Finite Group DatabasebetaFinite Group DatabasebetaOrder 27648.
6292 conjugacy classes of subgroups.
The subgroup lattice has 272 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C82×C9 | C24×C3 |
| socle, prime radical | C24×C3 | C82×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C162×C27 | C2 | direct (N × H) |
| 8. C162×C27 | C22 | direct (N × H) |
| 27. C22×C162 | C27 | direct (N × H) |
| 30. C22×C16×C27 | C16 | direct (N × H) |
| 54. C2×C16×C27 | C2×C16 | direct (N × H) |
| 78. C2×C162 | C2×C27 | direct (N × H) |
| 90. C16×C27 | C22×C16 | direct (N × H) |
| 115. C162 | C22×C27 | direct (N × H) |
| 156. C22×C27 | C162 | direct (N × H) |
| 183. C22×C16 | C16×C27 | direct (N × H) |
| 193. C2×C27 | C2×C162 | direct (N × H) |
| 218. C2×C16 | C2×C16×C27 | direct (N × H) |
| 224. C27 | C22×C162 | direct (N × H) |
| 246. C16 | C22×C16×C27 | direct (N × H) |
| 263. C22 | C162×C27 | direct (N × H) |
| 269. C2 | C2×C162×C27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×C16×C27 | 2 |
| C2×C162×C27 | 2 |
| C22×C9×C162 | 3 |