Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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 | C22×C3 | C23×C4 |
| Fitting | C3×#390 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | #2326 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#887 | C2 | semidirect |
| 5. #4975 | C2 | semidirect |
| 6. #4946 | C2 | semidirect |
| 7. #4945 | C2 | semidirect |
| 8. #4932 | C2 | semidirect |
| 9. #4929 | C2 | semidirect |
| 19. #936 | C4 | semidirect |
| 56. C2×Dic6 | C2×C4 | semidirect |
| 61. #217 | C2×C4 | semidirect |
| 66. C4×Dic3 | C2×C4 | semidirect |
| 216. C3 | #2326 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#936 | 2 |
| C2×D4×Dic3 | 2 |
| C2×#887 | 2 |
| C3×#390 | 2 |
| #4975 | 2 |
| #4946 | 2 |
| #4945 | 2 |
| #4932 | 2 |
| #4929 | 2 |
| #2326 | 3 |