Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
476 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D9 |
| commutator, cosocle | C2×C9 | C24×C3 |
| Frattini | C2×C3 | C23×C3×D3 |
| Fitting | C2×C3×Q8×C9 | C2 |
| socle | C2×C32 | C23×D3 |
| hypercenter | C2×C3×Q8 | D9 |
| hypocenter | C9 | C3×2−1+4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×Q8×C9 | C2 | semidirect |
| 2. C3×#3701 | C2 | semidirect |
| 3. C3×Q8×D9 | C2 | semidirect |
| 4. C3×#3697 | C2 | semidirect |
| 6. #63713 | C3 | direct (N × H) |
| 18. C2×Q8×C9 | C2×C3 | semidirect |
| 19. #3701 | C2×C3 | semidirect |
| 20. Q8×D9 | C2×C3 | semidirect |
| 21. #3697 | C2×C3 | semidirect |
| 49. C2×C3×Q8 | D9 | semidirect |
| 65. C3×C9 | 2−1+4 | semidirect |
| 91. C2×Q8 | C3×D9 | semidirect |
| 104. C9 | C3×2−1+4 | semidirect |
| 124. C3 | #63713 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×Q8×C9 | 2 |
| C3×#3701 | 2 |
| C3×Q8×D9 | 2 |
| C3×#3697 | 2 |
| C3×#1007 | 3 |
| #63713 | 3 |