Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1219 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1219 conjugacy classes of subgroups — too many to draw. Only its 303 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 |
|---|---|---|
| 3. C2×C4×#1088 | C2 | direct (N × H) |
| 5. C22×#1088 | C4 | direct (N × H) |
| 7. C22×C4×H(3,𝔽3) | C4 | semidirect |
| 8. C22×#1084 | C4 | semidirect |
| 9. C2×C4×#272 | C4 | semidirect |
| 11. C4×#1088 | C22 | direct (N × H) |
| 17. C2×#1088 | C2×C4 | direct (N × H) |
| 19. C2×C4×H(3,𝔽3) | C2×C4 | semidirect |
| 20. C2×#1084 | C2×C4 | semidirect |
| 21. C4×#272 | C2×C4 | semidirect |
| 29. C22×H(3,𝔽3) | C42 | semidirect |
| 31. C2×#272 | C42 | semidirect |
| 35. #1088 | C22×C4 | direct (N × H) |
| 36. C4×H(3,𝔽3) | C22×C4 | semidirect |
| 37. #1084 | C22×C4 | semidirect |
| 51. C2×H(3,𝔽3) | C2×C42 | semidirect |
| 53. #272 | C2×C42 | semidirect |
| 79. H(3,𝔽3) | C22×C42 | semidirect |
| 108. C22×C4 | #1088 | direct (N × H) |
| 137. C2×C4 | C2×#1088 | direct (N × H) |
| 150. C22 | C4×#1088 | direct (N × H) |
| 154. C4 | C22×#1088 | direct (N × H) |
| 159. C2 | C2×C4×#1088 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#1088 | 2 |
| C22×C4×#272 | 2 |
| C2×C4×#1088 | 2 |
| C22×C3×C42 | 9 |