Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 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×#391 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | #2804 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#391 | C2 | semidirect |
| 2. #5231 | C2 | semidirect |
| 3. #4989 | C2 | semidirect |
| 5. #4929 | C2 | semidirect |
| 6. #4673 | C2 | semidirect |
| 7. #4668 | C2 | semidirect |
| 9. #4660 | C2 | semidirect |
| 36. #391 | D3 | semidirect |
| 55. C2×D12 | Q8 | semidirect |
| 248. C3 | #2804 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#391 | 2 |
| #5231 | 2 |
| #4989 | 2 |
| #4939 | 2 |
| #4929 | 2 |
| #4673 | 2 |
| #4668 | 2 |
| #4665 | 2 |
| #4660 | 2 |
| #2804 | 3 |