Finite Group DatabasebetaFinite Group DatabasebetaOrder 648.
110 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊D3)⋊C9)⋊C2 |
| commutator, Fitting, cosocle | C2×C32⋊C9 | C22 |
| Frattini, socle | C2×C32 | D32 |
| hypocenter | C32⋊C9 | D4 |
| LCS | G ▷ C2×C32⋊C9 ▷ C32⋊C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4169 | C2 | semidirect |
| 2. C2×(C3⋊D3)⋊C9 | C2 | semidirect |
| 3. #66328 | C2 | semidirect |
| 12. C2×C3⋊D9 | D3 | semidirect |
| 16. C32⋊C9 | D4 | semidirect |
| 29. C2×C3⋊D3 | D9 | semidirect |
| 45. C3×C9 | #67 | semidirect |
| 79. C32 | #611 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4169 | 2 |
| C2×(C3⋊D3)⋊C9 | 2 |
| #66328 | 2 |
| #6295 | 3 |
| #6195 | 3 |