Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
349 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 79 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 |
|---|---|---|
| 4. #11720773 | C2 | semidirect |
| 5. C8×#1088 | C2 | semidirect |
| 8. #88631 | C4 | semidirect |
| 12. #88627 | C4 | semidirect |
| 14. M16×H(3,𝔽3) | C4 | semidirect |
| 15. #88538 | C4 | semidirect |
| 16. #88527 | C4 | semidirect |
| 29. #6188 | C2×C4 | semidirect |
| 30. C8×H(3,𝔽3) | C2×C4 | semidirect |
| 31. #6180 | C2×C4 | semidirect |
| 43. #1088 | M16 | direct (N × H) |
| 98. H(3,𝔽3) | C4×M16 | semidirect |
| 131. M16 | #1088 | direct (N × H) |
| 168. C8 | C2×#1088 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#1088 | 2 |
| #11722445 | 2 |
| M16×#272 | 2 |
| #11720773 | 2 |
| C8×#1088 | 2 |
| C3×C4×M16 | 9 |