Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
546 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle | C22×C32 | C22×D3 |
| hypercenter | C23 | D3×D9 |
| hypocenter | C3×C9 | C2×#41 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6233 | C2 | semidirect |
| 2. C2×C3×#611 | C2 | semidirect |
| 3. C22×C9×Dic3 | C2 | semidirect |
| 4. C2×#6192 | C2 | semidirect |
| 5. C2×(C3⋊D9)⋊C4 | C2 | semidirect |
| 6. #88658 | C2 | direct (N × H) |
| 7. C2×Dic3×D9 | C2 | semidirect |
| 8. C2×#6189 | C2 | semidirect |
| 14. #6233 | C22 | semidirect |
| 16. C3×#611 | C22 | semidirect |
| 19. C2×C9×Dic3 | C22 | semidirect |
| 21. #6192 | C22 | semidirect |
| 22. (C3⋊D9)⋊C4 | C22 | semidirect |
| 23. Dic3×D9 | C22 | semidirect |
| 24. #6189 | C22 | semidirect |
| 36. C2×#611 | D3 | semidirect |
| 79. #611 | C2×D3 | semidirect |
| 88. C2×C3×C9 | #41 | semidirect |
| 100. C22×Dic3 | D9 | semidirect |
| 138. C3×C9 | C2×#41 | semidirect |
| 164. C2×Dic3 | C2×D9 | semidirect |
| 180. C2×C9 | #237 | semidirect |
| 223. C9 | C2×#237 | semidirect |
| 244. C2×C3 | #3697 | semidirect |
| 264. C3 | C2×#3697 | semidirect |
| 269. C2 | #88658 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6233 | 2 |
| C2×C3×#611 | 2 |
| C22×C9×Dic3 | 2 |
| C2×#6192 | 2 |
| C2×(C3⋊D9)⋊C4 | 2 |
| #88658 | 2 |
| C2×Dic3×D9 | 2 |
| C2×#6189 | 2 |
| C2×#3804 | 3 |
| C2×#3697 | 3 |