Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
532 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 135 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 | C2×#41 |
| 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. #88746 | C2 | semidirect |
| 2. C9×#239 | C2 | semidirect |
| 3. C3×Q8×D9 | C2 | semidirect |
| 4. D9×D12 | C2 | semidirect |
| 5. C4×D3×D9 | C2 | semidirect |
| 6. #88642 | C2 | semidirect |
| 7. #88639 | C2 | semidirect |
| 11. C3×Q8×C9 | C22 | semidirect |
| 12. C4⋊(C3⋊D9) | C22 | semidirect |
| 13. C4×C3⋊D9 | C22 | semidirect |
| 14. C9×D12 | C22 | semidirect |
| 15. C4×D3×C9 | C22 | semidirect |
| 17. C3×Dic18 | C22 | semidirect |
| 18. #6195 | C22 | semidirect |
| 19. D3×Dic9 | C22 | semidirect |
| 20. (C3⋊D9)⋊C4 | C22 | semidirect |
| 32. Q8×D9 | D3 | semidirect |
| 61. Q8×C9 | C2×D3 | semidirect |
| 73. Dic18 | C2×D3 | semidirect |
| 77. C3×D9 | #41 | semidirect |
| 85. #239 | D9 | direct (N × H) |
| 113. C3×C9 | C2×#41 | semidirect |
| 118. C3×Q8 | C2×D9 | semidirect |
| 130. D12 | C2×D9 | semidirect |
| 139. C4×D3 | C2×D9 | semidirect |
| 150. D9 | #239 | direct (N × H) |
| 179. C9 | C2×#239 | semidirect |
| 182. Q8 | D3×D9 | semidirect |
| 211. C3 | #41×D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #88746 | 2 |
| C9×#239 | 2 |
| C3×Q8×D9 | 2 |
| D9×D12 | 2 |
| C4×D3×D9 | 2 |
| #88642 | 2 |
| #88639 | 2 |
| D3×#239 | 3 |
| #41×D9 | 3 |