Finite Group DatabasebetaFinite Group DatabasebetaOrder 32768.
18420276 conjugacy classes of subgroups.
The subgroup lattice has 420 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C24×C4×C8 | C26 |
| socle, prime radical | C26 | C24×C4×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C24×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C43×C8×C16 | C4 | direct (N × H) |
| 25. C44×C16 | C8 | direct (N × H) |
| 37. C44×C8 | C16 | direct (N × H) |
| 55. C42×C8×C16 | C42 | direct (N × H) |
| 94. C43×C16 | C4×C8 | direct (N × H) |
| 140. C43×C8 | C4×C16 | direct (N × H) |
| 149. C4×C8×C16 | C43 | direct (N × H) |
| 199. C44 | C8×C16 | direct (N × H) |
| 205. C42×C16 | C42×C8 | direct (N × H) |
| 260. C42×C8 | C42×C16 | direct (N × H) |
| 269. C8×C16 | C44 | direct (N × H) |
| 313. C43 | C4×C8×C16 | direct (N × H) |
| 319. C4×C16 | C43×C8 | direct (N × H) |
| 357. C4×C8 | C43×C16 | direct (N × H) |
| 385. C42 | C42×C8×C16 | direct (N × H) |
| 391. C16 | C44×C8 | direct (N × H) |
| 405. C8 | C44×C16 | direct (N × H) |
| 412. C4 | C43×C8×C16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C44×C82 | 2 |
| C45×C16 | 2 |
| C2×C43×C8×C16 | 2 |