Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
560 conjugacy classes of subgroups; only the 193 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 193 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 | C22×C3 | C23×C4 |
| Fitting | C3×#382 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2357 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#382 | C2 | semidirect |
| 4. Dic3×C4⋊C4 | C2 | semidirect |
| 5. C4×C4⋊Dic3 | C2 | semidirect |
| 6. #4670 | C2 | semidirect |
| 7. #4666 | C2 | semidirect |
| 8. #4664 | C2 | semidirect |
| 21. C4×D12 | C4 | semidirect |
| 30. #382 | D3 | semidirect |
| 188. C3 | #2357 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D12 | 2 |
| C3×#382 | 2 |
| #4988 | 2 |
| Dic3×C4⋊C4 | 2 |
| C4×C4⋊Dic3 | 2 |
| #4670 | 2 |
| #4666 | 2 |
| #4664 | 2 |
| #2357 | 3 |