math> || <math>(*4_24{:}2)</math> |- align=center |134||P4₂/nnm|| P 4₂/n 2/n 2/m || <math>\Gamma_qD_{4h}^{12}</math> || ''62a'' || <math>(c:a:a)\cdot \widetilde{ab}:4_2\odot \widetilde{ac}</math> || <math>(*4_24{\cdot}2)</math> |- align=center |135||P4₂/mbc|| P 4₂/m 2₁/b 2/c || <math>\Gamma_qD_{4h}^{13}</math> || ''66a'' || <math>(c:a:a)\cdot m:4_2\odot \tilde a</math> || <math>[4_2{*}{:}2]</math> |- align=center |136||P4₂/mnm|| P 4₂/m 2₁/n 2/m || <math>\Gamma_qD_{4h}^{14}</math> || ''65a'' || <math>(c:a:a)\cdot m:4_2\odot \widetilde{ac}</math> || <math>[4_2{*}{\cdot}2]</math> |- align=center |137||P4₂/nmc|| P 4₂/n 2₁/m 2/c || <math>\Gamma_qD_{4h}^{15}</math> || ''67a'' || <math>(c:a:a)\cdot \widetilde{ab}:4_2\cdot m</math> || <math>(*4{\cdot}4{:}2)</math> |- align=center |138||P4₂/ncm|| P 4₂/n 2₁/c 2/m || <math>\Gamma_qD_{4h}^{16}</math> || ''65a'' || <math>(c:a:a)\cdot \widetilde{ab}:4_2\cdot \tilde c</math> || <math>(*4{:}4{\cdot}2)</math> |- align=center |139||I4/mmm|| I 4/m 2/m 2/m || <math>\Gamma_q^vD_{4h}^{17}</math> || ''37s'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot m</math> || <math>[*{\cdot}4{\cdot}4{:}2]</math> |- align=center |140||I4/mcm|| I 4/m 2/c 2/m || <math>\Gamma_q^vD_{4h}^{18}</math> || ''38h'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot \tilde c</math> || <math>[*{\cdot}4{:}4{:}2]</math> |- align=center |141||I4₁/amd|| I 4₁/a 2/m 2/d || <math>\Gamma_q^vD_{4h}^{19}</math> || ''59a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot m</math> || <math>(*4_14{\cdot}2)</math> |- align=center |142||I4₁/acd|| I 4₁/a 2/c 2/d || <math>\Gamma_q^vD_{4h}^{20}</math> || ''58a'' || <math>\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot \tilde c</math> || <math>(*4_14{:}2)</math> |}

==List of trigonal== {| class="wikitable floatright" |+ Trigonal Bravais lattice |- ! Rhombohedral (R) ! Hexagonal (P) |- style="vertical-align:top;" | [[File:Hexagonal latticeR.svg|100px]] | [[File:Hexagonal latticeFRONT.svg|100px]] |} {{sticky header}} {| class="wikitable sticky-header" |+ [[Trigonal crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] |- align=center |{{anchor|143-146}}143||rowspan=4|3||rowspan=4|<math>33</math>||P3|| P 3 || <math>\Gamma_hC_3^1</math> || ''38s'' || <math>(c:(a/a)):3</math> || <math>(3_03_03_0)</math> |- align=center |144||P3₁|| P 3₁ || <math>\Gamma_hC_3^2</math> || ''68a'' || <math>(c:(a/a)):3_1</math> || <math>(3_13_13_1)</math> |- align=center |145||P3₂|| P 3₂ || <math>\Gamma_hC_3^3</math> || ''69a'' || <math>(c:(a/a)):3_2</math> || <math>(3_13_13_1)</math> |- align=center |146||R3|| R 3 || <math>\Gamma_{rh}C_3^4</math> || ''39s'' || <math>(a/a/a)/3</math> || <math>(3_03_13_2)</math> |- align=center |{{anchor|147-148}}147||rowspan=2|{{overline|3}}||rowspan=2|<math>3\times</math>||P{{overline|3}}|| P {{overline|3}} || <math>\Gamma_hC_{3i}^1</math> || ''51s'' || <math>(c:(a/a)):\tilde 6</math> || <math>(63_02)</math> |- align=center |148||R{{overline|3}}|| R {{overline|3}} || <math>\Gamma_{rh}C_{3i}^2</math> || ''52s'' || <math>(a/a/a)/\tilde 6</math> || <math>(63_12)</math> |- align=center |{{anchor|149-155}}149||rowspan=7|32||rowspan=7|<math>223</math>||P312|| P 3 1 2 || <math>\Gamma_hD_3^1</math> || ''45s'' || <math>(c:(a/a)):2:3</math> || <math>(*3_03_03_0)</math> |- align=center |150||P321|| P 3 2 1 || <math>\Gamma_hD_3^2</math> || ''44s'' || <math>(c:(a/a))\cdot 2:3</math> || <math>(3_0{*}3_0)</math> |- align=center |151||P3₁12|| P 3₁ 1 2 || <math>\Gamma_hD_3^3</math> || ''72a'' || <math>(c:(a/a)):2:3_1</math> || <math>(*3_13_13_1)</math> |- align=center |152||P3₁21|| P 3₁ 2 1 || <math>\Gamma_hD_3^4</math> || ''70a'' || <math>(c:(a/a))\cdot 2:3_1</math> || <math>(3_1{*}3_1)</math> |- align=center |153||P3₂12|| P 3₂ 1 2 || <math>\Gamma_hD_3^5</math> || ''73a'' || <math>(c:(a/a)):2:3_2</math> || <math>(*3_13_13_1)</math> |- align=center |154||P3₂21|| P 3₂ 2 1 || <math>\Gamma_hD_3^6</math> || ''71a'' || <math>(c:(a/a))\cdot 2:3_2</math> || <math>(3_1{*}3_1)</math> |- align=center |155||R32|| R 3 2 || <math>\Gamma_{rh}D_3^7</math> || ''46s'' || <math>(a/a/a)/3:2</math> || <math>(*3_03_13_2)</math> |- align=center |{{anchor|156-161}}156||rowspan=6|3m||rowspan=6|<math>*33</math>||P3m1|| P 3 m 1 || <math>\Gamma_hC_{3v}^1</math> || ''40s'' || <math>(c:(a/a)):m\cdot 3</math> || <math>(*{\cdot}3{\cdot}3{\cdot}3)</math> |- align=center |157||P31m|| P 3 1 m || <math>\Gamma_hC_{3v}^2</math> || ''41s'' || <math>(c:(a/a))\cdot m\cdot 3</math> || <math>(3_0{*}{\cdot}3)</math> |- align=center |158||P3c1|| P 3 c 1 || <math>\Gamma_hC_{3v}^3</math> || ''39h'' || <math>(c:(a/a)):\tilde c:3</math> || <math>(*{:}3{:}3{:}3)</math> |- align=center |159||P31c|| P 3 1 c || <math>\Gamma_hC_{3v}^4</math> || ''40h'' || <math>(c:(a/a))\cdot\tilde c :3</math> || <math>(3_0{*}{:}3)</math> |- align=center |160||R3m|| R 3 m || <math>\Gamma_{rh}C_{3v}^5</math> || ''42s'' || <math>(a/a/a)/3\cdot m</math> || <math>(3_1{*}{\cdot}3)</math> |- align=center |161||R3c|| R 3 c || <math>\Gamma_{rh}C_{3v}^6</math> || ''41h'' || <math>(a/a/a)/3\cdot\tilde c</math> || <math>(3_1{*}{:}3)</math> |- align=center |{{anchor|162-167}}162||rowspan=6|{{overline|3}} 2/m ({{overline|3}}m)||rowspan=6|<math>2{*}3</math>||P{{overline|3}}1m|| P {{overline|3}} 1 2/m || <math>\Gamma_hD_{3d}^1</math> || ''56s'' || <math>(c:(a/a))\cdot m\cdot\tilde 6</math> || <math>(*{\cdot}63_02)</math> |- align=center |163||P{{overline|3}}1c|| P {{overline|3}} 1 2/c || <math>\Gamma_hD_{3d}^2</math> || ''46h'' || <math>(c:(a/a))\cdot\tilde c \cdot\tilde 6</math> || <math>(*{:}63_02)</math> |- align=center |164||P{{overline|3}}m1|| P {{overline|3}} 2/m 1 || <math>\Gamma_hD_{3d}^3</math> || ''55s'' || <math>(c:(a/a)):m\cdot\tilde 6</math> || <math>(*6{\cdot}3{\cdot}2)</math> |- align=center |165||P{{overline|3}}c1|| P {{overline|3}} 2/c 1 || <math>\Gamma_hD_{3d}^4</math> || ''45h'' || <math>(c:(a/a)):\tilde c \cdot\tilde 6</math> || <math>(*6{:}3{:}2)</math> |- align=center |166||R{{overline|3}}m|| R {{overline|3}} 2/m || <math>\Gamma_{rh}D_{3d}^5</math> || ''57s'' || <math>(a/a/a)/\tilde 6 \cdot m</math> || <math>(*{\cdot}63_12)</math> |- align=center |167||R{{overline|3}}c|| R {{overline|3}} 2/c || <math>\Gamma_{rh}D_{3d}^6</math> || ''47h'' || <math>(a/a/a)/\tilde 6 \cdot\tilde c</math> || <math>(*{:}63_12)</math> |}

==List of hexagonal== {| class="wikitable floatright" |+ Hexagonal Bravais lattice |- | [[File:Hexagonal latticeFRONT.svg|80px]] |} {{sticky header}} {| class="wikitable sticky-header" |+ [[Hexagonal crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! [[Fibrifold notation|Fibrifold]] |- align=center |{{anchor|168-173}}168||rowspan=6|6||rowspan=6|<math>66</math>||P6|| P 6 || <math>\Gamma_hC_6^1</math> || ''49s'' || <math>(c:(a/a)):6</math> || <math>(6_03_02_0)</math> |- align=center |169||P6₁|| P 6₁ || <math>\Gamma_hC_6^2</math> || ''74a'' || <math>(c:(a/a)):6_1</math> || <math>(6_13_12_1)</math> |- align=center |170||P6₅|| P 6₅ || <math>\Gamma_hC_6^3</math> || ''75a'' || <math>(c:(a/a)):6_5</math> || <math>(6_13_12_1)</math> |- align=center |171||P6₂|| P 6₂ || <math>\Gamma_hC_6^4</math> || ''76a'' || <math>(c:(a/a)):6_2</math> || <math>(6_23_22_0)</math> |- align=center |172||P6₄|| P 6₄ || <math>\Gamma_hC_6^5</math> || ''77a'' || <math>(c:(a/a)):6_4</math> || <math>(6_23_22_0)</math> |- align=center |173||P6₃|| P 6₃ || <math>\Gamma_hC_6^6</math> || ''78a'' || <math>(c:(a/a)):6_3</math> || <math>(6_33_02_1)</math> |- align=center |{{anchor|174-176}}174||{{overline|6}}||<math>3*</math>||P{{overline|6}}|| P {{overline|6}} || <math>\Gamma_hC_{3h}^1</math> || ''43s'' || <math>(c:(a/a)):3:m</math> || <math>[3_03_03_0]</math> |- align=center |175||rowspan=2|6/m||rowspan=2|<math>6*</math>||P6/m|| P 6/m || <math>\Gamma_hC_{6h}^1</math> || ''53s'' || <math>(c:(a/a))\cdot m :6</math> || <math>[6_03_02_0]</math> |- align=center |176||P6₃/m|| P 6₃/m || <math>\Gamma_hC_{6h}^2</math> || ''81a'' || <math>(c:(a/a))\cdot m :6_3</math> || <math>[6_33_02_1]</math> |- align=center |{{anchor|177-182}}177||rowspan=6|622||rowspan=6|<math>226</math>||P622|| P 6 2 2 || <math>\Gamma_hD_6^1</math> || ''54s'' || <math>(c:(a/a))\cdot 2 :6</math> || <math>(*6_03_02_0)</math> |- align=center |178||P6₁22|| P 6₁ 2 2 || <math>\Gamma_hD_6^2</math> || ''82a'' || <math>(c:(a/a))\cdot 2 :6_1</math> || <math>(*6_13_12_1)</math> |- align=center |179||P6₅22|| P 6₅ 2 2 || <math>\Gamma_hD_6^3</math> || ''83a'' || <math>(c:(a/a))\cdot 