Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
698 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 169 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. #11722559 | C2 | direct (N × H) |
| 2. C2×#88683 | C2 | semidirect |
| 3. C22×C9⋊SL(2,𝔽3) | C2 | semidirect |
| 5. C2×C9×2−1+4 | C3 | semidirect |
| 6. C2×2−1+4⋊C9 | C3 | semidirect |
| 8. #88683 | C22 | semidirect |
| 10. C2×C9⋊SL(2,𝔽3) | C22 | semidirect |
| 15. C9×2−1+4 | C2×C3 | semidirect |
| 18. C2×C9×#41 | C2×C3 | semidirect |
| 20. C22×Q8×C9 | C2×C3 | semidirect |
| 21. 2−1+4⋊C9 | C2×C3 | semidirect |
| 22. C2×#41⋊C9 | C2×C3 | semidirect |
| 23. C22×Q8⋊C9 | C2×C3 | semidirect |
| 39. C9×#41 | C22×C3 | semidirect |
| 46. C2×Q8×C9 | C22×C3 | semidirect |
| 54. #41⋊C9 | C22×C3 | semidirect |
| 56. C2×Q8⋊C9 | C22×C3 | semidirect |
| 98. C2×2−1+4 | 3−1+2 | semidirect |
| 143. 2−1+4 | C2×3−1+2 | semidirect |
| 146. C2×#41 | C2×3−1+2 | semidirect |
| 148. C22×Q8 | C2×3−1+2 | semidirect |
| 181. C2×C9 | 2−1+4⋊C3 | semidirect |
| 188. #41 | C22×3−1+2 | semidirect |
| 195. C2×Q8 | C22×3−1+2 | semidirect |
| 217. C9 | C2×2−1+4⋊C3 | semidirect |
| 257. C2 | #11722559 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722559 | 2 |
| C2×#88683 | 2 |
| C22×C9⋊SL(2,𝔽3) | 2 |
| C2×C3×2−1+4⋊C3 | 3 |
| C2×C9×2−1+4 | 3 |
| C2×2−1+4⋊C9 | 3 |
| C2×D4×3−1+2 | 4 |