Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1668 conjugacy classes of subgroups; only the 666 normal subgroups are shown.
This group has 1668 conjugacy classes of subgroups — too many to draw. Only its 666 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D9 |
| commutator, cosocle | C22×C9 | C25 |
| Frattini, socle | C22×C3 | C24×D3 |
| Fitting | Q82×C9 | C2 |
| prime radical | C22×D9 | C24 |
| hypercenter | Q82 | D9 |
| hypocenter | C9 | C2×Q82 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q82×C9 | C2 | semidirect |
| 3. #10589824 | C2 | semidirect |
| 5. Q8×Dic18 | C2 | semidirect |
| 27. Q8×D9 | Q8 | direct (N × H) |
| 53. Q8×C9 | C2×Q8 | semidirect |
| 61. Dic18 | C2×Q8 | semidirect |
| 64. Q82 | D9 | direct (N × H) |
| 113. D9 | Q82 | direct (N × H) |
| 135. C9 | C2×Q82 | semidirect |
| 140. Q8 | Q8×D9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C9 | 2 |
| D9×C4⋊Q8 | 2 |
| #10589824 | 2 |
| C4×Q8×D9 | 2 |
| Q8×Dic18 | 2 |
| D3×Q82 | 3 |