Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
702 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 702 conjugacy classes of subgroups — too many to draw. Only its 149 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, prime radical | C2×C3×#120 | C2 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2622 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#217 | C2 | semidirect |
| 3. #5247 | C2 | semidirect |
| 4. #5222 | C2 | semidirect |
| 5. #5220 | C2 | semidirect |
| 6. #4952 | C2 | semidirect |
| 7. #4946 | C2 | semidirect |
| 20. C2×#217 | C22 | semidirect |
| 21. C2×Dic3⋊C4 | C22 | semidirect |
| 22. C3⋊#95 | C22 | semidirect |
| 23. C3⋊#95 | C22 | semidirect |
| 188. C3 | #2622 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#120 | 2 |
| C22×#217 | 2 |
| #5247 | 2 |
| #5222 | 2 |
| #5220 | 2 |
| #4952 | 2 |
| #4946 | 2 |
| #2622 | 3 |