Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#381 | C2 |
| hypercenter | #381 | D3 |
| hypocenter | C3 | #2613 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D12 | C2 | semidirect |
| 2. C2×#929 | C2 | semidirect |
| 3. C2×#882 | C2 | semidirect |
| 4. C3×#381 | C2 | semidirect |
| 5. #4955 | C2 | semidirect |
| 6. #4672 | C2 | semidirect |
| 7. #4671 | C2 | semidirect |
| 15. C22×D12 | C22 | semidirect |
| 19. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 20. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 21. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 22. C2×Dic3⋊C4 | C22 | semidirect |
| 24. C3×#95 | C22 | semidirect |
| 30. #381 | D3 | semidirect |
| 36. C23×D3 | D4 | semidirect |
| 40. C22×C3×C4 | D4 | semidirect |
| 73. #95 | C2×D3 | semidirect |
| 119. C22×C4 | D12 | semidirect |
| 200. C3 | #2613 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D12 | 2 |
| C2×#929 | 2 |
| C2×#882 | 2 |
| C3×#381 | 2 |
| #4955 | 2 |
| #4672 | 2 |
| #4671 | 2 |
| #2613 | 3 |