Finite Group DatabasebetaFinite Group DatabasebetaOrder 504.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | AGL(1,𝔽8) | — | 9 | 1 | no | maximal, Hall |
| 2 | D9 | — | 28 | 1 | no | maximal |
| 3 | D7 | — | 36 | 1 | no | maximal |
| 4 | C9 | — | 28 | 1 | no | Sylow |
| 5 | C23 | — | 9 | 1 | no | Sylow |
| 6 | C7 | — | 36 | 1 | no | Sylow |
| 7 | D3 | — | 84 | 1 | no | — |
| 8 | C22 | — | 63 | 1 | no | — |
| 9 | C3 | — | 28 | 1 | no | — |
| 10 | C2 | — | 63 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| AGL(1,𝔽8) | 9 |
| D9 | 28 |
| D7 | 36 |