Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
353 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 51 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. D4×#1088 | C2 | semidirect |
| 3. #11720788 | C2 | semidirect |
| 4. #11720775 | C2 | semidirect |
| 5. C8×#1088 | C2 | semidirect |
| 12. D8×H(3,𝔽3) | C4 | semidirect |
| 13. #88545 | C4 | semidirect |
| 14. #88529 | C4 | semidirect |
| 22. D4×H(3,𝔽3) | C2×C4 | semidirect |
| 24. C4⋊#272 | C2×C4 | semidirect |
| 27. C8×H(3,𝔽3) | C2×C4 | semidirect |
| 28. #6180 | C2×C4 | semidirect |
| 42. #1088 | D8 | direct (N × H) |
| 91. H(3,𝔽3) | C4×D8 | semidirect |
| 120. D8 | #1088 | direct (N × H) |
| 147. D4 | C2×#1088 | semidirect |
| 155. C8 | C2×#1088 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×#1088 | 2 |
| D8×#272 | 2 |
| #11720788 | 2 |
| #11720775 | 2 |
| C8×#1088 | 2 |
| C3×C4×D8 | 9 |