Finite Group DatabasebetaFinite Group DatabasebetaOrder 1512.
25 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | AΓL(1,𝔽8) | — | 9 | 1 | no | maximal |
| 3 | AGL(1,𝔽8) | — | 9 | 1 | no | Hall |
| 4 | AGL(1,ℤ/9) | — | 28 | 1 | no | maximal |
| 5 | AGL(1,𝔽7) | — | 36 | 1 | no | maximal |
| 6 | 3−1+2 | — | 28 | 1 | no | Sylow |
| 7 | C2×A4 | — | 63 | 1 | no | — |
| 8 | ΓL(1,𝔽8) | — | 36 | 1 | no | — |
| 9 | C3×D3 | — | 84 | 1 | no | — |
| 10 | D9 | — | 28 | 1 | no | — |
| 11 | D7 | — | 36 | 1 | no | — |
| 12 | A4 | — | 63 | 1 | no | — |
| 13 | C32 | — | 28 | 1 | no | — |
| 14 | C9 | — | 28 | 1 | no | — |
| 15 | C9 | — | 56 | 1 | no | — |
| 16 | C23 | — | 9 | 1 | no | Sylow |
| 17 | C7 | — | 36 | 1 | no | Sylow |
| 18 | C2×C3 | — | 252 | 1 | no | — |
| 19 | D3 | — | 84 | 1 | no | — |
| 20 | C22 | — | 63 | 1 | no | — |
| 21 | C3 | — | 28 | 1 | no | — |
| 22 | C3 | — | 84 | 1 | no | — |
| 23 | C2 | — | 63 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | SL(2,𝔽8) | C3 |
| LCS | G ▷ SL(2,𝔽8) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽8) | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽8) | 3 |
| AΓL(1,𝔽8) | 9 |
| AGL(1,ℤ/9) | 28 |
| AGL(1,𝔽7) | 36 |