Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2503 conjugacy classes of subgroups; only the 954 normal subgroups are shown.
This group has 2503 conjugacy classes of subgroups — too many to draw. Only its 954 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C25×C3 |
| Frattini, socle | C2×C3 | C24×C3×D3 |
| Fitting | 3−1+2×2−1+4 | C2 |
| hypercenter | 2−1+4 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×#585 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 3−1+2×2−1+4 | C2 | semidirect |
| 2. #11722641 | C2 | semidirect |
| 3. #41×AGL(1,ℤ/9) | C2 | semidirect |
| 4. #11722630 | C2 | semidirect |
| 5. C2×#88724 | C2 | semidirect |
| 7. #10593143 | C3 | semidirect |
| 8. #41×3−1+2 | C22 | semidirect |
| 9. C2×Q8×3−1+2 | C22 | semidirect |
| 10. #88724 | C22 | semidirect |
| 11. #88724 | C22 | semidirect |
| 12. Q8×AGL(1,ℤ/9) | C22 | semidirect |
| 13. #3699⋊C3 | C22 | semidirect |
| 14. D4×AGL(1,ℤ/9) | C22 | semidirect |
| 15. #3697⋊C3 | C22 | semidirect |
| 16. C2×C4⋊AGL(1,ℤ/9) | C22 | semidirect |
| 17. C2×C4×AGL(1,ℤ/9) | C22 | semidirect |
| 24. C9×2−1+4 | C2×C3 | semidirect |
| 25. #63715 | C2×C3 | semidirect |
| 26. #41×D9 | C2×C3 | semidirect |
| 27. #63713 | C2×C3 | semidirect |
| 28. C2×#3701 | C2×C3 | semidirect |
| 52. C9×#41 | C22×C3 | semidirect |
| 54. C2×Q8×C9 | C22×C3 | semidirect |
| 55. #3701 | C22×C3 | semidirect |
| 56. #3701 | C22×C3 | semidirect |
| 57. Q8×D9 | C22×C3 | semidirect |
| 58. #3699 | C22×C3 | semidirect |
| 59. D4×D9 | C22×C3 | semidirect |
| 60. #3697 | C22×C3 | semidirect |
| 61. C2×D36 | C22×C3 | semidirect |
| 62. C2×C4×D9 | C22×C3 | semidirect |
| 141. 2−1+4 | AGL(1,ℤ/9) | semidirect |
| 145. 3−1+2 | #585 | semidirect |
| 178. #41 | C2×AGL(1,ℤ/9) | semidirect |
| 180. C2×Q8 | C2×AGL(1,ℤ/9) | semidirect |
| 198. C9 | C3×#585 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 3−1+2×2−1+4 | 2 |
| #11722641 | 2 |
| #41×AGL(1,ℤ/9) | 2 |
| #11722630 | 2 |
| C2×#88724 | 2 |
| C3×#5968 | 3 |
| #10593143 | 3 |