Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1250 conjugacy classes of subgroups; only the 810 normal subgroups are shown.
This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 810 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C4 | D9 |
| commutator, hypocenter | C9 | C23×C3×C8 |
| Frattini | C3×C4 | C23×C3×D3 |
| Fitting | C23×C3×C4×C9 | C2 |
| socle, prime radical | C24×C32 | Dic3 |
| cosocle | C4×C9 | C24×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C23×C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C9⋊C8 | C2 | direct (N × H) |
| 4. C23×C9⋊C8 | C3 | direct (N × H) |
| 8. C2×C3×C9⋊C8 | C22 | direct (N × H) |
| 13. C22×C9⋊C8 | C2×C3 | direct (N × H) |
| 14. C23×C3×C9 | C8 | semidirect |
| 18. C3×C9⋊C8 | C23 | direct (N × H) |
| 31. C2×C9⋊C8 | C22×C3 | direct (N × H) |
| 32. C22×C3×C9 | C2×C8 | semidirect |
| 47. C23×C9 | C3×C8 | semidirect |
| 54. C9⋊C8 | C23×C3 | direct (N × H) |
| 56. C2×C3×C9 | C22×C8 | semidirect |
| 75. C22×C9 | C2×C3×C8 | semidirect |
| 83. C3×C9 | C23×C8 | semidirect |
| 86. C23×C3 | C9⋊C8 | direct (N × H) |
| 102. C2×C9 | C22×C3×C8 | semidirect |
| 110. C22×C3 | C2×C9⋊C8 | direct (N × H) |
| 122. C9 | C23×C3×C8 | semidirect |
| 123. C23 | C3×C9⋊C8 | direct (N × H) |
| 130. C2×C3 | C22×C9⋊C8 | direct (N × H) |
| 134. C22 | C2×C3×C9⋊C8 | direct (N × H) |
| 139. C3 | C23×C9⋊C8 | direct (N × H) |
| 141. C2 | C22×C3×C9⋊C8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C4×C9 | 2 |
| C22×C3×C9⋊C8 | 2 |
| C23×C3×C3⋊C8 | 3 |
| C23×C9⋊C8 | 3 |