Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
430 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 117 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×#475 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #475 | D3 |
| hypocenter | C3 | #3332 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #475 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5726 | C2 | semidirect |
| 2. #5577 | C2 | semidirect |
| 3. C3×#475 | C2 | semidirect |
| 4. #5088 | C2 | semidirect |
| 5. C4×#215 | C2 | semidirect |
| 6. #4865 | C2 | semidirect |
| 7. #4863 | C2 | semidirect |
| 8. #4843 | C2 | semidirect |
| 9. #4838 | C2 | semidirect |
| 10. #4810 | C2 | semidirect |
| 11. #4803 | C2 | semidirect |
| 46. #475 | D3 | semidirect |
| 123. C3×Q8 | #41 | semidirect |
| 210. Q8 | #239 | semidirect |
| 262. C3 | #3332 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5726 | 2 |
| #5577 | 2 |
| C3×#475 | 2 |
| #5088 | 2 |
| C4×#215 | 2 |
| #4865 | 2 |
| #4863 | 2 |
| #4843 | 2 |
| #4838 | 2 |
| #4810 | 2 |
| #4803 | 2 |
| #3332 | 3 |