Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4895 conjugacy classes of subgroups; only the 853 normal subgroups are shown.
This group has 4895 conjugacy classes of subgroups — too many to draw. Only its 853 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×C4×H(3,𝔽3) | C2 | semidirect |
| 2. C22×#6235 | C2 | semidirect |
| 3. C2×#88740 | C2 | direct (N × H) |
| 4. C22×C4⋊#272 | C2 | semidirect |
| 5. C22×C4×#272 | C2 | semidirect |
| 6. C22×#6229 | C2 | semidirect |
| 12. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 13. C2×#6235 | C22 | semidirect |
| 15. #88740 | C22 | direct (N × H) |
| 16. C2×C4⋊#272 | C22 | semidirect |
| 17. C2×C4×#272 | C22 | semidirect |
| 18. C2×#6229 | C22 | semidirect |
| 19. C23×C32×C4 | D3 | semidirect |
| 32. C2×C4×H(3,𝔽3) | C23 | semidirect |
| 33. #6235 | C23 | semidirect |
| 35. C4⋊#272 | C23 | semidirect |
| 36. C4×#272 | C23 | semidirect |
| 37. #6229 | C23 | semidirect |
| 44. C22×C32×C4 | C2×D3 | semidirect |
| 52. C22×H(3,𝔽3) | #41 | semidirect |
| 79. C2×C32×C4 | C22×D3 | semidirect |
| 87. C2×H(3,𝔽3) | C2×#41 | semidirect |
| 113. C22×C32 | #235 | semidirect |
| 126. C23×C4 | #272 | semidirect |
| 127. H(3,𝔽3) | C22×#41 | semidirect |
| 155. C2×C32 | C2×#235 | semidirect |
| 168. C22×C4 | C2×#272 | semidirect |
| 187. C32 | C22×#235 | semidirect |
| 200. C2×C4 | C22×#272 | semidirect |
| 212. C22 | #88740 | direct (N × H) |
| 222. C2 | C2×#88740 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C22×#6235 | 2 |
| C2×#88740 | 2 |
| C22×C4⋊#272 | 2 |
| C22×C4×#272 | 2 |
| C22×#6229 | 2 |
| C22×C3×#235 | 3 |