Finite Group DatabasebetaFinite Group DatabasebetaOrder 13824.
7688 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×C42×C9 | C24×C3 |
| socle, prime radical | C24×C3 | C2×C42×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×C82×C27 | C2 | direct (N × H) |
| 10. C2×C82×C27 | C4 | direct (N × H) |
| 19. C2×C4×C8×C27 | C8 | direct (N × H) |
| 26. C82×C27 | C2×C4 | direct (N × H) |
| 46. C4×C8×C27 | C2×C8 | direct (N × H) |
| 62. C2×C4×C82 | C27 | direct (N × H) |
| 70. C2×C8×C27 | C4×C8 | direct (N × H) |
| 100. C4×C82 | C2×C27 | direct (N × H) |
| 104. C2×C4×C27 | C82 | direct (N × H) |
| 111. C8×C27 | C2×C4×C8 | direct (N × H) |
| 142. C2×C82 | C4×C27 | direct (N × H) |
| 148. C4×C27 | C2×C82 | direct (N × H) |
| 179. C2×C4×C8 | C8×C27 | direct (N × H) |
| 186. C82 | C2×C4×C27 | direct (N × H) |
| 187. C2×C27 | C4×C82 | direct (N × H) |
| 223. C4×C8 | C2×C8×C27 | direct (N × H) |
| 225. C27 | C2×C4×C82 | direct (N × H) |
| 247. C2×C8 | C4×C8×C27 | direct (N × H) |
| 265. C2×C4 | C82×C27 | direct (N × H) |
| 272. C8 | C2×C4×C8×C27 | direct (N × H) |
| 280. C4 | C2×C82×C27 | direct (N × H) |
| 284. C2 | C4×C82×C27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×C8×C27 | 2 |
| C22×C82×C27 | 2 |
| C4×C82×C27 | 2 |
| C2×C4×C82×C9 | 3 |