Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
644 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 186 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×#919 | C2 | semidirect |
| 2. C32×D3×D8 | C2 | semidirect |
| 3. C33×SD32 | C2 | semidirect |
| 4. C32×#828 | C2 | semidirect |
| 5. C32×#826 | C2 | semidirect |
| 6. C32×D48 | C2 | semidirect |
| 7. C32×#798 | C2 | semidirect |
| 9. C3×#4914 | C3 | direct (N × H) |
| 12. C33×Q16 | C22 | semidirect |
| 17. C32×D24 | C22 | semidirect |
| 19. C33×C16 | C22 | semidirect |
| 20. C32×C3⋊C16 | C22 | semidirect |
| 23. C3×#919 | C2×C3 | semidirect |
| 24. C3×D3×D8 | C2×C3 | semidirect |
| 27. C32×SD32 | D3 | semidirect |
| 28. C32×SD32 | C2×C3 | semidirect |
| 32. C3×#828 | C2×C3 | semidirect |
| 33. C3×#826 | C2×C3 | semidirect |
| 34. C3×D48 | C2×C3 | semidirect |
| 35. C3×#798 | C2×C3 | semidirect |
| 48. #4914 | C32 | direct (N × H) |
| 54. C32×Q16 | C2×D3 | semidirect |
| 55. C32×Q16 | C22×C3 | semidirect |
| 67. C3×D24 | C22×C3 | semidirect |
| 71. C32×C16 | C2×D3 | semidirect |
| 72. C32×C16 | C22×C3 | semidirect |
| 73. C3×C3⋊C16 | C22×C3 | semidirect |
| 84. #919 | C2×C32 | semidirect |
| 85. D3×D8 | C2×C32 | semidirect |
| 88. C3×SD32 | C3×D3 | semidirect |
| 89. C3×SD32 | C2×C32 | semidirect |
| 93. #828 | C2×C32 | semidirect |
| 94. #826 | C2×C32 | semidirect |
| 95. D48 | C2×C32 | semidirect |
| 96. #798 | C2×C32 | semidirect |
| 133. C3×Q16 | C2×C3×D3 | semidirect |
| 134. C3×Q16 | C22×C32 | semidirect |
| 146. D24 | C22×C32 | semidirect |
| 150. C3×C16 | C2×C3×D3 | semidirect |
| 151. C3×C16 | C22×C32 | semidirect |
| 152. C3⋊C16 | C22×C32 | semidirect |
| 175. SD32 | C32×D3 | semidirect |
| 179. C33 | #509 | semidirect |
| 219. Q16 | C2×C32×D3 | semidirect |
| 227. C16 | C2×C32×D3 | semidirect |
| 248. C32 | #4914 | direct (N × H) |
| 249. C32 | C3×#509 | semidirect |
| 282. C3 | C3×#4914 | direct (N × H) |
| 283. C3 | C32×#509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#919 | 2 |
| C32×D3×D8 | 2 |
| C33×SD32 | 2 |
| C32×#828 | 2 |
| C32×#826 | 2 |
| C32×D48 | 2 |
| C32×#798 | 2 |
| C32×#509 | 3 |
| C3×#4914 | 3 |