Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
780 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C9 | C23×C32 |
| commutator, socle | C22×C3 | C23×C3×C9 |
| prime radical | C32×#120 | C9 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C27⋊C3 | C2 | semidirect |
| 2. C2×D4×C27⋊C3 | C2 | semidirect |
| 3. C22⋊C4×C27⋊C3 | C2 | semidirect |
| 5. C27×#120 | C3 | semidirect |
| 8. C23×C27⋊C3 | C22 | semidirect |
| 10. C2×C4×C27⋊C3 | C22 | semidirect |
| 13. C24×C27 | C2×C3 | semidirect |
| 15. C2×D4×C27 | C2×C3 | semidirect |
| 16. C22⋊C4×C27 | C2×C3 | semidirect |
| 20. C22×C27⋊C3 | D4 | semidirect |
| 33. C23×C27 | C22×C3 | semidirect |
| 35. C2×C4×C27 | C22×C3 | semidirect |
| 57. C22×C27 | C3×D4 | semidirect |
| 62. C27⋊C3 | #120 | direct (N × H) |
| 108. #120 | C27⋊C3 | direct (N × H) |
| 110. C27 | C3×#120 | semidirect |
| 130. C24 | C2×C27⋊C3 | semidirect |
| 131. C2×D4 | C2×C27⋊C3 | semidirect |
| 132. C22⋊C4 | C2×C27⋊C3 | semidirect |
| 150. C23 | C22×C27⋊C3 | semidirect |
| 152. C2×C4 | C22×C27⋊C3 | semidirect |
| 162. C22 | D4×C27⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C27⋊C3 | 2 |
| C2×D4×C27⋊C3 | 2 |
| C22⋊C4×C27⋊C3 | 2 |
| C3×C9×#120 | 3 |
| C27×#120 | 3 |