Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
700 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 700 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. C2×C32×D24 | C2 | semidirect |
| 2. C2×C33×C16 | C2 | semidirect |
| 3. C32×D48 | C2 | direct (N × H) |
| 5. C2×C3×D48 | C3 | direct (N × H) |
| 9. C32×D24 | C22 | semidirect |
| 10. C33×C16 | C22 | semidirect |
| 12. C2×C3×D24 | C2×C3 | semidirect |
| 15. C2×C32×C16 | D3 | semidirect |
| 16. C2×C32×C16 | C2×C3 | semidirect |
| 17. C3×D48 | C2×C3 | direct (N × H) |
| 25. C2×D48 | C32 | direct (N × H) |
| 34. C3×D24 | C22×C3 | semidirect |
| 36. C32×C16 | C2×D3 | semidirect |
| 37. C32×C16 | C22×C3 | semidirect |
| 44. C2×D24 | C2×C32 | semidirect |
| 47. C2×C3×C16 | C3×D3 | semidirect |
| 48. C2×C3×C16 | C2×C32 | semidirect |
| 49. D48 | C2×C32 | direct (N × H) |
| 66. C2×C33 | D16 | semidirect |
| 77. D24 | C22×C32 | semidirect |
| 79. C3×C16 | C2×C3×D3 | semidirect |
| 80. C3×C16 | C22×C32 | semidirect |
| 96. C2×C16 | C32×D3 | semidirect |
| 97. C33 | C2×D16 | semidirect |
| 116. C2×C32 | D48 | direct (N × H) |
| 117. C2×C32 | C3×D16 | semidirect |
| 125. C16 | C2×C32×D3 | semidirect |
| 140. C32 | C2×D48 | direct (N × H) |
| 141. C32 | C2×C3×D16 | semidirect |
| 151. C2×C3 | C3×D48 | direct (N × H) |
| 152. C2×C3 | C32×D16 | semidirect |
| 162. C3 | C2×C3×D48 | direct (N × H) |
| 163. C3 | C2×C32×D16 | semidirect |
| 165. C2 | C32×D48 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×D24 | 2 |
| C2×C33×C16 | 2 |
| C32×D48 | 2 |
| C2×C32×D16 | 3 |
| C2×C3×D48 | 3 |