Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
648 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 173 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×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C2×#940 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2380 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#386 | C2 | semidirect |
| 3. #5220 | C2 | semidirect |
| 4. #5213 | C2 | semidirect |
| 5. #4958 | C2 | semidirect |
| 6. #4956 | C2 | semidirect |
| 7. #4956 | C2 | semidirect |
| 8. #4665 | C2 | semidirect |
| 9. #4651 | C2 | semidirect |
| 22. #940 | C4 | semidirect |
| 40. #386 | D3 | semidirect |
| 47. C2×C3×D4 | C2×C4 | semidirect |
| 57. C2×#67 | C2×C4 | semidirect |
| 58. C2×#67 | C2×C4 | semidirect |
| 66. C2×D12 | C2×C4 | semidirect |
| 77. C4×Dic3 | C2×C4 | semidirect |
| 157. C2×D4 | C4×D3 | semidirect |
| 286. C3 | #2380 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#940 | 2 |
| C3×#386 | 2 |
| #5220 | 2 |
| #5213 | 2 |
| #4958 | 2 |
| #4956 | 2 |
| #4665 | 2 |
| #4651 | 2 |
| #2380 | 3 |