Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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, prime radical | C2×C3×#120 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | #2375 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5247 | C2 | semidirect |
| 3. #5221 | C2 | semidirect |
| 4. #5220 | C2 | semidirect |
| 5. #4951 | C2 | semidirect |
| 9. C23×Dic3 | C22 | semidirect |
| 10. C3×#120 | C4 | semidirect |
| 14. C2×#217 | C22 | semidirect |
| 15. C2×C4⋊Dic3 | C22 | semidirect |
| 16. C2×C4×Dic3 | C22 | semidirect |
| 17. C3⋊#95 | C22 | semidirect |
| 25. C24×C3 | C2×C4 | semidirect |
| 35. C3×C22⋊C4 | C2×C4 | semidirect |
| 44. #120 | Dic3 | semidirect |
| 75. C24 | C2×Dic3 | semidirect |
| 89. C22⋊C4 | C2×Dic3 | semidirect |
| 147. C3 | #2375 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#120 | 2 |
| #5247 | 2 |
| #5221 | 2 |
| #5220 | 2 |
| #4951 | 2 |
| #2375 | 3 |