Finite Group DatabasebetaFinite Group DatabasebetaOrder 13824.
12552 conjugacy classes of subgroups.
The subgroup lattice has 360 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C8×C9 | C25×C3 |
| socle, prime radical | C25×C3 | C2×C8×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4×C16×C27 | C2 | direct (N × H) |
| 10. C23×C16×C27 | C4 | direct (N × H) |
| 12. C2×C4×C16×C27 | C22 | direct (N × H) |
| 26. C22×C16×C27 | C2×C4 | direct (N × H) |
| 30. C4×C16×C27 | C23 | direct (N × H) |
| 40. C23×C4×C27 | C16 | direct (N × H) |
| 54. C2×C16×C27 | C22×C4 | direct (N × H) |
| 75. C23×C4×C16 | C27 | direct (N × H) |
| 80. C22×C4×C27 | C2×C16 | direct (N × H) |
| 94. C16×C27 | C23×C4 | direct (N × H) |
| 124. C22×C4×C16 | C2×C27 | direct (N × H) |
| 125. C23×C27 | C4×C16 | direct (N × H) |
| 130. C2×C4×C27 | C22×C16 | direct (N × H) |
| 175. C23×C16 | C4×C27 | direct (N × H) |
| 179. C2×C4×C16 | C22×C27 | direct (N × H) |
| 180. C22×C27 | C2×C4×C16 | direct (N × H) |
| 185. C4×C27 | C23×C16 | direct (N × H) |
| 230. C22×C16 | C2×C4×C27 | direct (N × H) |
| 234. C4×C16 | C23×C27 | direct (N × H) |
| 235. C2×C27 | C22×C4×C16 | direct (N × H) |
| 266. C23×C4 | C16×C27 | direct (N × H) |
| 280. C2×C16 | C22×C4×C27 | direct (N × H) |
| 284. C27 | C23×C4×C16 | direct (N × H) |
| 306. C22×C4 | C2×C16×C27 | direct (N × H) |
| 320. C16 | C23×C4×C27 | direct (N × H) |
| 330. C23 | C4×C16×C27 | direct (N × H) |
| 335. C2×C4 | C22×C16×C27 | direct (N × H) |
| 347. C22 | C2×C4×C16×C27 | direct (N × H) |
| 351. C4 | C23×C16×C27 | direct (N × H) |
| 356. C2 | C22×C4×C16×C27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×C8×C27 | 2 |
| C24×C16×C27 | 2 |
| C22×C4×C16×C27 | 2 |
| C23×C4×C9×C16 | 3 |