Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3351 conjugacy classes of subgroups; only the 595 normal subgroups are shown.
This group has 3351 conjugacy classes of subgroups — too many to draw. Only its 595 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×#1084 | C2 | semidirect |
| 2. C22×#638⋊C3 | C2 | semidirect |
| 3. C22×#6198 | C2 | semidirect |
| 4. C2×#88672 | C2 | direct (N × H) |
| 5. C22×((C3⋊D3)⋊C3)⋊C4 | C2 | semidirect |
| 6. C22×#6196 | C2 | semidirect |
| 12. C22×#1084 | C22 | semidirect |
| 13. C2×#638⋊C3 | C22 | semidirect |
| 15. C2×#6198 | C22 | semidirect |
| 16. #88672 | C22 | direct (N × H) |
| 17. C2×((C3⋊D3)⋊C3)⋊C4 | C22 | semidirect |
| 18. C2×#6196 | C22 | semidirect |
| 19. C22×#638 | D3 | semidirect |
| 35. C2×#1084 | C23 | semidirect |
| 36. #638⋊C3 | C23 | semidirect |
| 38. #6198 | C23 | semidirect |
| 39. ((C3⋊D3)⋊C3)⋊C4 | C23 | semidirect |
| 40. #6196 | C23 | semidirect |
| 47. C2×#638 | C2×D3 | semidirect |
| 63. C22×H(3,𝔽3) | #41 | semidirect |
| 96. #638 | C22×D3 | semidirect |
| 112. C2×H(3,𝔽3) | C2×#41 | semidirect |
| 150. C22×C32 | #237 | semidirect |
| 169. H(3,𝔽3) | C22×#41 | semidirect |
| 211. C2×C32 | C2×#237 | semidirect |
| 254. C32 | C22×#237 | semidirect |
| 283. C22 | #88672 | direct (N × H) |
| 294. C2 | C2×#88672 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#1084 | 2 |
| C22×#638⋊C3 | 2 |
| C22×#6198 | 2 |
| C2×#88672 | 2 |
| C22×((C3⋊D3)⋊C3)⋊C4 | 2 |
| C22×#6196 | 2 |
| C22×#3804 | 3 |