Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
766 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #953 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×#585 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | #3025 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585 | C2 | semidirect |
| 2. #5828 | C2 | semidirect |
| 3. #5244 | C2 | semidirect |
| 4. #5243 | C2 | semidirect |
| 5. #5240 | C2 | semidirect |
| 12. C22×C4×D3 | C22 | semidirect |
| 15. #937 | C22 | semidirect |
| 16. #935 | C22 | semidirect |
| 18. #926 | C22 | semidirect |
| 19. #834 | C22 | semidirect |
| 20. #585 | D3 | semidirect |
| 32. C2×C3×Q8 | D4 | semidirect |
| 35. C2×C3×D4 | D4 | semidirect |
| 38. C22×C3×C4 | D4 | semidirect |
| 91. C2×Q8 | #67 | semidirect |
| 95. C2×D4 | #67 | semidirect |
| 101. C22×C4 | #67 | semidirect |
| 160. C3 | #3025 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| #5828 | 2 |
| #5244 | 2 |
| #5243 | 2 |
| #5240 | 2 |
| #3025 | 3 |