Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1290 conjugacy classes of subgroups; only the 615 normal subgroups are shown.
This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 615 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C43 |
| Frattini | C23×C3 | C22×C3⋊D3 |
| Fitting | C2×C3×C42×C9 | C2 |
| socle, prime radical | C23×C32 | C22×D3 |
| cosocle | C23×C3×C9 | C23 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C3×C42×C9 | C4 | semidirect |
| 8. C4×C3⋊Dic9 | C4 | direct (N × H) |
| 25. C42×C9 | Dic3 | semidirect |
| 29. C3×C4×C9 | C42 | semidirect |
| 30. C3⋊Dic9 | C42 | direct (N × H) |
| 50. C3×C42 | Dic9 | semidirect |
| 60. C4×C9 | C4×Dic3 | semidirect |
| 64. C3×C9 | C43 | semidirect |
| 80. C42 | C3⋊Dic9 | direct (N × H) |
| 85. C3×C4 | C4×Dic9 | semidirect |
| 90. C9 | C42×Dic3 | semidirect |
| 102. C4 | C4×C3⋊Dic9 | direct (N × H) |
| 103. C3 | C42×Dic9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C42×C9 | 2 |
| C2×C4×C3⋊Dic9 | 2 |
| C42×C3⋊Dic3 | 3 |
| C42×Dic9 | 3 |