Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3355 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5558 | C2 | semidirect |
| 2. C3×C4×D8 | C2 | semidirect |
| 3. #5159 | C2 | semidirect |
| 4. #5009 | C2 | semidirect |
| 5. #4881 | C2 | semidirect |
| 6. #4784 | C2 | semidirect |
| 7. #4774 | C2 | semidirect |
| 8. #4688 | C2 | semidirect |
| 9. #4683 | C2 | semidirect |
| 42. C4×D8 | D3 | semidirect |
| 199. C3 | #3355 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5558 | 2 |
| C3×C4×D8 | 2 |
| #5159 | 2 |
| #5009 | 2 |
| #4881 | 2 |
| #4784 | 2 |
| #4774 | 2 |
| #4688 | 2 |
| #4683 | 2 |
| #3355 | 3 |