Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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×#390 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C2×#940 | C2 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | C4×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×D4×Dic3 | C2 | semidirect |
| 3. C2×#893 | C2 | semidirect |
| 4. C2×#67⋊C4 | C2 | semidirect |
| 5. C3×#390 | C2 | semidirect |
| 6. #4975 | C2 | semidirect |
| 7. #4957 | C2 | semidirect |
| 8. #4939 | C2 | semidirect |
| 9. C42×Dic3 | C2 | semidirect |
| 20. #940 | C4 | semidirect |
| 38. #390 | D3 | semidirect |
| 45. C2×C3×D4 | C2×C4 | semidirect |
| 51. C2×#67 | C2×C4 | semidirect |
| 60. C2×D12 | C2×C4 | semidirect |
| 72. C4×Dic3 | C2×C4 | semidirect |
| 73. C4×Dic3 | D4 | semidirect |
| 74. C4×Dic3 | D4 | semidirect |
| 139. C2×D4 | C4×D3 | semidirect |
| 180. C3×C4 | C4×D4 | semidirect |
| 183. Dic3 | C4×D4 | semidirect |
| 184. Dic3 | #127 | semidirect |
| 246. C4 | #67⋊C4 | semidirect |
| 247. C3 | C4×#127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#940 | 2 |
| C2×D4×Dic3 | 2 |
| C2×#893 | 2 |
| C2×#67⋊C4 | 2 |
| C3×#390 | 2 |
| #4975 | 2 |
| #4957 | 2 |
| #4939 | 2 |
| C42×Dic3 | 2 |
| C4×#127 | 3 |