Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
652 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 213 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×#391 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | C4×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q8×Dic3 | C2 | semidirect |
| 4. C3×#391 | C2 | semidirect |
| 5. #4977 | C2 | semidirect |
| 7. C42×Dic3 | C2 | semidirect |
| 8. #4670 | C2 | semidirect |
| 9. #4666 | C2 | semidirect |
| 16. #947 | C4 | semidirect |
| 34. #391 | D3 | semidirect |
| 43. C2×C3×Q8 | C2×C4 | semidirect |
| 61. (C2×D3)⋊C4 | C2×C4 | semidirect |
| 67. C4×Dic3 | C2×C4 | semidirect |
| 121. C2×Q8 | C4×D3 | semidirect |
| 164. Dic3 | #124 | semidirect |
| 218. C3 | C4×#124 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#947 | 2 |
| C2×Q8×Dic3 | 2 |
| C2×#893 | 2 |
| C3×#391 | 2 |
| #4977 | 2 |
| #4939 | 2 |
| C42×Dic3 | 2 |
| #4670 | 2 |
| #4666 | 2 |
| C4×#124 | 3 |