Finite Group DatabasebetaFinite Group DatabasebetaOrder 1008.
64 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AGL(1,𝔽8) | C2×C3 |
| Fitting, socle | C23×C3 | C2×ΓL(1,𝔽8) |
| LCS | G ▷ C3×AGL(1,𝔽8) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×AΓL(1,𝔽8) | C2 | semidirect |
| 2. D3×AGL(1,𝔽8) | C3 | semidirect |
| 4. C3×AGL(1,𝔽8) | C2×C3 | semidirect |
| 6. AΓL(1,𝔽8) | D3 | direct (N × H) |
| 14. AGL(1,𝔽8) | C3×D3 | semidirect |
| 15. C23×D3 | ΓL(1,𝔽8) | semidirect |
| 21. C23×C3 | C2×ΓL(1,𝔽8) | semidirect |
| 45. C23 | D3×ΓL(1,𝔽8) | semidirect |
| 53. D3 | AΓL(1,𝔽8) | direct (N × H) |
| 59. C3 | C2×AΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×AΓL(1,𝔽8) | 2 |
| D3×AGL(1,𝔽8) | 3 |
| C2×AΓL(1,𝔽8) | 3 |
| C2×D3×A4 | 7 |
| D3×ΓL(1,𝔽8) | 8 |