Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
388 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #5450 |
| commutator, hypocenter | C24 | C22×C3×C9 |
| Frattini | C22×C3 | C22×C3×A4 |
| Fitting, prime radical | C32×H(3,𝔽4) | C3 |
| socle | C22×C32 | C22×A4 |
| cosocle | C24×C3 | C22×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#120⋊C9 | C2 | semidirect |
| 3. #10589552 | C3 | direct (N × H) |
| 4. C3×C22⋊(C22⋊C9) | C22 | semidirect |
| 9. #120⋊C9 | C2×C3 | semidirect |
| 11. C3×H(3,𝔽4) | C9 | semidirect |
| 17. C22⋊(C22⋊C9) | C22×C3 | semidirect |
| 29. C3×#120 | C2×C9 | semidirect |
| 39. H(3,𝔽4) | C3×C9 | semidirect |
| 41. C24×C3 | C22⋊C9 | semidirect |
| 42. C24×C3 | C22×C9 | semidirect |
| 68. #120 | C2×C3×C9 | semidirect |
| 88. C24 | C3×C22⋊C9 | semidirect |
| 89. C24 | C22×C3×C9 | semidirect |
| 134. C3 | #10589552 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#120⋊C9 | 2 |
| C32×H(3,𝔽4) | 3 |
| #10589552 | 3 |
| C22×C3×C22⋊C9 | 4 |