Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
730 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×C3⋊D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C23×C3⋊D3 |
| Fitting | C3×C9×#41 | C2 |
| socle | C2×C32 | C23×D3 |
| hypercenter | #41 | C3⋊D9 |
| hypocenter | C3×C9 | 2−1+4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C9×#41 | C2 | semidirect |
| 2. Q8×C3⋊D9 | C2 | semidirect |
| 3. #88742 | C2 | semidirect |
| 4. #88736 | C2 | semidirect |
| 5. C2×#6226 | C2 | semidirect |
| 22. C9×#41 | D3 | semidirect |
| 55. C3×#41 | D9 | semidirect |
| 76. C3×C9 | 2−1+4 | semidirect |
| 103. #41 | C3⋊D9 | semidirect |
| 117. C9 | #1010 | semidirect |
| 138. C3 | #63716 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×#41 | 2 |
| Q8×C3⋊D9 | 2 |
| #88742 | 2 |
| #88736 | 2 |
| C2×#6226 | 2 |
| #64367 | 3 |
| #63716 | 3 |