Finite Group DatabasebetaFinite Group DatabasebetaOrder 12642.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C43×#3862 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×D301 | — | 3 | 1 | no | maximal, Hall |
| 3 | C72×C43 | D3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | ΓL(1,𝔽8)×C43 | — | 14 | 1 | no | — |
| 5 | D301 | — | 3 | 1 | no | — |
| 6 | C7×D43 | — | 21 | 1 | no | — |
| 7 | C7×C43 | — | 2 | 1 | no | — |
| 8 | C7×C43 | — | 3 | 1 | no | — |
| 9 | C7×C43 | — | 3 | 1 | no | — |
| 10 | #64418 | — | 43 | 1 | no | maximal, Hall |
| 11 | D129 | — | 49 | 1 | no | maximal, Hall |
| 12 | #3862 | D43 | 1 | 1 | yes | characteristic, complemented, Hall |
| 13 | C3×C43 | — | 49 | 1 | no | Hall |
| 14 | C7×D7 | — | 129 | 1 | no | Hall |
| 15 | D43 | — | 21 | 1 | no | Hall |
| 16 | C72 | D129 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C43 | #64418 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 18 | ΓL(1,𝔽8) | — | 14 | 1 | no | — |
| 19 | C2×C7 | — | 903 | 1 | no | — |
| 20 | D7 | — | 129 | 1 | no | — |
| 21 | C7 | — | 2 | 1 | no | — |
| 22 | C7 | — | 3 | 1 | no | — |
| 23 | C7 | — | 3 | 1 | no | — |
| 24 | D3 | — | 2107 | 1 | no | Hall |
| 25 | C3 | — | 49 | 1 | no | Sylow |
| 26 | C2 | — | 903 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C43×#3862 | C2 |
| Fitting, socle | C72×C43 | D3 |
| LCS | G ▷ C43×#3862 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C43×#3862 | C2 | semidirect |
| 3. C72×C43 | D3 | semidirect |
| 12. #3862 | D43 | semidirect |
| 16. C72 | D129 | semidirect |
| 17. C43 | #64418 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C43×#3862 | 2 |
| C7×D301 | 3 |
| #64418 | 43 |
| D129 | 49 |