Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1298 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D32 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C22×C3×D32 |
| Fitting, socle | C23×C33 | C22 |
| hypocenter | C32 | C22×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×C3⋊D3 | C2 | semidirect |
| 2. C23×C32×D3 | C2 | semidirect |
| 3. C22×C32×Dic3 | C2 | semidirect |
| 4. C2×C3×#626 | C2 | direct (N × H) |
| 7. C22×#626 | C3 | direct (N × H) |
| 10. C22×C3×C3⋊D3 | C22 | semidirect |
| 12. C22×C32×D3 | C22 | semidirect |
| 13. C2×C32×Dic3 | C22 | semidirect |
| 14. C3×#626 | C22 | direct (N × H) |
| 16. C23×C3⋊D3 | C2×C3 | semidirect |
| 21. C23×C3×D3 | D3 | semidirect |
| 22. C23×C3×D3 | C2×C3 | semidirect |
| 26. C22×C3×Dic3 | D3 | semidirect |
| 27. C22×C3×Dic3 | C2×C3 | semidirect |
| 29. C2×#626 | C2×C3 | direct (N × H) |
| 30. C22×C33 | D4 | semidirect |
| 33. C2×C3×C3⋊D3 | C23 | semidirect |
| 35. C2×C32×D3 | C23 | semidirect |
| 36. C32×Dic3 | C23 | semidirect |
| 50. C22×C3⋊D3 | C22×C3 | semidirect |
| 61. C22×C3×D3 | C2×D3 | semidirect |
| 62. C22×C3×D3 | C22×C3 | semidirect |
| 66. C2×C3×Dic3 | C2×D3 | semidirect |
| 67. C2×C3×Dic3 | C22×C3 | semidirect |
| 69. #626 | C22×C3 | direct (N × H) |
| 70. C2×C33 | C2×D4 | semidirect |
| 81. C23×D3 | C3×D3 | semidirect |
| 87. C22×Dic3 | C3×D3 | semidirect |
| 99. C22×C32 | D12 | semidirect |
| 100. C22×C32 | #67 | semidirect |
| 101. C22×C32 | C3×D4 | semidirect |
| 106. C2×C3⋊D3 | C23×C3 | semidirect |
| 117. C2×C3×D3 | C22×D3 | semidirect |
| 118. C2×C3×D3 | C23×C3 | semidirect |
| 121. C3×Dic3 | C22×D3 | semidirect |
| 122. C3×Dic3 | C23×C3 | semidirect |
| 124. C33 | C22×D4 | semidirect |
| 147. C22×D3 | C2×C3×D3 | semidirect |
| 153. C2×Dic3 | C2×C3×D3 | semidirect |
| 164. C2×C32 | C2×D12 | semidirect |
| 165. C2×C32 | C2×#67 | semidirect |
| 166. C2×C32 | C2×C3×D4 | semidirect |
| 197. C22×C3 | #626 | direct (N × H) |
| 198. C22×C3 | C3×D12 | semidirect |
| 199. C22×C3 | C3×#67 | semidirect |
| 210. C2×D3 | C22×C3×D3 | semidirect |
| 214. Dic3 | C22×C3×D3 | semidirect |
| 219. C32 | C22×D12 | semidirect |
| 220. C32 | C22×#67 | semidirect |
| 221. C32 | C22×C3×D4 | semidirect |
| 241. C2×C3 | C2×#626 | direct (N × H) |
| 242. C2×C3 | C2×C3×D12 | semidirect |
| 243. C2×C3 | C2×C3×#67 | semidirect |
| 255. C22 | C3×#626 | direct (N × H) |
| 262. C3 | C22×#626 | direct (N × H) |
| 263. C3 | C22×C3×D12 | semidirect |
| 264. C3 | C22×C3×#67 | semidirect |
| 267. C2 | C2×C3×#626 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊D3 | 2 |
| C23×C32×D3 | 2 |
| C22×C32×Dic3 | 2 |
| C2×C3×#626 | 2 |
| C22×C3×#67 | 3 |
| C22×C3×D12 | 3 |
| C22×#626 | 3 |