Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
At least 7766 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has at least 7766 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72235024 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | #72235022 | A4 | 1 | 3 | yes | complemented |
| 3 | #72235019 | A4 | 1 | 1 | yes | characteristic, complemented |
| 4 | H(5,𝔽4) | A4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C22×H(3,𝔽4) | C22⋊A4 | 1 | 1 | yes | characteristic |
| 6 | C22×#120 | #997 | 1 | 3 | yes | — |
| 7 | C26 | #5464 | 1 | 1 | yes | characteristic |
| 8 | C26 | #5465 | 1 | 1 | yes | characteristic |
| 9 | C25 | #2679⋊C3 | 1 | 3 | yes | — |
| 10 | C24 | #11709313 | 1 | 1 | yes | characteristic |
| 11 | C22 | #72210409 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #72235024 | C3 |
| Frattini | C22×H(3,𝔽4) | C22⋊A4 |
| socle | C22 | #72210409 |
| LCS | G ▷ #72235024 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72235024 | C3 | semidirect |
| 2. #72235022 | A4 | semidirect |
| 3. #72235019 | A4 | semidirect |
| 4. H(5,𝔽4) | A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72235030 | 4 |
| #72235028 | 4 |
| #72235026 | 4 |
| #72235024 | 3 |