Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4322 conjugacy classes of subgroups; only the 1322 normal subgroups are shown.
This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 1322 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. C24×C3×D9 | C2 | semidirect |
| 2. C23×C3×C4×C9 | C2 | semidirect |
| 3. C22×C3×D36 | C2 | direct (N × H) |
| 5. C23×D36 | C3 | direct (N × H) |
| 8. C23×C3×D9 | C22 | semidirect |
| 9. C22×C3×C4×C9 | C22 | semidirect |
| 10. C2×C3×D36 | C22 | direct (N × H) |
| 14. C24×D9 | C2×C3 | semidirect |
| 16. C23×C4×C9 | C2×C3 | semidirect |
| 17. C22×D36 | C2×C3 | direct (N × H) |
| 18. C23×C3×C9 | D4 | semidirect |
| 21. C22×C3×D9 | C23 | semidirect |
| 22. C2×C3×C4×C9 | C23 | semidirect |
| 23. C3×D36 | C23 | direct (N × H) |
| 34. C23×D9 | C22×C3 | semidirect |
| 36. C22×C4×C9 | C22×C3 | semidirect |
| 37. C2×D36 | C22×C3 | direct (N × H) |
| 38. C22×C3×C9 | C2×D4 | semidirect |
| 41. C2×C3×D9 | C24 | semidirect |
| 42. C3×C4×C9 | C24 | semidirect |
| 47. C23×C3×C4 | D9 | semidirect |
| 58. C23×C9 | C3×D4 | semidirect |
| 61. C22×D9 | C23×C3 | semidirect |
| 63. C2×C4×C9 | C23×C3 | semidirect |
| 64. D36 | C23×C3 | direct (N × H) |
| 66. C2×C3×C9 | C22×D4 | semidirect |
| 78. C22×C3×C4 | C2×D9 | semidirect |
| 88. C22×C9 | C2×C3×D4 | semidirect |
| 91. C2×D9 | C24×C3 | semidirect |
| 93. C4×C9 | C24×C3 | semidirect |
| 96. C23×C4 | C3×D9 | semidirect |
| 97. C3×C9 | C23×D4 | semidirect |
| 102. C23×C3 | D36 | direct (N × H) |
| 110. C2×C3×C4 | C22×D9 | semidirect |
| 118. C2×C9 | C22×C3×D4 | semidirect |
| 125. C22×C4 | C2×C3×D9 | semidirect |
| 130. C22×C3 | C2×D36 | direct (N × H) |
| 137. C3×C4 | C23×D9 | semidirect |
| 141. C9 | C23×C3×D4 | semidirect |
| 144. C23 | C3×D36 | direct (N × H) |
| 147. C2×C4 | C22×C3×D9 | semidirect |
| 151. C2×C3 | C22×D36 | direct (N × H) |
| 158. C22 | C2×C3×D36 | direct (N × H) |
| 160. C4 | C23×C3×D9 | semidirect |
| 162. C3 | C23×D36 | direct (N × H) |
| 165. C2 | C22×C3×D36 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×D9 | 2 |
| C23×C3×C4×C9 | 2 |
| C22×C3×D36 | 2 |
| C23×C3×D12 | 3 |
| C23×D36 | 3 |