Finite Group DatabasebetaFinite Group DatabasebetaOrder 504.
35 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | GL(3,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C3×S4 | — | 7 | 2 | no | maximal, Hall |
| 3 | C3×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal, Hall |
| 4 | C3×A4 | — | 7 | 2 | no | — |
| 5 | S4 | — | 7 | 2 | no | — |
| 6 | C3×D4 | — | 21 | 1 | no | — |
| 7 | C3×C7 | — | 8 | 1 | no | — |
| 8 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 9 | ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 10 | C3×D3 | — | 28 | 1 | no | — |
| 11 | C22×C3 | — | 7 | 2 | no | — |
| 12 | A4 | — | 7 | 2 | no | — |
| 13 | A4 | — | 14 | 2 | no | — |
| 14 | C3×C4 | — | 21 | 1 | no | — |
| 15 | C32 | — | 28 | 1 | no | Sylow |
| 16 | D4 | — | 21 | 1 | no | Sylow |
| 17 | C7 | — | 8 | 1 | no | Sylow |
| 18 | C2×C3 | — | 21 | 1 | no | — |
| 19 | D3 | — | 28 | 1 | no | — |
| 20 | C22 | — | 7 | 2 | no | — |
| 21 | C4 | — | 21 | 1 | no | — |
| 22 | C3 | — | 28 | 1 | no | — |
| 23 | C3 | — | 56 | 1 | no | — |
| 24 | C3 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 25 | C2 | — | 21 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C3 |
| Frattini | C1 | G |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. GL(3,𝔽2) | C3 | direct (N × H) |
| 24. C3 | GL(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| GL(3,𝔽2) | 3 |
| C3×S4 | 7 |
| C3×ΓL(1,𝔽8) | 8 |