Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#427 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #427 | D3 |
| hypocenter | C3 | #3058 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D24 | C2 | semidirect |
| 2. C2×#901 | C2 | semidirect |
| 3. #5525 | C2 | semidirect |
| 4. C3×#427 | C2 | semidirect |
| 5. #5001 | C2 | semidirect |
| 6. #4883 | C2 | semidirect |
| 7. #4865 | C2 | semidirect |
| 18. C2×ΓL(1,𝔽25) | C22 | semidirect |
| 19. C2×D3×C8 | C22 | semidirect |
| 20. #895 | C22 | semidirect |
| 24. C3×#107 | C22 | semidirect |
| 25. C3×#106 | C22 | semidirect |
| 26. #809 | C22 | semidirect |
| 32. #427 | D3 | semidirect |
| 47. C2×C3×C8 | D4 | semidirect |
| 75. #107 | C2×D3 | semidirect |
| 76. #106 | C2×D3 | semidirect |
| 120. C2×C8 | D12 | semidirect |
| 181. C3 | #3058 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D24 | 2 |
| C2×#901 | 2 |
| #5525 | 2 |
| C3×#427 | 2 |
| #5001 | 2 |
| #4883 | 2 |
| #4865 | 2 |
| #3058 | 3 |