Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1610 conjugacy classes of subgroups; only the 277 normal subgroups are shown.
This group has 1610 conjugacy classes of subgroups — too many to draw. Only its 277 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×C3⋊D9 |
| commutator, cosocle | C23×C3×C9 | C23 |
| Frattini | C23×C3 | C22×C3⋊D3 |
| Fitting | C3×C9×#95 | C2 |
| socle | C23×C32 | C22×D3 |
| hypercenter | #95 | C3⋊D9 |
| hypocenter | C3×C9 | #392 |
| LCS | G ▷ C23×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊(C3⋊D9) | C2 | semidirect |
| 2. C2×#88536 | C2 | semidirect |
| 3. C3×C9×#95 | C2 | semidirect |
| 17. C9×#95 | D3 | semidirect |
| 23. C22×C3⋊D9 | D4 | semidirect |
| 25. C2×C3×C4×C9 | D4 | semidirect |
| 56. C3×#95 | D9 | semidirect |
| 75. C2×C4×C9 | D12 | semidirect |
| 117. #95 | C3⋊D9 | semidirect |
| 118. C3×C9 | #392 | semidirect |
| 133. C2×C3×C4 | D36 | semidirect |
| 174. C9 | #4671 | semidirect |
| 184. C2×C4 | C4⋊(C3⋊D9) | semidirect |
| 203. C3 | #10588351 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊(C3⋊D9) | 2 |
| C2×#88536 | 2 |
| C3×C9×#95 | 2 |
| #10592370 | 3 |
| #10588351 | 3 |