Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
274 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #5450 |
| commutator, hypocenter | C24 | C22×C3×C9 |
| Frattini, socle | C22×C3 | C22×C3×A4 |
| Fitting | C9×H(3,𝔽4) | C3 |
| prime radical | C3×#5450 | C3 |
| cosocle | C24×C3 | C22×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#120⋊C3 | C2 | semidirect |
| 3. C9×H(3,𝔽4) | C3 | semidirect |
| 4. #10589552 | C3 | semidirect |
| 5. C9×C22⋊A4 | C22 | semidirect |
| 10. C9×#120 | C2×C3 | semidirect |
| 11. #120⋊C9 | C2×C3 | semidirect |
| 14. #5450 | C9 | direct (N × H) |
| 17. C24×C9 | A4 | semidirect |
| 18. C24×C9 | C22×C3 | semidirect |
| 19. C22⋊(C22⋊C9) | C22×C3 | semidirect |
| 32. #120⋊C3 | C2×C9 | semidirect |
| 42. H(3,𝔽4) | C3×C9 | semidirect |
| 46. C22⋊A4 | C22×C9 | semidirect |
| 69. #120 | C2×C3×C9 | semidirect |
| 84. C24 | C9×A4 | semidirect |
| 85. C24 | C22×C3×C9 | semidirect |
| 104. C9 | #5450 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#120⋊C3 | 2 |
| C3×#5450 | 3 |
| C9×H(3,𝔽4) | 3 |
| #10589552 | 3 |
| C22×C9×A4 | 4 |