Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
430 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 119 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×#460 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #460 | D3 |
| hypocenter | C3 | #3115 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #460 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5716 | C2 | semidirect |
| 2. C4×#239 | C2 | semidirect |
| 3. C3×#460 | C2 | semidirect |
| 4. #5172 | C2 | semidirect |
| 5. #5164 | C2 | semidirect |
| 6. #5072 | C2 | semidirect |
| 7. #5020 | C2 | semidirect |
| 8. #4846 | C2 | semidirect |
| 9. #4835 | C2 | semidirect |
| 10. #4785 | C2 | semidirect |
| 11. #4784 | C2 | semidirect |
| 49. #460 | D3 | semidirect |
| 130. C3×Q8 | #41 | semidirect |
| 149. D12 | #41 | semidirect |
| 226. Q8 | #237 | semidirect |
| 282. C3 | #3115 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5716 | 2 |
| C4×#239 | 2 |
| C3×#460 | 2 |
| #5172 | 2 |
| #5164 | 2 |
| #5072 | 2 |
| #5020 | 2 |
| #4846 | 2 |
| #4835 | 2 |
| #4785 | 2 |
| #4784 | 2 |
| #3115 | 3 |