Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
709 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 709 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×C22⋊A4 |
| commutator, hypocenter | C22×C3×Q8 | C2×C9 |
| Fitting, cosocle | C22×C32×Q8 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| prime radical | C22×C3×#239 | C3 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C22⋊(Q8⋊C9) | C2 | semidirect |
| 12. C22⋊(Q8⋊C9) | D3 | semidirect |
| 16. C22×#239 | C9 | semidirect |
| 82. #239 | C22⋊C9 | semidirect |
| 161. C22×C3 | #41⋊C9 | semidirect |
| 205. C22 | C3⋊(#41⋊C9) | semidirect |
| 210. C3 | #10593123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C22⋊(Q8⋊C9) | 2 |
| C22×C3×#239 | 3 |
| #10593123 | 3 |
| Dic3×C22⋊C9 | 4 |
| C3⋊(#41⋊C9) | 4 |