Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
8808 conjugacy classes of subgroups; only the 3606 normal subgroups are shown.
This group has 8808 conjugacy classes of subgroups — too many to draw. Only its 3606 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D9 |
| commutator, cosocle | C2×C9 | C26 |
| Frattini | C2×C3 | C25×D3 |
| Fitting | C23×Q8×C9 | C2 |
| socle | C23×C3 | C23×D3 |
| hypercenter | C23×Q8 | D9 |
| hypocenter | C9 | C22×2−1+4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C9 | C2 | semidirect |
| 2. C2×#63713 | C2 | direct (N × H) |
| 3. C22×#3701 | C2 | semidirect |
| 4. C22×Q8×D9 | C2 | semidirect |
| 5. C22×#3697 | C2 | semidirect |
| 8. C22×Q8×C9 | C22 | semidirect |
| 11. #63713 | C22 | direct (N × H) |
| 12. C2×#3701 | C22 | semidirect |
| 13. C2×Q8×D9 | C22 | semidirect |
| 14. C2×#3697 | C22 | semidirect |
| 26. C2×Q8×C9 | C23 | semidirect |
| 31. #3701 | C23 | semidirect |
| 32. Q8×D9 | C23 | semidirect |
| 33. #3697 | C23 | semidirect |
| 66. C23×Q8 | D9 | semidirect |
| 82. C22×C9 | 2−1+4 | semidirect |
| 95. C22×Q8 | C2×D9 | semidirect |
| 114. C2×C9 | C2×2−1+4 | semidirect |
| 124. C2×Q8 | C22×D9 | semidirect |
| 139. C9 | C22×2−1+4 | semidirect |
| 160. C22 | #63713 | direct (N × H) |
| 168. C2 | C2×#63713 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C9 | 2 |
| C2×#63713 | 2 |
| C22×#3701 | 2 |
| C22×Q8×D9 | 2 |
| C22×#3697 | 2 |
| C22×#1007 | 3 |