2 :6_5</math> || <math>(*6_13_12_1)</math> |- align=center |180||P6₂22|| P 6₂ 2 2 || <math>\Gamma_hD_6^4</math> || ''84a'' || <math>(c:(a/a))\cdot 2 :6_2</math> || <math>(*6_23_22_0)</math> |- align=center |181||P6₄22|| P 6₄ 2 2 || <math>\Gamma_hD_6^5</math> || ''85a'' || <math>(c:(a/a))\cdot 2 :6_4</math> || <math>(*6_23_22_0)</math> |- align=center |182||P6₃22|| P 6₃ 2 2 || <math>\Gamma_hD_6^6</math> || ''86a'' || <math>(c:(a/a))\cdot 2 :6_3</math> || <math>(*6_33_02_1)</math> |- align=center |{{anchor|183-186}}183||rowspan=4|6mm||rowspan=4|<math>*66</math>||P6mm|| P 6 m m || <math>\Gamma_hC_{6v}^1</math> || ''50s'' || <math>(c:(a/a)):m\cdot 6</math> || <math>(*{\cdot}6{\cdot}3{\cdot}2)</math> |- align=center |184||P6cc|| P 6 c c || <math>\Gamma_hC_{6v}^2</math> || ''44h'' || <math>(c:(a/a)):\tilde c \cdot 6</math> || <math>(*{:}6{:}3{:}2)</math> |- align=center |185||P6₃cm|| P 6₃ c m || <math>\Gamma_hC_{6v}^3</math> || ''80a'' || <math>(c:(a/a)):\tilde c \cdot 6_3</math> || <math>(*{\cdot}6{:}3{:}2)</math> |- align=center |186||P6₃mc|| P 6₃ m c || <math>\Gamma_hC_{6v}^4</math> || ''79a'' || <math>(c:(a/a)):m\cdot 6_3</math> || <math>(*{:}6{\cdot}3{\cdot}2)</math> |- align=center |{{anchor|187-190}}187||rowspan=4|{{overline|6}}m2||rowspan=4|<math>*223</math>||P{{overline|6}}m2|| P {{overline|6}} m 2 || <math>\Gamma_hD_{3h}^1</math> || ''48s'' || <math>(c:(a/a)):m\cdot 3:m</math> || <math>[*{\cdot}3{\cdot}3{\cdot}3]</math> |- align=center |188||P{{overline|6}}c2|| P {{overline|6}} c 2 || <math>\Gamma_hD_{3h}^2</math> || ''43h'' || <math>(c:(a/a)):\tilde c \cdot 3:m</math> || <math>[*{:}3{:}3{:}3]</math> |- align=center |189||P{{overline|6}}2m|| P {{overline|6}} 2 m || <math>\Gamma_hD_{3h}^3</math> || ''47s'' || <math>(c:(a/a))\cdot m:3\cdot m</math> || <math>[3_0{*}{\cdot}3]</math> |- align=center |190||P{{overline|6}}2c|| P {{overline|6}} 2 c || <math>\Gamma_hD_{3h}^4</math> || ''42h'' || <math>(c:(a/a))\cdot m:3\cdot \tilde c</math> || <math>[3_0{*}{:}3]</math> |- align=center |{{anchor|191-194}}191||rowspan=4|6/m 2/m 2/m (6/mmm) ||rowspan=4|<math>*226</math>||P6/mmm|| P 6/m 2/m 2/m || <math>\Gamma_hD_{6h}^1</math> || ''58s'' || <math>(c:(a/a))\cdot m:6\cdot m</math> || <math>[*{\cdot}6{\cdot}3{\cdot}2]</math> |- align=center |192||P6/mcc|| P 6/m 2/c 2/c || <math>\Gamma_hD_{6h}^2</math> || ''48h'' || <math>(c:(a/a))\cdot m:6\cdot\tilde c</math> || <math>[*{:}6{:}3{:}2]</math> |- align=center |193||P6₃/mcm|| P 6₃/m 2/c 2/m || <math>\Gamma_hD_{6h}^3</math> || ''87a'' || <math>(c:(a/a))\cdot m:6_3\cdot\tilde c</math> || <math>[*{\cdot}6{:}3{:}2]</math> |- align=center |194||P6₃/mmc|| P 6₃/m 2/m 2/c || <math>\Gamma_hD_{6h}^4</math> || ''88a'' || <math>(c:(a/a))\cdot m:6_3\cdot m</math> || <math>[*{:}6{\cdot}3{\cdot}2]</math> |}

