Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4834 conjugacy classes of subgroups; only the 1169 normal subgroups are shown.
This group has 4834 conjugacy classes of subgroups — too many to draw. Only its 1169 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×D9 |
| commutator, hypocenter | C3×C9 | C24×C4 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C24×C32 | C2×D3 |
| prime radical | C24×C3⋊D9 | C2 |
| cosocle | C2×C3×C9 | C25 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C9×Dic3 | C2 | semidirect |
| 3. C23×C3×Dic9 | C2 | semidirect |
| 4. C22×(C3⋊D9)⋊C4 | C2 | direct (N × H) |
| 8. C23×C3⋊D9 | C4 | semidirect |
| 10. C22×C9×Dic3 | C22 | semidirect |
| 11. C22×C3×Dic9 | C22 | semidirect |
| 12. C2×(C3⋊D9)⋊C4 | C22 | direct (N × H) |
| 20. C23×Dic9 | D3 | semidirect |
| 22. C23×C3×C9 | C2×C4 | semidirect |
| 24. C22×C3⋊D9 | C2×C4 | semidirect |
| 26. C2×C9×Dic3 | C23 | semidirect |
| 27. C2×C3×Dic9 | C23 | semidirect |
| 28. (C3⋊D9)⋊C4 | C23 | direct (N × H) |
| 44. C22×Dic9 | C2×D3 | semidirect |
| 46. C22×C3×C9 | C22×C4 | semidirect |
| 48. C2×C3⋊D9 | C22×C4 | semidirect |
| 50. C9×Dic3 | C24 | semidirect |
| 51. C3×Dic9 | C24 | semidirect |
| 58. C23×Dic3 | D9 | semidirect |
| 70. C23×C9 | C4×D3 | semidirect |
| 77. C2×Dic9 | C22×D3 | semidirect |
| 80. C2×C3×C9 | C23×C4 | semidirect |
| 82. C3⋊D9 | C23×C4 | semidirect |
| 95. C22×Dic3 | C2×D9 | semidirect |
| 106. C22×C9 | C2×C4×D3 | semidirect |
| 113. Dic9 | C23×D3 | semidirect |
| 118. C3×C9 | C24×C4 | semidirect |
| 121. C23×C3 | C4×D9 | semidirect |
| 134. C2×Dic3 | C22×D9 | semidirect |
| 142. C2×C9 | C22×C4×D3 | semidirect |
| 154. C22×C3 | C2×C4×D9 | semidirect |
| 167. Dic3 | C23×D9 | semidirect |
| 170. C9 | C23×C4×D3 | semidirect |
| 173. C23 | (C3⋊D9)⋊C4 | direct (N × H) |
| 180. C2×C3 | C22×C4×D9 | semidirect |
| 189. C22 | C2×(C3⋊D9)⋊C4 | direct (N × H) |
| 194. C3 | C23×C4×D9 | semidirect |
| 197. C2 | C22×(C3⋊D9)⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3⋊D9 | 2 |
| C23×C9×Dic3 | 2 |
| C23×C3×Dic9 | 2 |
| C22×(C3⋊D9)⋊C4 | 2 |
| C23×#624 | 3 |
| C23×C4×D9 | 3 |