Finite Group DatabasebetaFinite Group DatabasebetaOrder 13824.
2568 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×C9×C16 | C24×C3 |
| socle, prime radical | C24×C3 | C2×C9×C16 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×C27×C32 | C2 | direct (N × H) |
| 10. C22×C27×C32 | C4 | direct (N × H) |
| 12. C4×C27×C32 | C22 | direct (N × H) |
| 27. C2×C27×C32 | C2×C4 | direct (N × H) |
| 57. C27×C32 | C22×C4 | direct (N × H) |
| 75. C22×C4×C32 | C27 | direct (N × H) |
| 77. C22×C4×C27 | C32 | direct (N × H) |
| 124. C2×C4×C32 | C2×C27 | direct (N × H) |
| 129. C2×C4×C27 | C2×C32 | direct (N × H) |
| 175. C22×C32 | C4×C27 | direct (N × H) |
| 179. C4×C32 | C22×C27 | direct (N × H) |
| 180. C22×C27 | C4×C32 | direct (N × H) |
| 185. C4×C27 | C22×C32 | direct (N × H) |
| 231. C2×C32 | C2×C4×C27 | direct (N × H) |
| 235. C2×C27 | C2×C4×C32 | direct (N × H) |
| 283. C32 | C22×C4×C27 | direct (N × H) |
| 284. C27 | C22×C4×C32 | direct (N × H) |
| 303. C22×C4 | C27×C32 | direct (N × H) |
| 334. C2×C4 | C2×C27×C32 | direct (N × H) |
| 347. C22 | C4×C27×C32 | direct (N × H) |
| 351. C4 | C22×C27×C32 | direct (N × H) |
| 356. C2 | C2×C4×C27×C32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C16×C27 | 2 |
| C23×C27×C32 | 2 |
| C2×C4×C27×C32 | 2 |
| C22×C4×C9×C32 | 3 |