Finite Group DatabasebetaFinite Group DatabasebetaOrder 30492.
392 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C112 | A4×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C32×C112 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C112 | C32 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C112 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C112×ΓL(1,𝔽8) | C3 | semidirect |
| 2. C7×C112×A4 | C3 | semidirect |
| 3. C112×C22⋊ΓL(1,𝔽8) | C3 | semidirect |
| 7. C22×C7×C112 | C32 | semidirect |
| 8. C11×A4×ΓL(1,𝔽8) | C11 | direct (N × H) |
| 10. C112×ΓL(1,𝔽8) | A4 | direct (N × H) |
| 15. C112×A4 | ΓL(1,𝔽8) | direct (N × H) |
| 17. C11×C22⋊ΓL(1,𝔽8) | C3×C11 | semidirect |
| 18. C7×C11×A4 | C3×C11 | semidirect |
| 19. C22×C11×ΓL(1,𝔽8) | C3×C11 | semidirect |
| 20. C7×C112 | C3×A4 | semidirect |
| 23. C22×C112 | C3×ΓL(1,𝔽8) | semidirect |
| 29. C22×C7×C11 | C32×C11 | semidirect |
| 30. A4×ΓL(1,𝔽8) | C112 | direct (N × H) |
| 34. C11×ΓL(1,𝔽8) | C11×A4 | direct (N × H) |
| 38. C11×A4 | C11×ΓL(1,𝔽8) | direct (N × H) |
| 39. C112 | A4×ΓL(1,𝔽8) | direct (N × H) |
| 41. C22⋊ΓL(1,𝔽8) | C3×C112 | semidirect |
| 42. C7×A4 | C3×C112 | semidirect |
| 43. C22×ΓL(1,𝔽8) | C3×C112 | semidirect |
| 44. C7×C11 | C3×C11×A4 | semidirect |
| 47. C22×C11 | C3×C11×ΓL(1,𝔽8) | semidirect |
| 53. C22×C7 | C32×C112 | semidirect |
| 57. ΓL(1,𝔽8) | C112×A4 | direct (N × H) |
| 61. A4 | C112×ΓL(1,𝔽8) | direct (N × H) |
| 62. C11 | C11×A4×ΓL(1,𝔽8) | direct (N × H) |
| 64. C7 | C3×C112×A4 | semidirect |
| 66. C22 | C3×C112×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C112×ΓL(1,𝔽8) | 3 |
| C7×C112×A4 | 3 |
| C112×C22⋊ΓL(1,𝔽8) | 3 |
| C3×C112×ΓL(1,𝔽8) | 4 |
| C3×C112×A4 | 7 |
| C11×A4×ΓL(1,𝔽8) | 11 |