==List of cubic== {| class="wikitable" style="text-align:center;" |+ Cubic Bravais lattice |- ! Simple (P) ! Body centered (I) ! Face centered (F) |- | [[File:Cubic.svg|100px]] | [[File:Cubic-body-centered.svg|100px]] | [[File:Cubic-face-centered.svg|100px]] |} {{Gallery | title=Example cubic structures | width=160 |height=170 | File:CsCl crystal.svg | (221) [[Caesium chloride]]. Different colors for the two atom types. | File:Sphalerite-unit-cell-depth-fade-3D-balls.png|(216) [[Sphalerite]] | File:12-14-hedral honeycomb.png | (223) [[Weaire–Phelan structure]] }} {{sticky header}} {| class="wikitable sticky-header" |+ [[Cubic crystal system]] !Number ! [[Crystallographic point group|Point group]] ! [[Orbifold notation|Orbifold]] ! Short name ! Full name ! [[Schoenflies notation|Schoenflies]] ! [[Evgraf Fedorov|Fedorov]] ! [[Shubnikov group|Shubnikov]] ! Conway ! Fibrifold (preserving <math>z</math>) ! Fibrifold (preserving <math>x</math>, <math>y</math>, <math>z</math>) |- align=center |{{anchor|195-199}}195||rowspan=5|23||rowspan=5|<math>332</math>||P23|| P 2 3 || <math>\Gamma_cT^1</math> || ''59s'' || <math>\left ( a:a:a\right ) :2/3</math> || <math>2^\circ</math> || <math>(*2_02_02_02_0){:}3</math> || <math>(*2_02_02_02_0){:}3</math> |- align=center |196||F23|| F 2 3 || <math>\Gamma_c^fT^2</math> || ''61s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :2/3</math> || <math>1^\circ</math> || <math>(*2_02_12_02_1){:}3</math> || <math>(*2_02_12_02_1){:}3</math> |- align=center |197||I23|| I 2 3 || <math>\Gamma_c^vT^3</math> || ''60s'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2/3</math>|| <math>4^{\circ\circ}</math> || <math>(2_1{*}2_02_0){:}3</math> || <math>(2_1{*}2_02_0){:}3</math> |- align=center |198||P2₁3|| P 2₁ 3 || <math>\Gamma_cT^4</math> || ''89a'' || <math>\left ( a:a:a\right ) :2_1/3</math>|| <math>1^\circ/4</math> || <math>(2_12_1\bar{\times}){:}3</math> || <math>(2_12_1\bar{\times}){:}3</math> |- align=center |199||I2₁3|| I 2₁ 3 || <math>\Gamma_c^vT^5</math> || ''90a'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2_1/3</math>|| <math>2^\circ/4</math> || <math>(2_0{*}2_12_1){:}3</math> || <math>(2_0{*}2_12_1){:}3</math> |- align=center |{{anchor|200-206}}200||rowspan=7|2/m {{overline|3}} (m{{overline|3}}) ||rowspan=7|<math>3{*}2</math>||Pm{{overline|3}}|| P 2/m {{overline|3}} || <math>\Gamma_cT_h^1</math> || ''62s'' || <math>\left ( a:a:a\right ) \cdot m/ \tilde 6</math> || <math>4^-</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3</math> |- align=center |201||Pn{{overline|3}}|| P 2/n {{overline|3}} || <math>\Gamma_cT_h^2</math> || ''49h'' ||<math>\left ( a:a:a\right ) \cdot \widetilde{ab} / \tilde 6</math> || <math>4^{\circ+}</math> || <math>(2\bar{*}_12_02_0){:}3</math> || <math>(2\bar{*}_12_02_0){:}3</math> |- align=center |202||Fm{{overline|3}}|| F 