Finite Group DatabasebetaFinite Group DatabasebetaOrder 13824.
4536 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C32 | C23×C33 |
| socle, prime radical | C23×C33 | C2×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C33×C4×C64 | C2 | direct (N × H) |
| 4. C2×C32×C4×C64 | C3 | direct (N × H) |
| 10. C2×C33×C64 | C4 | direct (N × H) |
| 14. C32×C4×C64 | C2×C3 | direct (N × H) |
| 15. C2×C3×C4×C64 | C32 | direct (N × H) |
| 26. C33×C64 | C2×C4 | direct (N × H) |
| 30. C2×C32×C64 | C3×C4 | direct (N × H) |
| 50. C3×C4×C64 | C2×C32 | direct (N × H) |
| 61. C32×C64 | C2×C3×C4 | direct (N × H) |
| 62. C2×C4×C64 | C33 | direct (N × H) |
| 80. C2×C3×C64 | C32×C4 | direct (N × H) |
| 100. C4×C64 | C2×C33 | direct (N × H) |
| 102. C2×C33×C4 | C64 | direct (N × H) |
| 122. C3×C64 | C2×C32×C4 | direct (N × H) |
| 140. C2×C64 | C33×C4 | direct (N × H) |
| 148. C33×C4 | C2×C64 | direct (N × H) |
| 166. C2×C32×C4 | C3×C64 | direct (N × H) |
| 186. C64 | C2×C33×C4 | direct (N × H) |
| 187. C2×C33 | C4×C64 | direct (N × H) |
| 208. C32×C4 | C2×C3×C64 | direct (N × H) |
| 225. C33 | C2×C4×C64 | direct (N × H) |
| 227. C2×C3×C4 | C32×C64 | direct (N × H) |
| 237. C2×C32 | C3×C4×C64 | direct (N × H) |
| 258. C3×C4 | C2×C32×C64 | direct (N × H) |
| 261. C32 | C2×C3×C4×C64 | direct (N × H) |
| 263. C2×C4 | C33×C64 | direct (N × H) |
| 273. C2×C3 | C32×C4×C64 | direct (N × H) |
| 279. C4 | C2×C33×C64 | direct (N × H) |
| 283. C3 | C2×C32×C4×C64 | direct (N × H) |
| 284. C2 | C33×C4×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×C4×C32 | 2 |
| C22×C33×C64 | 2 |
| C33×C4×C64 | 2 |
| C2×C32×C4×C64 | 3 |