Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#550 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #550 | D3 |
| hypocenter | C3 | #3358 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ #550 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#550 | C2 | semidirect |
| 2. Q8×D12 | C2 | semidirect |
| 3. #5072 | C2 | semidirect |
| 4. #5070 | C2 | semidirect |
| 5. Q8×C3⋊C8 | C2 | semidirect |
| 6. C4×#213 | C2 | semidirect |
| 7. #5007 | C2 | semidirect |
| 25. #550 | D3 | semidirect |
| 70. C3×Q8 | D8 | semidirect |
| 82. D12 | #41 | semidirect |
| 123. Q8 | #213 | semidirect |
| 157. C3 | #3358 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#550 | 2 |
| Q8×D12 | 2 |
| #5072 | 2 |
| #5070 | 2 |
| Q8×C3⋊C8 | 2 |
| C4×#213 | 2 |
| #5007 | 2 |
| #3358 | 3 |