Editing Subgroup (section)

==Example: Subgroups of S<sub>4</sub>{{anchor|Subgroups of S4}}==
{{math|S<sub>4</sub>}} is the [[symmetric group]] whose elements correspond to the [[permutation]]s of 4 elements.<br>
Below are all its subgroups, ordered by cardinality.<br>
Each group <small>(except those of cardinality 1 and 2)</small> is represented by its [[Cayley table]].

===24 elements===
Like each group, {{math|S<sub>4</sub>}} is a subgroup of itself.

{| style="width:100%"
| style="vertical-align:top;"|[[File:Symmetric group 4; Cayley table; numbers.svg|thumb|left|595px|Symmetric group {{math|S<sub>4</sub>}}]]
| style="vertical-align:top;"|
{{multiple image
 | align = right
 | image1 = Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg
 | width1 = 250
 | caption1 = All 30 subgroups
 | image2 = Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg
 | width2 = 185
 | caption2 = Simplified
 | footer = [[Hasse diagram]]s of the [[lattice of subgroups]] of {{math|S<sub>4</sub>}}
}}
|}

===12 elements===
The [[alternating group]] contains only the [[w:parity of a permutation|even permutations]].<br>
It is one of the two nontrivial proper [[normal subgroup]]s of {{math|S<sub>4</sub>}}. <small>(The other one is its Klein subgroup.)</small>
[[File:Alternating group 4; Cayley table; numbers.svg|thumb|left|323px|Alternating group {{math|A<sub>4</sub>}}<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]]<br>[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|60px]][[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|60px]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|60px]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]]
{{clear}}

===8 elements===
{| 
|-
| <!-- LEFT -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,2,2,4,4,2); subgroup of S4.svg|thumb|233px|[[w:Dihedral group|Dihedral group]] [[Dihedral group of order 8|of order 8]]<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|70px]]]] || &nbsp; || <!-- CENTRAL -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|70px]]]] || &nbsp; || <!-- RIGHT -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|70px]]]]
|}
{{clear}}

===6 elements===
{| 
|-
| <!-- 1 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,2,3,4,5).svg|thumb|187px|[[w:Symmetric group|Symmetric group]] {{math|[[w:Dihedral group of order 6|S<sub>3</sub>]]}}<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|60px]]]] ||  <!-- 2 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,5,6,11,19,21).svg|thumb|187px|Symmetric group {{math|S<sub>3</sub>}}<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|60px]]]]  || <!-- 3 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,14,15,20,21).svg|thumb|187px|Symmetric group {{math|S<sub>3</sub>}}<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|60px]]]] ||  <!-- 4 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,2,6,8,12,14).svg|thumb|187px|Symmetric group {{math|S<sub>3</sub>}}<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]]
|}
{{clear}}

===4 elements===
{|
|- style="vertical-align: top;"
| [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|thumb|142px|[[w:Klein four-group|Klein four-group]]]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|thumb|142px|Klein four-group]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|thumb|142px|Klein four-group]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|thumb|142px|Klein four-group<br><small>([[normal subgroup]])</small>]]
|}
{{clear}}
{|
|-
| [[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|thumb|142px|[[w:Cyclic group|Cyclic group]] {{math|Z<sub>4</sub>}}]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|thumb|142px|Cyclic group {{math|Z<sub>4</sub>}}]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|thumb|142px|Cyclic group {{math|Z<sub>4</sub>}}]]
|}
{{clear}}

===3 elements===
{|
|-
| [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|thumb|120px|[[w:Cyclic group|Cyclic group]] {{math|Z<sub>3</sub>}}]] || [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|thumb|120px|Cyclic group {{math|Z<sub>3</sub>}}]] || [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|thumb|120px|Cyclic group {{math|Z<sub>3</sub>}}]] || 
[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|thumb|120px|Cyclic group {{math|Z<sub>3</sub>}}]]
|}
{{clear}}

===2 elements===
Each permutation {{mvar|p}} of order 2 generates a subgroup {{math|{1, ''p''}}}.
These are the permutations that have only 2-cycles:<br>
* There are the 6 [[Cyclic permutation#Transpositions|transpositions]] with one 2-cycle. &nbsp; <small>(green background)</small>
* And 3 permutations with two 2-cycles. &nbsp; <small>(white background, bold numbers)</small>

===1 element===
The [[trivial group|trivial subgroup]] is the unique subgroup of order 1.