Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C4×Q8 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #2580 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C4×Q8 | C2 | semidirect |
| 3. #5231 | C2 | semidirect |
| 4. #4993 | C2 | semidirect |
| 5. #4989 | C2 | semidirect |
| 7. #4936 | C2 | semidirect |
| 8. #4928 | C2 | semidirect |
| 28. C2×C4×Q8 | D3 | semidirect |
| 48. C2×D12 | Q8 | semidirect |
| 198. C3 | #2580 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×Q8 | 2 |
| C2×C4×D12 | 2 |
| #5231 | 2 |
| #4993 | 2 |
| #4989 | 2 |
| #4939 | 2 |
| #4936 | 2 |
| #4928 | 2 |
| #2580 | 3 |