Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
488 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#432 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #432 | D3 |
| hypocenter | C3 | #3109 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #432 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5477 | C2 | semidirect |
| 2. C2×C4×D12 | C2 | semidirect |
| 3. C3×#432 | C2 | semidirect |
| 4. #4843 | C2 | semidirect |
| 5. #4842 | C2 | semidirect |
| 6. #4735 | C2 | semidirect |
| 7. #4732 | C2 | semidirect |
| 8. #4711 | C2 | semidirect |
| 20. #874 | C22 | semidirect |
| 26. C4×D12 | C22 | semidirect |
| 27. #869 | C22 | semidirect |
| 28. C3×C4⋊C8 | C22 | semidirect |
| 30. C3×#97 | C22 | semidirect |
| 32. #821 | C22 | semidirect |
| 33. #817 | C22 | semidirect |
| 41. #432 | D3 | semidirect |
| 93. C4⋊C8 | C2×D3 | semidirect |
| 97. #97 | C2×D3 | semidirect |
| 122. D12 | #41 | semidirect |
| 234. C3 | #3109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5477 | 2 |
| C2×C4×D12 | 2 |
| C3×#432 | 2 |
| #4843 | 2 |
| #4842 | 2 |
| #4735 | 2 |
| #4732 | 2 |
| #4711 | 2 |
| #3109 | 3 |