Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
11164 conjugacy classes of subgroups; only the 1468 normal subgroups are shown.
This group has 11164 conjugacy classes of subgroups — too many to draw. Only its 1468 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×C3⋊D3 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×D3×C3⋊D3 |
| Fitting, socle | C24×C33 | C22 |
| hypocenter | C33 | C23×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3⋊(C3⋊D3) | C2 | semidirect |
| 2. C24×C32×D3 | C2 | semidirect |
| 3. C23×C3×C3⋊Dic3 | C2 | semidirect |
| 4. C22×#6291 | C2 | direct (N × H) |
| 9. C23×C3⋊(C3⋊D3) | C22 | semidirect |
| 11. C23×C32×D3 | C22 | semidirect |
| 12. C22×C3×C3⋊Dic3 | C22 | semidirect |
| 13. C2×#6291 | C22 | direct (N × H) |
| 18. C24×C3×D3 | D3 | semidirect |
| 20. C23×C3⋊Dic3 | D3 | semidirect |
| 23. C23×C33 | D4 | semidirect |
| 26. C22×C3⋊(C3⋊D3) | C23 | semidirect |
| 28. C22×C32×D3 | C23 | semidirect |
| 29. C2×C3×C3⋊Dic3 | C23 | semidirect |
| 30. #6291 | C23 | direct (N × H) |
| 45. C23×C3×D3 | C2×D3 | semidirect |
| 47. C22×C3⋊Dic3 | C2×D3 | semidirect |
| 50. C22×C33 | C2×D4 | semidirect |
| 53. C2×C3⋊(C3⋊D3) | C24 | semidirect |
| 55. C2×C32×D3 | C24 | semidirect |
| 56. C3×C3⋊Dic3 | C24 | semidirect |
| 61. C24×D3 | C3⋊D3 | semidirect |
| 70. C23×C32 | D12 | semidirect |
| 71. C23×C32 | #67 | semidirect |
| 81. C22×C3×D3 | C22×D3 | semidirect |
| 83. C2×C3⋊Dic3 | C22×D3 | semidirect |
| 87. C2×C33 | C22×D4 | semidirect |
| 103. C23×D3 | C2×C3⋊D3 | semidirect |
| 112. C22×C32 | C2×D12 | semidirect |
| 113. C22×C32 | C2×#67 | semidirect |
| 123. C2×C3×D3 | C23×D3 | semidirect |
| 124. C3⋊Dic3 | C23×D3 | semidirect |
| 130. C33 | C23×D4 | semidirect |
| 135. C23×C3 | #626 | semidirect |
| 136. C23×C3 | #638 | semidirect |
| 146. C22×D3 | C22×C3⋊D3 | semidirect |
| 154. C2×C32 | C22×D12 | semidirect |
| 155. C2×C32 | C22×#67 | semidirect |
| 173. C22×C3 | C2×#626 | semidirect |
| 174. C22×C3 | C2×#638 | semidirect |
| 184. C2×D3 | C23×C3⋊D3 | semidirect |
| 188. C32 | C23×D12 | semidirect |
| 189. C32 | C23×#67 | semidirect |
| 194. C23 | #6291 | direct (N × H) |
| 201. C2×C3 | C22×#626 | semidirect |
| 202. C2×C3 | C22×#638 | semidirect |
| 213. C22 | C2×#6291 | direct (N × H) |
| 217. C3 | C23×#626 | semidirect |
| 218. C3 | C23×#638 | semidirect |
| 221. C2 | C22×#6291 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3⋊(C3⋊D3) | 2 |
| C24×C32×D3 | 2 |
| C23×C3×C3⋊Dic3 | 2 |
| C22×#6291 | 2 |
| C23×#638 | 3 |
| C23×#626 | 3 |