2/m {{overline|3}} || <math>\Gamma_c^fT_h^3</math> || ''64s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot m/ \tilde 6</math> || <math>2^-</math> || <math>[*{\cdot}2{\cdot}2{:}2{:}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{:}2{:}2]{:}3</math> |- align=center |203||Fd{{overline|3}}|| F 2/d {{overline|3}} || <math>\Gamma_c^fT_h^4</math> || ''50h'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot \tfrac{1}{2}\widetilde{ab} / \tilde 6</math> || <math>2^{\circ+}</math> || <math>(2\bar{*}2_02_1){:}3</math> || <math>(2\bar{*}2_02_1){:}3</math> |- align=center |204||Im{{overline|3}}|| I 2/m {{overline|3}} || <math>\Gamma_c^vT_h^5</math> || ''63s'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot m/\tilde 6</math> || <math>8^{-\circ}</math> || <math>[2_1{*}{\cdot}2{\cdot}2]{:}3</math> || <math>[2_1{*}{\cdot}2{\cdot}2]{:}3</math> |- align=center |205||Pa{{overline|3}}|| P 2₁/a {{overline|3}} || <math>\Gamma_cT_h^6</math> || ''91a'' || <math>\left ( a:a:a\right ) \cdot \tilde a /\tilde 6</math> || <math>2^-/4</math> || <math>(2_12\bar{*}{:}){:}3</math>|| <math>(2_12\bar{*}{:}){:}3</math> |- align=center |206||Ia{{overline|3}}|| I 2₁/a {{overline|3}} || <math>\Gamma_c^vT_h^7</math> || ''92a'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot \tilde a /\tilde 6</math> || <math>4^-/4</math> || <math>(*2_12{:}2{:}2){:}3</math> || <math>(*2_12{:}2{:}2){:}3</math> |- align=center |{{anchor|207-214}}207||rowspan=8|432||rowspan=8|<math>432</math>||P432|| P 4 3 2 || <math>\Gamma_cO^1</math> || ''68s'' || <math>\left ( a:a:a\right ) :4/3</math> || <math>4^{\circ-}</math> || <math>(*4_04_02_0){:}3</math> || <math>(*2_02_02_02_0){:}6</math> |- align=center |208||P4₂32|| P 4₂ 3 2 || <math>\Gamma_cO^2</math> || ''98a'' || <math>\left ( a:a:a\right ) :4_2//3</math> || <math>4^+</math> || <math>(*4_24_22_0){:}3</math> || <math>(*2_02_02_02_0){:}6</math> |- align=center |209||F432|| F 4 3 2 || <math>\Gamma_c^fO^3</math> || ''70s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/3</math> || <math>2^{\circ-}</math> || <math>(*4_24_02_1){:}3</math> || <math>(*2_02_12_02_1){:}6</math> |- align=center |210||F4₁32|| F 4₁ 3 2 || <math>\Gamma_c^fO^4</math> || ''97a'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//3</math> || <math>2^+</math> || <math>(*4_34_12_0){:}3</math> || <math>(*2_02_12_02_1){:}6</math> |- align=center |211||I432|| I 4 3 2 || <math>\Gamma_c^vO^5</math> || ''69s'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/3</math> || <math>8^{+\circ}</math> || <math>(4_24_02_1){:}3</math>|| <math>(2_1{*}2_02_0){:}6</math> |- align=center |212||P4₃32|| P 4₃ 3 2 || <math>\Gamma_cO^6</math> || ''94a'' || <math>\left ( a:a:a\right ) :4_3//3</math> || <math>2^+/4</math> || <math>(4_1{*}2_1){:}3</math> || <math>(2_12_1\bar{\times}){:}6</math> |- align=center |213||P4₁32|| P 4₁ 3 2 || <math>\Gamma_cO^7</math> || ''95a'' || <math>\left ( a:a:a\right ) :4_1//3</math> || <math>2^+/4</math> || <math>(4_1{*}2_1){:}3</math> || <math>(2_12_1\bar{\times}){:}6</math> |- align=center |214||I4₁32|| I 4₁ 3 2 || <math>\Gamma_c^vO^8</math> ||''96a'' || <math>\left ( \tfrac{a+b+c}{2}/:a:a:a\right ) :4_1//3</math> || <math>4^+/4</math> || <math>(*4_34_12_0){:}3</math> || <math>(2_0{*}2_12_1){:}6</math> |- align=center |{{anchor|215-220}}215||rowspan=6|{{overline|4}}3m||rowspan=6|<math>*332</math>||P{{overline|4}}3m|| P {{overline|4}} 3 m || <math>\Gamma_cT_d^1</math> || ''65s'' || <math>\left ( a:a:a\right ) :\tilde 4 /3</math> || <math>2^\circ{:}2</math> || <math>(*4{\cdot}42_0){:}3</math> || <math>(*2_02_02_02_0){:}6</math> |- align=center |216||F{{overline|4}}3m|| F {{overline|4}} 3 m || <math>\Gamma_c^fT_d^2</math> || ''67s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 /3</math> || <math>1^\circ{:}2</math> || <math>(*4{\cdot}42_1){:}3</math> || <math>(*2_02_12_02_1){:}6</math> |- align=center |217||I{{overline|4}}3m|| I {{overline|4}} 3 m || <math>\Gamma_c^vT_d^3</math> || ''66s'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 /3</math> || <math>4^\circ{:}2</math> || <math>(*{\cdot}44{:}2){:}3</math> || <math>(2_1{*}2_02_0){:}6</math> |- align=center |218||P{{overline|4}}3n|| P {{overline|4}} 3 n || <math>\Gamma_cT_d^4</math> || ''51h'' || <math>\left ( a:a:a\right ) :\tilde 4 //3</math> || <math>4^\circ</math> || <math>(*4{:}42_0){:}3</math> || <math>(*2_02_02_02_0){:}6</math> |- align=center |219||F{{overline|4}}3c|| F {{overline|4}} 3 c || <math>\Gamma_c^fT_d^5</math> || ''52h'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 //3</math> || <math>2^{\circ\circ}</math> || <math>(*4{:}42_1){:}3</math> || <math>(*2_02_12_02_1){:}6</math> |- align=center |220||I{{overline|4}}3d|| I {{overline|4}} 3 d || <math>\Gamma_c^vT_d^6</math> || ''93a'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 //3</math> || <math>4^\circ/4</math> || <math>(4\bar{*}2_1){:}3</math> || <math>(2_0{*}2_12_1){:}6</math> |- align=center |{{anchor|221-230}}221||rowspan=10|4/m {{overline|3}} 2/m (m{{overline|3}}m)||rowspan=10|<math>*432</math>||Pm{{overline|3}}m|| P 4/m {{overline|3}} 2/m || <math>\Gamma_cO_h^1</math> || ''71s'' || <math>\left ( a:a:a\right ) :4/\tilde 6 \cdot m</math> || <math>4^-{:}2</math> || <math>[*{\cdot}4{\cdot}4{\cdot}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6</math> |- align=center |222||Pn{{overline|3}}n|| P 4/n {{overline|3}} 2/n || <math>\Gamma_cO_h^2</math> || ''53h'' || <math>\left ( a:a:a\right ) :4/\tilde 6 \cdot \widetilde{abc}</math> || <math>8^{\circ\circ}</math> || <math>(*4_04{:}2){:}3</math> || <math>(2\bar{*}_12_02_0){:}6</math> |- align=center |223||Pm{{overline|3}}n|| P 4₂/m {{overline|3}} 