Finite Group DatabasebetaFinite Group DatabasebetaOrder 32768.
45940 conjugacy classes of subgroups.
The subgroup lattice has 216 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C322 | C25 |
| socle, prime radical | C25 | C322 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C322 ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C642 | C2 | direct (N × H) |
| 7. C2×C642 | C22 | direct (N × H) |
| 16. C642 | C23 | direct (N × H) |
| 56. C23×C64 | C64 | direct (N × H) |
| 89. C22×C64 | C2×C64 | direct (N × H) |
| 127. C2×C64 | C22×C64 | direct (N × H) |
| 162. C64 | C23×C64 | direct (N × H) |
| 199. C23 | C642 | direct (N × H) |
| 208. C22 | C2×C642 | direct (N × H) |
| 213. C2 | C22×C642 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32×C64 | 2 |
| C22×C642 | 2 |