Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2530 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 2530 conjugacy classes of subgroups — too many to draw. Only its 650 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | C23 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C23×C32 | C23×C3 |
| prime radical | C25×C32 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C9×C22⋊C4 | C2 | semidirect |
| 4. C9×#379 | C3 | direct (N × H) |
| 6. C24×C3×C9 | C4 | semidirect |
| 8. C22×C3×C4×C9 | C22 | semidirect |
| 13. C2×C9×C22⋊C4 | C2×C3 | semidirect |
| 21. C3×#379 | C9 | direct (N × H) |
| 28. C24×C9 | C3×C4 | semidirect |
| 31. C22×C4×C9 | C22×C3 | semidirect |
| 36. C22×C3×C9 | C22⋊C4 | semidirect |
| 42. C2×C3×C22⋊C4 | C2×C9 | semidirect |
| 58. #379 | C3×C9 | direct (N × H) |
| 64. C24×C3 | C4×C9 | semidirect |
| 68. C22×C3×C4 | C22×C9 | semidirect |
| 82. C22×C9 | C3×C22⋊C4 | semidirect |
| 87. C2×C22⋊C4 | C2×C3×C9 | semidirect |
| 88. C3×C9 | #379 | direct (N × H) |
| 110. C24 | C3×C4×C9 | semidirect |
| 112. C22×C4 | C22×C3×C9 | semidirect |
| 120. C22×C3 | C9×C22⋊C4 | semidirect |
| 129. C9 | C3×#379 | direct (N × H) |
| 146. C22 | C3×C9×C22⋊C4 | semidirect |
| 150. C3 | C9×#379 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C3×C9 | 2 |
| C2×C3×C9×C22⋊C4 | 2 |
| C32×#379 | 3 |
| C9×#379 | 3 |