Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
644 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 210 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×#904 | C2 | semidirect |
| 2. C2×C32×D24 | C2 | semidirect |
| 3. C33×M32 | C2 | semidirect |
| 4. C32×#800 | C2 | semidirect |
| 5. C32×D48 | C2 | semidirect |
| 7. C3×#4908 | C3 | direct (N × H) |
| 11. C32×Dic12 | C22 | semidirect |
| 14. C32×D24 | C22 | semidirect |
| 16. C33×C16 | C22 | semidirect |
| 19. C3×#904 | C2×C3 | semidirect |
| 20. C2×C3×D24 | C2×C3 | semidirect |
| 24. C32×M32 | D3 | semidirect |
| 25. C32×M32 | C2×C3 | semidirect |
| 26. C3×#800 | C2×C3 | semidirect |
| 27. C3×D48 | C2×C3 | semidirect |
| 39. #4908 | C32 | direct (N × H) |
| 52. C3×Dic12 | C22×C3 | semidirect |
| 55. C3×D24 | C22×C3 | semidirect |
| 58. C32×C16 | C2×D3 | semidirect |
| 59. C32×C16 | C22×C3 | semidirect |
| 69. #904 | C2×C32 | semidirect |
| 70. C2×D24 | C2×C32 | semidirect |
| 74. C3×M32 | C3×D3 | semidirect |
| 75. C3×M32 | C2×C32 | semidirect |
| 76. #800 | C2×C32 | semidirect |
| 77. D48 | C2×C32 | semidirect |
| 118. Dic12 | C22×C32 | semidirect |
| 121. D24 | C22×C32 | semidirect |
| 124. C3×C16 | C2×C3×D3 | semidirect |
| 125. C3×C16 | C22×C32 | semidirect |
| 147. M32 | C32×D3 | semidirect |
| 148. C33 | #509 | semidirect |
| 190. C16 | C2×C32×D3 | semidirect |
| 209. C32 | #4908 | direct (N × H) |
| 210. C32 | C3×#509 | semidirect |
| 239. C3 | C3×#4908 | direct (N × H) |
| 240. C3 | C32×#509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#904 | 2 |
| C2×C32×D24 | 2 |
| C33×M32 | 2 |
| C32×#800 | 2 |
| C32×D48 | 2 |
| C32×#509 | 3 |
| C3×#4908 | 3 |