Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4460 conjugacy classes of subgroups; only the 2484 normal subgroups are shown.
This group has 4460 conjugacy classes of subgroups — too many to draw. Only its 2484 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C24×C3×C4 |
| Frattini | C2×C3 | C24×C3×D3 |
| Fitting | C25×3−1+2 | C2 |
| socle | C25×C3 | C3×D3 |
| prime radical | C25×C32 | D3 |
| cosocle | C2×C9 | C25×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×3−1+2⋊C4 | C2 | direct (N × H) |
| 4. C24×Dic9 | C3 | semidirect |
| 5. C24×3−1+2 | C4 | semidirect |
| 7. C22×3−1+2⋊C4 | C22 | direct (N × H) |
| 12. C23×Dic9 | C2×C3 | semidirect |
| 13. C23×3−1+2 | C2×C4 | semidirect |
| 15. C2×3−1+2⋊C4 | C23 | direct (N × H) |
| 23. C24×C9 | C3×C4 | semidirect |
| 25. C22×Dic9 | C22×C3 | semidirect |
| 26. C22×3−1+2 | C22×C4 | semidirect |
| 28. 3−1+2⋊C4 | C24 | direct (N × H) |
| 38. C23×C9 | C2×C3×C4 | semidirect |
| 40. C2×Dic9 | C23×C3 | semidirect |
| 42. C2×3−1+2 | C23×C4 | semidirect |
| 55. C22×C9 | C22×C3×C4 | semidirect |
| 57. Dic9 | C24×C3 | semidirect |
| 60. 3−1+2 | C24×C4 | semidirect |
| 71. C2×C9 | C23×C3×C4 | semidirect |
| 73. C24 | 3−1+2⋊C4 | direct (N × H) |
| 84. C9 | C24×C3×C4 | semidirect |
| 85. C23 | C2×3−1+2⋊C4 | direct (N × H) |
| 92. C22 | C22×3−1+2⋊C4 | direct (N × H) |
| 97. C2 | C23×3−1+2⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×3−1+2 | 2 |
| C23×3−1+2⋊C4 | 2 |
| C24×C3×Dic3 | 3 |
| C24×Dic9 | 3 |