Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
476 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C3×Q8×C9 | C22 |
| socle | C2×C32 | C23×D3 |
| hypercenter | Q8 | D3×D9 |
| hypocenter | C3×C9 | 2−1+4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C3⋊D9 | C2 | semidirect |
| 2. D3×Q8×C9 | C2 | semidirect |
| 3. C3×#3701 | C2 | semidirect |
| 4. #88641 | C2 | semidirect |
| 5. #88640 | C2 | semidirect |
| 6. #88634 | C2 | semidirect |
| 7. D9×Dic6 | C2 | semidirect |
| 31. #3701 | D3 | semidirect |
| 85. D3×Q8 | D9 | semidirect |
| 110. C3×C9 | 2−1+4 | semidirect |
| 174. C9 | #1010 | semidirect |
| 205. C3 | #63713 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C3⋊D9 | 2 |
| D3×Q8×C9 | 2 |
| C3×#3701 | 2 |
| #88641 | 2 |
| #88640 | 2 |
| #88634 | 2 |
| D9×Dic6 | 2 |
| #64315 | 3 |
| #63713 | 3 |