Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1218 conjugacy classes of subgroups; only the 456 normal subgroups are shown.
This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 456 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, hypocenter | C32 | C23×C3×C4 |
| Frattini | C2 | C22×C3×D32 |
| Fitting, socle | C23×C33 | C22 |
| prime radical | C23×C32×D3 | C2 |
| cosocle | C2×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C3⋊Dic3 | C2 | semidirect |
| 3. C22×C32×Dic3 | C2 | semidirect |
| 4. C2×C3×D3×Dic3 | C2 | direct (N × H) |
| 7. C22×D3×Dic3 | C3 | direct (N × H) |
| 9. C22×C32×D3 | C4 | semidirect |
| 11. C2×C3×C3⋊Dic3 | C22 | semidirect |
| 12. C2×C32×Dic3 | C22 | semidirect |
| 13. C3×D3×Dic3 | C22 | direct (N × H) |
| 19. C22×C3⋊Dic3 | C2×C3 | semidirect |
| 25. C22×C3×Dic3 | D3 | direct (N × H) |
| 26. C22×C3×Dic3 | C2×C3 | semidirect |
| 28. C2×D3×Dic3 | C2×C3 | direct (N × H) |
| 29. C22×C33 | C2×C4 | semidirect |
| 31. C2×C32×D3 | C2×C4 | semidirect |
| 33. C3×C3⋊Dic3 | C23 | semidirect |
| 34. C32×Dic3 | C23 | semidirect |
| 49. C22×C3×D3 | Dic3 | direct (N × H) |
| 50. C22×C3×D3 | C3×C4 | semidirect |
| 54. C2×C3⋊Dic3 | C22×C3 | semidirect |
| 60. C2×C3×Dic3 | C2×D3 | direct (N × H) |
| 61. C2×C3×Dic3 | C22×C3 | semidirect |
| 63. D3×Dic3 | C22×C3 | direct (N × H) |
| 64. C2×C33 | C22×C4 | semidirect |
| 66. C32×D3 | C22×C4 | semidirect |
| 80. C22×Dic3 | C3×D3 | direct (N × H) |
| 92. C22×C32 | C4×D3 | semidirect |
| 93. C22×C32 | C2×Dic3 | semidirect |
| 94. C22×C32 | C2×C3×C4 | semidirect |
| 100. C2×C3×D3 | C2×Dic3 | direct (N × H) |
| 101. C2×C3×D3 | C2×C3×C4 | semidirect |
| 105. C3⋊Dic3 | C23×C3 | semidirect |
| 110. C3×Dic3 | C22×D3 | direct (N × H) |
| 111. C3×Dic3 | C23×C3 | semidirect |
| 113. C33 | C23×C4 | semidirect |
| 127. C22×D3 | C3×Dic3 | direct (N × H) |
| 138. C2×Dic3 | C2×C3×D3 | direct (N × H) |
| 149. C2×C32 | C2×C4×D3 | semidirect |
| 150. C2×C32 | C22×Dic3 | semidirect |
| 151. C2×C32 | C22×C3×C4 | semidirect |
| 156. C3×D3 | C22×Dic3 | direct (N × H) |
| 157. C3×D3 | C22×C3×C4 | semidirect |
| 176. C22×C3 | D3×Dic3 | direct (N × H) |
| 177. C22×C3 | C3×C4×D3 | semidirect |
| 178. C22×C3 | C2×C3×Dic3 | semidirect |
| 182. C2×D3 | C2×C3×Dic3 | direct (N × H) |
| 191. Dic3 | C22×C3×D3 | direct (N × H) |
| 196. C32 | C22×C4×D3 | semidirect |
| 197. C32 | C23×Dic3 | semidirect |
| 198. C32 | C23×C3×C4 | semidirect |
| 216. C2×C3 | C2×D3×Dic3 | direct (N × H) |
| 217. C2×C3 | C2×C3×C4×D3 | semidirect |
| 218. C2×C3 | C22×C3×Dic3 | semidirect |
| 222. D3 | C22×C3×Dic3 | direct (N × H) |
| 225. C22 | C3×D3×Dic3 | direct (N × H) |
| 233. C3 | C22×D3×Dic3 | direct (N × H) |
| 234. C3 | C22×C3×C4×D3 | semidirect |
| 235. C3 | C23×C3×Dic3 | semidirect |
| 237. C2 | C2×C3×D3×Dic3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32×D3 | 2 |
| C22×C3×C3⋊Dic3 | 2 |
| C22×C32×Dic3 | 2 |
| C2×C3×D3×Dic3 | 2 |
| C23×C3×Dic3 | 3 |
| C22×C3×C4×D3 | 3 |
| C22×D3×Dic3 | 3 |