Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1089 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 127 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. C22×C4⋊#272 | C2 | semidirect |
| 2. H(3,𝔽3)×#121 | C2 | semidirect |
| 3. C2×#88545 | C2 | semidirect |
| 4. #11720573 | C2 | semidirect |
| 5. #11720531 | C2 | semidirect |
| 9. C2×D4×H(3,𝔽3) | C22 | semidirect |
| 14. C4⋊C4×H(3,𝔽3) | C22 | semidirect |
| 17. C2×#6180 | C22 | semidirect |
| 19. C32×#121 | D3 | semidirect |
| 37. C4⋊#272 | D4 | semidirect |
| 43. C2×C32×D4 | C2×D3 | semidirect |
| 48. C32×C4⋊C4 | C2×D3 | semidirect |
| 57. C22×H(3,𝔽3) | D8 | semidirect |
| 126. C22×C32 | #213 | semidirect |
| 137. #121 | #272 | semidirect |
| 140. H(3,𝔽3) | #447 | semidirect |
| 184. C2×D4 | C2×#272 | semidirect |
| 188. C4⋊C4 | C2×#272 | semidirect |
| 218. C32 | #5036 | semidirect |
| 256. C22 | #88545 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊#272 | 2 |
| H(3,𝔽3)×#121 | 2 |
| C2×#88545 | 2 |
| #11720573 | 2 |
| #11720531 | 2 |
| C3×#5036 | 3 |