Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
560 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 185 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 | #2372 |
| 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. C3×#391 | C2 | semidirect |
| 3. #5229 | C2 | semidirect |
| 4. #4990 | C2 | semidirect |
| 5. C4×(C2×D3)⋊C4 | C2 | semidirect |
| 6. #4670 | C2 | semidirect |
| 7. #4666 | C2 | semidirect |
| 9. #4648 | C2 | semidirect |
| 16. #947 | C4 | semidirect |
| 35. #391 | D3 | semidirect |
| 45. C2×C3×Q8 | C2×C4 | semidirect |
| 65. (C2×D3)⋊C4 | C2×C4 | semidirect |
| 66. (C2×D3)⋊C4 | C2×C4 | semidirect |
| 71. C4×Dic3 | C2×C4 | semidirect |
| 132. C2×Q8 | C4×D3 | semidirect |
| 240. C3 | #2372 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#947 | 2 |
| C3×#391 | 2 |
| #5229 | 2 |
| #4990 | 2 |
| C4×(C2×D3)⋊C4 | 2 |
| #4670 | 2 |
| #4666 | 2 |
| #4665 | 2 |
| #4648 | 2 |
| #2372 | 3 |