Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
406 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 406 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×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3397 |
| 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. #5553 | C2 | semidirect |
| 2. #5542 | C2 | semidirect |
| 3. C3×C4×D8 | C2 | semidirect |
| 4. #5154 | C2 | semidirect |
| 5. #5019 | C2 | semidirect |
| 6. #5015 | C2 | semidirect |
| 7. #4889 | C2 | semidirect |
| 8. #4787 | C2 | semidirect |
| 9. #4762 | C2 | semidirect |
| 10. #4693 | C2 | semidirect |
| 11. #4679 | C2 | semidirect |
| 52. C4×D8 | D3 | semidirect |
| 146. C3×C8 | #41 | semidirect |
| 232. C8 | #235 | semidirect |
| 258. C3 | #3397 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5553 | 2 |
| #5542 | 2 |
| C3×C4×D8 | 2 |
| #5154 | 2 |
| #5019 | 2 |
| #5015 | 2 |
| #4889 | 2 |
| #4787 | 2 |
| #4762 | 2 |
| #4693 | 2 |
| #4679 | 2 |
| #3397 | 3 |