Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2507 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 2507 conjugacy classes of subgroups — too many to draw. Only its 223 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×#1088 | C2 | semidirect |
| 3. C2×C22⋊#1088 | C2 | direct (N × H) |
| 4. C24×H(3,𝔽3) | C4 | semidirect |
| 6. C23×#272 | C4 | semidirect |
| 7. C23×#272 | C4 | semidirect |
| 11. C22×#1088 | C22 | semidirect |
| 12. C22⋊#1088 | C22 | direct (N × H) |
| 16. C23×H(3,𝔽3) | C2×C4 | semidirect |
| 21. C22×#272 | C2×C4 | semidirect |
| 22. C22×#272 | C2×C4 | semidirect |
| 27. C2×#1088 | C23 | semidirect |
| 36. C22×H(3,𝔽3) | C22⋊C4 | semidirect |
| 38. C22×H(3,𝔽3) | C22×C4 | semidirect |
| 42. C2×#272 | C22⋊C4 | semidirect |
| 44. C2×#272 | C22×C4 | semidirect |
| 66. C2×H(3,𝔽3) | C2×C22⋊C4 | semidirect |
| 69. #272 | C2×C22⋊C4 | semidirect |
| 101. H(3,𝔽3) | C22×C22⋊C4 | semidirect |
| 139. C24 | #1088 | semidirect |
| 180. C23 | C2×#1088 | semidirect |
| 204. C22 | C22⋊#1088 | direct (N × H) |
| 206. C22 | C22×#1088 | semidirect |
| 214. C2 | C2×C22⋊#1088 | 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×#1088 | 2 |
| C2×C22⋊#1088 | 2 |
| C22×C3×C22⋊C4 | 9 |