Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2644 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#396 | C2 | semidirect |
| 2. #4992 | C2 | semidirect |
| 3. #4976 | C2 | semidirect |
| 5. #4952 | C2 | semidirect |
| 6. #4676 | C2 | semidirect |
| 7. #4648 | C2 | semidirect |
| 24. #396 | D3 | semidirect |
| 147. C3 | #2644 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#396 | 2 |
| #4992 | 2 |
| #4976 | 2 |
| #4957 | 2 |
| #4952 | 2 |
| #4676 | 2 |
| #4648 | 2 |
| #2644 | 3 |