Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5515 conjugacy classes of subgroups; only the 1051 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 1051 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 |
|---|---|---|
| 2. C23×(C3⋊Dic3)⋊C3 | C2 | semidirect |
| 3. C22×#6199 | C2 | direct (N × H) |
| 6. C23×#272 | C4 | semidirect |
| 8. C22×(C3⋊Dic3)⋊C3 | C22 | semidirect |
| 9. C2×#6199 | C22 | direct (N × H) |
| 12. C23×C3⋊Dic3 | D3 | semidirect |
| 15. C23×H(3,𝔽3) | C2×C4 | semidirect |
| 17. C22×#272 | C2×C4 | semidirect |
| 19. C2×(C3⋊Dic3)⋊C3 | C23 | semidirect |
| 20. #6199 | C23 | direct (N × H) |
| 29. C22×C3⋊Dic3 | C2×D3 | semidirect |
| 32. C22×H(3,𝔽3) | C22×C4 | semidirect |
| 34. C2×#272 | C22×C4 | semidirect |
| 36. (C3⋊Dic3)⋊C3 | C24 | semidirect |
| 47. C23×C32 | C4×D3 | semidirect |
| 53. C2×C3⋊Dic3 | C22×D3 | semidirect |
| 57. C2×H(3,𝔽3) | C23×C4 | semidirect |
| 59. #272 | C23×C4 | semidirect |
| 76. C22×C32 | C2×C4×D3 | semidirect |
| 82. C3⋊Dic3 | C23×D3 | semidirect |
| 87. H(3,𝔽3) | C24×C4 | semidirect |
| 107. C2×C32 | C22×C4×D3 | semidirect |
| 132. C32 | C23×C4×D3 | semidirect |
| 135. C23 | #6199 | direct (N × H) |
| 150. C22 | C2×#6199 | direct (N × H) |
| 157. C2 | C22×#6199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#272 | 2 |
| C23×(C3⋊Dic3)⋊C3 | 2 |
| C22×#6199 | 2 |
| C23×D3×Dic3 | 3 |