Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2766 conjugacy classes of subgroups; only the 646 normal subgroups are shown.
This group has 2766 conjugacy classes of subgroups — too many to draw. Only its 646 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C32×A4 | C2 | semidirect |
| 2. C32×D4×A4 | C2 | direct (N × H) |
| 3. C2×C32×C4×A4 | C2 | semidirect |
| 4. C23×C32×D4 | C3 | semidirect |
| 5. C2×C3×D4×A4 | C3 | direct (N × H) |
| 8. C22×C32×A4 | C22 | semidirect |
| 10. C32×C4×A4 | C22 | semidirect |
| 11. C25×C32 | C2×C3 | semidirect |
| 12. C23×C3×A4 | C2×C3 | semidirect |
| 17. C22×C32×D4 | C2×C3 | semidirect |
| 18. C23×C32×C4 | C2×C3 | semidirect |
| 19. C3×D4×A4 | C2×C3 | direct (N × H) |
| 20. C2×C3×C4×A4 | C2×C3 | semidirect |
| 23. C2×C32×A4 | D4 | direct (N × H) |
| 27. C23×C3×D4 | C32 | semidirect |
| 28. C2×D4×A4 | C32 | direct (N × H) |
| 35. C24×C32 | C22×C3 | semidirect |
| 38. C22×C3×A4 | C22×C3 | semidirect |
| 49. C2×C32×D4 | A4 | direct (N × H) |
| 53. C22×C32×C4 | C22×C3 | semidirect |
| 54. C3×C4×A4 | C22×C3 | semidirect |
| 58. C32×A4 | C2×D4 | direct (N × H) |
| 60. C25×C3 | C2×C32 | semidirect |
| 61. C23×A4 | C2×C32 | semidirect |
| 66. C22×C3×D4 | C2×C32 | semidirect |
| 67. C23×C3×C4 | C2×C32 | semidirect |
| 68. D4×A4 | C2×C32 | direct (N × H) |
| 69. C2×C4×A4 | C2×C32 | semidirect |
| 87. C23×C32 | C3×D4 | semidirect |
| 88. C23×C32 | C2×A4 | semidirect |
| 91. C2×C3×A4 | C3×D4 | direct (N × H) |
| 97. C32×D4 | C2×A4 | direct (N × H) |
| 104. C2×C32×C4 | C2×A4 | semidirect |
| 105. C23×D4 | C33 | semidirect |
| 115. C24×C3 | C22×C32 | semidirect |
| 118. C22×A4 | C22×C32 | semidirect |
| 129. C2×C3×D4 | C3×A4 | direct (N × H) |
| 133. C22×C3×C4 | C22×C32 | semidirect |
| 134. C4×A4 | C22×C32 | semidirect |
| 154. C22×C32 | C2×C3×D4 | semidirect |
| 156. C22×C32 | C22×A4 | semidirect |
| 157. C3×A4 | C2×C3×D4 | direct (N × H) |
| 160. C32×C4 | C22×A4 | semidirect |
| 161. C25 | C2×C33 | semidirect |
| 166. C22×D4 | C2×C33 | semidirect |
| 167. C23×C4 | C2×C33 | semidirect |
| 186. C23×C3 | C32×D4 | semidirect |
| 187. C23×C3 | C2×C3×A4 | semidirect |
| 190. C2×A4 | C32×D4 | direct (N × H) |
| 196. C3×D4 | C2×C3×A4 | direct (N × H) |
| 203. C2×C3×C4 | C2×C3×A4 | semidirect |
| 212. C2×C32 | D4×A4 | direct (N × H) |
| 220. C24 | C22×C33 | semidirect |
| 230. C2×D4 | C32×A4 | direct (N × H) |
| 234. C22×C4 | C22×C33 | semidirect |
| 254. C22×C3 | C2×C32×D4 | semidirect |
| 256. C22×C3 | C22×C3×A4 | semidirect |
| 257. A4 | C2×C32×D4 | direct (N × H) |
| 260. C3×C4 | C22×C3×A4 | semidirect |
| 262. C32 | C2×D4×A4 | direct (N × H) |
| 279. C23 | C33×D4 | semidirect |
| 280. C23 | C2×C32×A4 | semidirect |
| 285. D4 | C2×C32×A4 | direct (N × H) |
| 291. C2×C4 | C2×C32×A4 | semidirect |
| 300. C2×C3 | C3×D4×A4 | direct (N × H) |
| 318. C22 | C2×C33×D4 | semidirect |
| 320. C22 | C22×C32×A4 | semidirect |
| 322. C4 | C22×C32×A4 | semidirect |
| 324. C3 | C2×C3×D4×A4 | direct (N × H) |
| 330. C2 | C32×D4×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32×A4 | 2 |
| C32×D4×A4 | 2 |
| C2×C32×C4×A4 | 2 |
| C23×C32×D4 | 3 |
| C2×C3×D4×A4 | 3 |
| C2×C33×D4 | 4 |