Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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×#462 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #462 | D3 |
| hypocenter | C3 | #3212 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #462 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5725 | C2 | semidirect |
| 2. #5580 | C2 | semidirect |
| 3. C3×#462 | C2 | semidirect |
| 4. #5191 | C2 | semidirect |
| 5. #5187 | C2 | semidirect |
| 6. #5083 | C2 | semidirect |
| 7. #5027 | C2 | semidirect |
| 8. #4843 | C2 | semidirect |
| 9. #4833 | C2 | semidirect |
| 10. #4810 | C2 | semidirect |
| 11. #4800 | C2 | semidirect |
| 48. #462 | D3 | semidirect |
| 282. C3 | #3212 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5725 | 2 |
| #5580 | 2 |
| C3×#462 | 2 |
| #5191 | 2 |
| #5187 | 2 |
| #5083 | 2 |
| #5027 | 2 |
| #4843 | 2 |
| #4833 | 2 |
| #4810 | 2 |
| #4800 | 2 |
| #3212 | 3 |