2/n || <math>\Gamma_cO_h^3</math> || ''102a'' || <math>\left ( a:a:a\right ) :4_2//\tilde 6 \cdot \widetilde{abc}</math> || <math>8^\circ</math> || <math>[*{\cdot}4{:}4{\cdot}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6</math> |- align=center |224||Pn{{overline|3}}m|| P 4₂/n {{overline|3}} 2/m || <math>\Gamma_cO_h^4</math> || ''103a'' || <math>\left ( a:a:a\right ) :4_2//\tilde 6 \cdot m</math> || <math>4^+{:}2</math> || <math>(*4_24{\cdot}2){:}3</math> || <math>(2\bar{*}_12_02_0){:}6</math> |- align=center |225||Fm{{overline|3}}m|| F 4/m {{overline|3}} 2/m || <math>\Gamma_c^fO_h^5</math> || ''73s'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot m</math> || <math>2^-{:}2</math> || <math>[*{\cdot}4{\cdot}4{:}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{:}2{:}2]{:}6</math> |- align=center |226||Fm{{overline|3}}c|| F 4/m {{overline|3}} 2/c || <math>\Gamma_c^fO_h^6</math> || ''54h'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot \tilde c</math> || <math>4^{--}</math> || <math>[*{\cdot}4{:}4{:}2]{:}3</math> || <math>[*{\cdot}2{\cdot}2{:}2{:}2]{:}6</math> |- align=center |227||Fd{{overline|3}}m|| F 4₁/d {{overline|3}} 2/m || <math>\Gamma_c^fO_h^7</math> || ''100a'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot m</math> || <math>2^+{:}2</math> || <math>(*4_14{\cdot}2){:}3</math> || <math>(2\bar{*}2_02_1){:}6</math> |- align=center |228||Fd{{overline|3}}c|| F 4₁/d {{overline|3}} 2/c || <math>\Gamma_c^fO_h^8</math> || ''101a'' || <math>\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot \tilde c</math> || <math>4^{++}</math> || <math>(*4_14{:}2){:}3</math> || <math>(2\bar{*}2_02_1){:}6</math> |- align=center |229||Im{{overline|3}}m|| I 4/m {{overline|3}} 2/m || <math>\Gamma_c^vO_h^9</math> || ''72s'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/\tilde 6 \cdot m</math> || <math>8^\circ{:}2</math> || <math>[*{\cdot}4{\cdot}4{:}2]{:}3</math> || <math>[2_1{*}{\cdot}2{\cdot}2]{:}6</math> |- align=center |230||Ia{{overline|3}}d|| I 4₁/a {{overline|3}} 2/d || <math>\Gamma_c^vO_h^{10}</math> || ''99a'' || <math>\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4_1//\tilde 6 \cdot \tfrac{1}{2}\widetilde{abc}</math> || <math>8^\circ/4</math> || <math>(*4_14{:}2){:}3</math> || <math>(*2_12{:}2{:}2){:}6</math> |}

==Notes== {{Reflist|group=note}}

==References== {{Reflist}}

==External links== {{Commons category|Space groups}} * [https://www.iucr.org/ International Union of Crystallography] * [http://neon.mems.cmu.edu/degraef/pointgroups/ Point Groups and Bravais Lattices] * [http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Full list of 230 crystallographic space groups] * [https://www.emis.de/journals/BAG/vol.42/no.2/b42h2con.pdf Conway et al. on fibrifold notation]

[[Category:Symmetry]] [[Category:Crystallography]]