Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#443 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #443 | D3 |
| hypocenter | C3 | #3293 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5771 | C2 | semidirect |
| 2. #5753 | C2 | semidirect |
| 3. C3×#443 | C2 | semidirect |
| 4. #5143 | C2 | semidirect |
| 5. #4896 | C2 | semidirect |
| 6. #4826 | C2 | semidirect |
| 7. #4700 | C2 | semidirect |
| 8. #4699 | C2 | semidirect |
| 39. #443 | D3 | semidirect |
| 194. C3 | #3293 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5771 | 2 |
| #5753 | 2 |
| C3×#443 | 2 |
| #5143 | 2 |
| #4896 | 2 |
| #4826 | 2 |
| #4700 | 2 |
| #4699 | 2 |
| #3293 | 3 |