Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5003 conjugacy classes of subgroups; only the 635 normal subgroups are shown.
This group has 5003 conjugacy classes of subgroups — too many to draw. Only its 635 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. C23×((C3⋊D3)⋊C3)⋊C2 | C2 | semidirect |
| 2. C22×C4×#272 | C2 | semidirect |
| 3. C22×C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 4. C22×#6198 | C2 | semidirect |
| 5. C2×#88655 | C2 | direct (N × H) |
| 8. C23×(C3⋊D3)⋊C3 | C22 | semidirect |
| 10. C22×((C3⋊D3)⋊C3)⋊C2 | C22 | semidirect |
| 11. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 12. C22×#1084 | C22 | semidirect |
| 13. C2×C4×#272 | C22 | semidirect |
| 14. C2×C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 15. C2×#6198 | C22 | semidirect |
| 16. #88655 | C22 | direct (N × H) |
| 18. C22×C4⋊(C3⋊D3) | D3 | semidirect |
| 25. C22×#272 | D4 | semidirect |
| 28. C22×(C3⋊D3)⋊C3 | C23 | semidirect |
| 30. C2×((C3⋊D3)⋊C3)⋊C2 | C23 | semidirect |
| 31. C2×C4×H(3,𝔽3) | C23 | semidirect |
| 32. C2×#1084 | C23 | semidirect |
| 33. C4×#272 | C23 | semidirect |
| 34. C4⋊((C3⋊D3)⋊C3) | C23 | semidirect |
| 35. #6198 | C23 | semidirect |
| 38. C23×C3⋊D3 | C2×D3 | semidirect |
| 45. C22×C32×C4 | C2×D3 | semidirect |
| 46. C2×C4⋊(C3⋊D3) | C2×D3 | semidirect |
| 54. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 56. C2×#272 | C2×D4 | semidirect |
| 59. C2×(C3⋊D3)⋊C3 | C24 | semidirect |
| 61. C4×H(3,𝔽3) | C24 | semidirect |
| 62. #1084 | C24 | semidirect |
| 78. C22×C3⋊D3 | C22×D3 | semidirect |
| 88. C2×C32×C4 | C22×D3 | semidirect |
| 89. C4⋊(C3⋊D3) | C22×D3 | semidirect |
| 97. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 99. #272 | C22×D4 | semidirect |
| 129. C22×C32 | D3×D4 | semidirect |
| 132. C2×C3⋊D3 | C23×D3 | semidirect |
| 141. C32×C4 | C23×D3 | semidirect |
| 149. H(3,𝔽3) | C23×D4 | semidirect |
| 185. C2×C32 | C2×D3×D4 | semidirect |
| 200. C22×C4 | ((C3⋊D3)⋊C3)⋊C2 | semidirect |
| 228. C32 | C22×D3×D4 | semidirect |
| 240. C2×C4 | C2×((C3⋊D3)⋊C3)⋊C2 | semidirect |
| 258. C22 | #88655 | direct (N × H) |
| 261. C4 | C22×((C3⋊D3)⋊C3)⋊C2 | semidirect |
| 267. C2 | C2×#88655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×((C3⋊D3)⋊C3)⋊C2 | 2 |
| C22×C4×#272 | 2 |
| C22×C4⋊((C3⋊D3)⋊C3) | 2 |
| C22×#6198 | 2 |
| C2×#88655 | 2 |
| C22×D3×D12 | 3 |