Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| prime radical | C22×D3 | C24 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3556 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#564 | C2 | semidirect |
| 4. C3⋊#564 | C2 | semidirect |
| 5. #5685 | C2 | semidirect |
| 25. #564 | D3 | semidirect |
| 108. C3 | #3556 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#564 | 2 |
| #5696 | 2 |
| #5688 | 2 |
| C3⋊#564 | 2 |
| #5685 | 2 |
| #3556 | 3 |