Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D12 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#336 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | #336 | D3 |
| hypocenter | C3 | #2011 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #336 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D24 | C2 | semidirect |
| 2. C2×#874 | C2 | semidirect |
| 3. C2×#821 | C2 | semidirect |
| 4. #4939 | C2 | semidirect |
| 5. C3×#336 | C2 | semidirect |
| 23. #336 | D3 | semidirect |
| 41. C2×D12 | D4 | semidirect |
| 44. C2×C3×C8 | D4 | semidirect |
| 74. C2×C3×C4 | D8 | semidirect |
| 110. C2×C8 | D12 | semidirect |
| 155. C2×C4 | D24 | semidirect |
| 187. C3 | #2011 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D24 | 2 |
| C2×#874 | 2 |
| C2×#821 | 2 |
| #4939 | 2 |
| C3×#336 | 2 |
| #2011 | 3 |