Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
772 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 126 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 |
|---|---|---|
| 2. C33×#99 | C2 | semidirect |
| 3. C32×#833 | C2 | semidirect |
| 5. C3×#4471 | C3 | direct (N × H) |
| 6. C23×C32×D3 | C4 | semidirect |
| 10. C32×#217 | C4 | semidirect |
| 17. C32×#99 | D3 | semidirect |
| 18. C32×#99 | C2×C3 | semidirect |
| 19. C3×#833 | C2×C3 | semidirect |
| 32. #4471 | C32 | direct (N × H) |
| 34. C23×C3×D3 | C3×C4 | semidirect |
| 46. C3×#217 | C3×C4 | semidirect |
| 66. C3×#99 | C3×D3 | semidirect |
| 67. C3×#99 | C2×C32 | semidirect |
| 68. #833 | C2×C32 | semidirect |
| 103. C23×D3 | C32×C4 | semidirect |
| 115. #217 | C32×C4 | semidirect |
| 153. #99 | C32×D3 | semidirect |
| 154. C33 | AΣL(1,𝔽16) | semidirect |
| 237. C32 | #4471 | direct (N × H) |
| 238. C32 | C3×AΣL(1,𝔽16) | semidirect |
| 278. C3 | C3×#4471 | direct (N × H) |
| 279. C3 | C32×AΣL(1,𝔽16) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#937 | 2 |
| C33×#99 | 2 |
| C32×#833 | 2 |
| C32×AΣL(1,𝔽16) | 3 |
| C3×#4471 | 3 |