Finite Group DatabasebetaFinite Group DatabasebetaOrder 13824.
3220 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C32 | C24×C32 |
| socle, prime radical | C24×C32 | C3×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C9×C64 | C2 | direct (N × H) |
| 4. C23×C9×C64 | C3 | direct (N × H) |
| 8. C2×C3×C9×C64 | C22 | direct (N × H) |
| 12. C22×C9×C64 | C2×C3 | direct (N × H) |
| 13. C23×C3×C64 | C9 | direct (N × H) |
| 20. C3×C9×C64 | C23 | direct (N × H) |
| 28. C2×C9×C64 | C22×C3 | direct (N × H) |
| 38. C22×C3×C64 | C2×C9 | direct (N × H) |
| 51. C9×C64 | C23×C3 | direct (N × H) |
| 52. C23×C64 | C3×C9 | direct (N × H) |
| 66. C2×C3×C64 | C22×C9 | direct (N × H) |
| 83. C22×C64 | C2×C3×C9 | direct (N × H) |
| 84. C23×C3×C9 | C64 | direct (N × H) |
| 100. C3×C64 | C23×C9 | direct (N × H) |
| 119. C2×C64 | C22×C3×C9 | direct (N × H) |
| 120. C22×C3×C9 | C2×C64 | direct (N × H) |
| 142. C23×C9 | C3×C64 | direct (N × H) |
| 155. C64 | C23×C3×C9 | direct (N × H) |
| 156. C2×C3×C9 | C22×C64 | direct (N × H) |
| 176. C22×C9 | C2×C3×C64 | direct (N × H) |
| 187. C3×C9 | C23×C64 | direct (N × H) |
| 188. C23×C3 | C9×C64 | direct (N × H) |
| 202. C2×C9 | C22×C3×C64 | direct (N × H) |
| 211. C22×C3 | C2×C9×C64 | direct (N × H) |
| 220. C9 | C23×C3×C64 | direct (N × H) |
| 221. C23 | C3×C9×C64 | direct (N × H) |
| 227. C2×C3 | C22×C9×C64 | direct (N × H) |
| 231. C22 | C2×C3×C9×C64 | direct (N × H) |
| 235. C3 | C23×C9×C64 | direct (N × H) |
| 237. C2 | C22×C3×C9×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C9×C32 | 2 |
| C22×C3×C9×C64 | 2 |
| C23×C32×C64 | 3 |
| C23×C9×C64 | 3 |