Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1396 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 234 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. C32×#1004 | C2 | semidirect |
| 2. C32×#937 | C2 | semidirect |
| 3. C32×#882 | C2 | semidirect |
| 4. C33×#99 | C2 | semidirect |
| 5. C32×#806 | C2 | semidirect |
| 7. C3×#4741 | C3 | direct (N × H) |
| 8. C23×C32×D3 | C22 | semidirect |
| 17. C33×C22⋊C4 | C22 | semidirect |
| 18. C32×#217 | C22 | semidirect |
| 22. C3×#1004 | C2×C3 | semidirect |
| 25. C3×#937 | C2×C3 | semidirect |
| 26. C3×#882 | C2×C3 | semidirect |
| 29. C32×#99 | D3 | semidirect |
| 30. C32×#99 | C2×C3 | semidirect |
| 31. C3×#806 | C2×C3 | semidirect |
| 32. C23×C33 | D4 | semidirect |
| 38. C22×C32×D3 | D4 | semidirect |
| 41. C2×C33×C4 | D4 | semidirect |
| 48. C2×C32×Dic3 | D4 | semidirect |
| 54. #4741 | C32 | direct (N × H) |
| 56. C23×C3×D3 | C22×C3 | semidirect |
| 78. C32×C22⋊C4 | C2×D3 | semidirect |
| 79. C32×C22⋊C4 | C22×C3 | semidirect |
| 80. C3×#217 | C22×C3 | semidirect |
| 101. #1004 | C2×C32 | semidirect |
| 104. #937 | C2×C32 | semidirect |
| 105. #882 | C2×C32 | semidirect |
| 108. C3×#99 | C3×D3 | semidirect |
| 109. C3×#99 | C2×C32 | semidirect |
| 110. #806 | C2×C32 | semidirect |
| 118. C23×C32 | D12 | semidirect |
| 119. C23×C32 | C3×D4 | semidirect |
| 126. C22×C3×D3 | C3×D4 | semidirect |
| 145. C2×C32×C4 | D12 | semidirect |
| 146. C2×C32×C4 | C3×D4 | semidirect |
| 153. C2×C3×Dic3 | C3×D4 | semidirect |
| 166. C23×D3 | C22×C32 | semidirect |
| 188. C3×C22⋊C4 | C2×C3×D3 | semidirect |
| 189. C3×C22⋊C4 | C22×C32 | semidirect |
| 190. #217 | C22×C32 | semidirect |
| 238. #99 | C32×D3 | semidirect |
| 239. C33 | B(4,𝔽2) | semidirect |
| 247. C23×C3 | C3×D12 | semidirect |
| 248. C23×C3 | C32×D4 | semidirect |
| 255. C22×D3 | C32×D4 | semidirect |
| 274. C2×C3×C4 | C3×D12 | semidirect |
| 275. C2×C3×C4 | C32×D4 | semidirect |
| 282. C2×Dic3 | C32×D4 | semidirect |
| 315. C22⋊C4 | C2×C32×D3 | semidirect |
| 358. C32 | #4741 | direct (N × H) |
| 359. C32 | C3×B(4,𝔽2) | semidirect |
| 365. C23 | C32×D12 | semidirect |
| 380. C2×C4 | C32×D12 | semidirect |
| 415. C3 | C3×#4741 | direct (N × H) |
| 416. C3 | C32×B(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#1004 | 2 |
| C32×#937 | 2 |
| C32×#882 | 2 |
| C33×#99 | 2 |
| C32×#806 | 2 |
| C32×B(4,𝔽2) | 3 |
| C3×#4741 | 3 |