Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
9780 conjugacy classes of subgroups; only the 3726 normal subgroups are shown.
This group has 9780 conjugacy classes of subgroups — too many to draw. Only its 3726 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D9 |
| commutator, cosocle | C2×C9 | C26 |
| Frattini | C2×C3 | C25×D3 |
| Fitting | C23×Q8×C9 | C2 |
| socle | C24×C3 | C22×D3 |
| hypercenter | C23×Q8 | D9 |
| hypocenter | C9 | C23×#41 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C9 | C2 | semidirect |
| 2. C22×#3701 | C2 | direct (N × H) |
| 3. C23×D36 | C2 | semidirect |
| 4. C23×C4×D9 | C2 | semidirect |
| 7. C22×Q8×C9 | C22 | semidirect |
| 10. C2×#3701 | C22 | direct (N × H) |
| 11. C22×D36 | C22 | semidirect |
| 12. C22×C4×D9 | C22 | semidirect |
| 20. C2×Q8×C9 | C23 | semidirect |
| 23. #3701 | C23 | direct (N × H) |
| 24. C2×D36 | C23 | semidirect |
| 25. C2×C4×D9 | C23 | semidirect |
| 34. C23×C9 | #41 | semidirect |
| 38. Q8×C9 | C24 | semidirect |
| 41. D36 | C24 | semidirect |
| 42. C4×D9 | C24 | semidirect |
| 44. C23×Q8 | D9 | semidirect |
| 56. C22×C9 | C2×#41 | semidirect |
| 64. C22×Q8 | C2×D9 | semidirect |
| 78. C2×C9 | C22×#41 | semidirect |
| 85. C2×Q8 | C22×D9 | semidirect |
| 96. C9 | C23×#41 | semidirect |
| 99. C23 | #3701 | direct (N × H) |
| 101. Q8 | C23×D9 | semidirect |
| 111. C22 | C2×#3701 | direct (N × H) |
| 117. C2 | C22×#3701 | 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 |
| C22×#3701 | 2 |
| C23×D36 | 2 |
| C23×C4×D9 | 2 |
| C23×#239 | 3 |