Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
417 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #393 |
| commutator | C24 | C23×C9 |
| Frattini, cosocle | C24×C3 | C23×C3 |
| socle | C2×C3 | C3×#393 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C24×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#543 | C2 | semidirect |
| 3. C9×#342 | C2 | semidirect |
| 13. C24×C9 | Q8 | semidirect |
| 20. C22×C4×C9 | Q8 | semidirect |
| 26. #2032 | C9 | direct (N × H) |
| 46. #543 | C2×C9 | semidirect |
| 48. #342 | C2×C9 | semidirect |
| 97. C24 | Q8×C9 | semidirect |
| 104. C22×C4 | Q8×C9 | semidirect |
| 121. C9 | #2032 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#543 | 2 |
| C2×C9×#99 | 2 |
| C9×#342 | 2 |
| C3×#2032 | 3 |