Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
402 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 109 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×#466 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #466 | D3 |
| hypocenter | C3 | #3203 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #466 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5608 | C2 | semidirect |
| 2. C3×#466 | C2 | semidirect |
| 3. #5158 | C2 | semidirect |
| 4. #5116 | C2 | semidirect |
| 5. #5110 | C2 | semidirect |
| 6. #5042 | C2 | semidirect |
| 7. #4884 | C2 | semidirect |
| 8. #4783 | C2 | semidirect |
| 9. #4774 | C2 | semidirect |
| 41. #466 | D3 | semidirect |
| 232. C3 | #3203 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5608 | 2 |
| C3×#466 | 2 |
| #5158 | 2 |
| #5116 | 2 |
| #5110 | 2 |
| #5042 | 2 |
| #4884 | 2 |
| #4783 | 2 |
| #4774 | 2 |
| #3203 | 3 |