Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
567 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C9 |
| Frattini, socle | C2×C3 | C2×C3×C22⋊A4 |
| Fitting | C9×#585 | C3 |
| prime radical | C3×#5950 | C3 |
| cosocle | C3×H(5,𝔽2) | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#997 | C2 | semidirect |
| 3. C9×#585 | C3 | semidirect |
| 4. #10593125 | C3 | semidirect |
| 17. #5950 | C9 | direct (N × H) |
| 24. C9×#41 | A4 | semidirect |
| 33. C22×C4×C9 | A4 | semidirect |
| 43. #997 | C2×C9 | semidirect |
| 62. #585 | C3×C9 | semidirect |
| 123. #41 | C9×A4 | semidirect |
| 132. C22×C4 | C9×A4 | semidirect |
| 145. C9 | #5950 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#997 | 2 |
| C3×#5950 | 3 |
| C9×#585 | 3 |
| #10593125 | 3 |
| C9×#41⋊C3 | 4 |
| C4×C9×A4 | 4 |