Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
120 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #4626 |
| commutator, hypocenter | C42⋊C3 | C4×C9 |
| Frattini | C23×C3 | C3×S4 |
| Fitting | C9×#127 | D3 |
| socle | C22×C3 | C3×SL(2,ℤ/4) |
| prime radical, cosocle | C3×#127⋊C3 | C2×C3 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C9×C42⋊C3 | C4 | semidirect |
| 12. #4626 | C9 | direct (N × H) |
| 20. C42×C9 | Dic3 | semidirect |
| 47. C42⋊C3 | C4×C9 | semidirect |
| 80. C42 | C9×Dic3 | semidirect |
| 93. C9 | #4626 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#127⋊C3 | 2 |
| C3×#4626 | 3 |
| C9×#353 | 3 |
| C9×SL(2,ℤ/4) | 4 |