Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 508 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, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#390 | C2 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | #2828 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#390 | C2 | semidirect |
| 2. #5223 | C2 | semidirect |
| 3. #4991 | C2 | semidirect |
| 4. #4986 | C2 | semidirect |
| 5. #4961 | C2 | semidirect |
| 6. #4960 | C2 | semidirect |
| 7. #4959 | C2 | semidirect |
| 8. #4952 | C2 | semidirect |
| 9. #4940 | C2 | semidirect |
| 39. #390 | D3 | semidirect |
| 261. C3 | #2828 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#390 | 2 |
| #5223 | 2 |
| #4991 | 2 |
| #4986 | 2 |
| #4961 | 2 |
| #4960 | 2 |
| #4959 | 2 |
| #4952 | 2 |
| #4940 | 2 |
| #2828 | 3 |