Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5515 conjugacy classes of subgroups; only the 1204 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 1204 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. C25×H(3,𝔽3) | C2 | semidirect |
| 2. C24×(C3⋊D3)⋊C3 | C2 | semidirect |
| 3. C22×#638⋊C3 | C2 | direct (N × H) |
| 4. C23×(C3⋊Dic3)⋊C3 | C2 | semidirect |
| 5. C23×#638 | C3 | semidirect |
| 9. C24×H(3,𝔽3) | C22 | semidirect |
| 11. C23×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. C2×#638⋊C3 | C22 | direct (N × H) |
| 13. C22×(C3⋊Dic3)⋊C3 | C22 | semidirect |
| 15. C25×C32 | D3 | semidirect |
| 16. C25×C32 | C2×C3 | semidirect |
| 17. C24×C3⋊D3 | C2×C3 | semidirect |
| 19. C22×#638 | C2×C3 | semidirect |
| 20. C23×C3⋊Dic3 | C2×C3 | semidirect |
| 24. C23×H(3,𝔽3) | D4 | semidirect |
| 26. C23×H(3,𝔽3) | C23 | semidirect |
| 28. C22×(C3⋊D3)⋊C3 | C23 | semidirect |
| 29. #638⋊C3 | C23 | direct (N × H) |
| 30. C2×(C3⋊Dic3)⋊C3 | C23 | semidirect |
| 40. C24×C32 | C2×D3 | semidirect |
| 42. C24×C32 | C22×C3 | semidirect |
| 44. C23×C3⋊D3 | C22×C3 | semidirect |
| 47. C2×#638 | C22×C3 | semidirect |
| 48. C22×C3⋊Dic3 | C22×C3 | semidirect |
| 52. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 54. C22×H(3,𝔽3) | C24 | semidirect |
| 56. C2×(C3⋊D3)⋊C3 | C24 | semidirect |
| 57. (C3⋊Dic3)⋊C3 | C24 | semidirect |
| 77. C23×C32 | #67 | semidirect |
| 78. C23×C32 | C3×D4 | semidirect |
| 80. C23×C32 | C22×D3 | semidirect |
| 82. C23×C32 | C23×C3 | semidirect |
| 84. C22×C3⋊D3 | C23×C3 | semidirect |
| 87. #638 | C23×C3 | semidirect |
| 88. C2×C3⋊Dic3 | C23×C3 | semidirect |
| 93. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 126. C22×C32 | C2×#67 | semidirect |
| 127. C22×C32 | C2×C3×D4 | semidirect |
| 129. C22×C32 | C23×D3 | semidirect |
| 131. C22×C32 | C24×C3 | semidirect |
| 133. C2×C3⋊D3 | C24×C3 | semidirect |
| 136. C3⋊Dic3 | C24×C3 | semidirect |
| 139. C25 | (C3⋊D3)⋊C3 | semidirect |
| 142. H(3,𝔽3) | C23×D4 | semidirect |
| 176. C2×C32 | C22×#67 | semidirect |
| 177. C2×C32 | C22×C3×D4 | semidirect |
| 186. C24 | C2×(C3⋊D3)⋊C3 | semidirect |
| 215. C32 | C23×#67 | semidirect |
| 216. C32 | C23×C3×D4 | semidirect |
| 220. C23 | #638⋊C3 | direct (N × H) |
| 222. C23 | C22×(C3⋊D3)⋊C3 | semidirect |
| 243. C22 | C2×#638⋊C3 | direct (N × H) |
| 245. C22 | C23×(C3⋊D3)⋊C3 | semidirect |
| 253. C2 | C22×#638⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×H(3,𝔽3) | 2 |
| C24×(C3⋊D3)⋊C3 | 2 |
| C22×#638⋊C3 | 2 |
| C23×(C3⋊Dic3)⋊C3 | 2 |
| C23×#638 | 3 |
| C23×C3×#67 | 3 |