Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
793 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 793 conjugacy classes of subgroups — too many to draw. Only its 159 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. Q8×#1088 | C2 | semidirect |
| 3. D4×#1088 | C2 | semidirect |
| 4. #11722448 | C2 | semidirect |
| 5. C2×C4×#1088 | C2 | semidirect |
| 7. #41×H(3,𝔽3) | C4 | semidirect |
| 8. #88748 | C4 | semidirect |
| 10. #88744 | C4 | semidirect |
| 12. #88740 | C4 | semidirect |
| 24. Q8×H(3,𝔽3) | C2×C4 | semidirect |
| 25. D4×H(3,𝔽3) | C2×C4 | semidirect |
| 26. C2×C4×H(3,𝔽3) | C2×C4 | semidirect |
| 27. #6235 | C2×C4 | semidirect |
| 28. C2×#1084 | C2×C4 | semidirect |
| 29. C4⋊#272 | C2×C4 | semidirect |
| 30. C4×#272 | C2×C4 | semidirect |
| 31. C4×#272 | C2×C4 | semidirect |
| 34. #6229 | C2×C4 | semidirect |
| 48. #1088 | #41 | direct (N × H) |
| 109. H(3,𝔽3) | C4×#41 | semidirect |
| 144. #41 | #1088 | direct (N × H) |
| 175. Q8 | C2×#1088 | semidirect |
| 178. D4 | C2×#1088 | semidirect |
| 187. C2×C4 | C2×#1088 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×#272 | 2 |
| Q8×#1088 | 2 |
| D4×#1088 | 2 |
| #11722448 | 2 |
| C2×C4×#1088 | 2 |
| C3×C4×#41 | 9 |