Finite Group DatabasebetaFinite Group DatabasebetaOrder 32768.
42 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q16384 | C2 | 1 | 2 | yes | maximal |
| 2 | Q8192 | — | 2 | 2 | no | — |
| 3 | Q4096 | — | 4 | 2 | no | — |
| 4 | Q2048 | — | 8 | 2 | no | — |
| 5 | Q1024 | — | 16 | 2 | no | — |
| 6 | C16384 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C8192 | C22 | 1 | 1 | yes | characteristic |
| 8 | C4096 | D4 | 1 | 1 | yes | characteristic |
| 9 | C2048 | D8 | 1 | 1 | yes | characteristic |
| 10 | C1024 | D16 | 1 | 1 | yes | characteristic |
| 11 | Q512 | — | 32 | 2 | no | — |
| 12 | C512 | D32 | 1 | 1 | yes | characteristic |
| 13 | Q256 | — | 64 | 2 | no | — |
| 14 | C256 | D64 | 1 | 1 | yes | characteristic |
| 15 | Q128 | — | 128 | 2 | no | — |
| 16 | C128 | D128 | 1 | 1 | yes | characteristic |
| 17 | Q64 | — | 256 | 2 | no | — |
| 18 | C64 | D256 | 1 | 1 | yes | characteristic |
| 19 | Q32 | — | 512 | 2 | no | — |
| 20 | C32 | D512 | 1 | 1 | yes | characteristic |
| 21 | Q16 | — | 1024 | 2 | no | — |
| 22 | C16 | D1024 | 1 | 1 | yes | characteristic |
| 23 | Q8 | — | 2048 | 2 | no | — |
| 24 | C8 | D2048 | 1 | 1 | yes | characteristic |
| 25 | C4 | — | 4096 | 2 | no | — |
| 26 | C4 | D4096 | 1 | 1 | yes | characteristic |
| 27 | C2 | D8192 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2 | D8192 |
| commutator, Frattini, cosocle | C8192 | C22 |
| LCS | G ▷ C8192 ▷ C4096 ▷ C2048 ▷ C1024 ▷ C512 ▷ C256 ▷ C128 ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ C128 ◁ C256 ◁ C512 ◁ C1024 ◁ C2048 ◁ C4096 ◁ C8192 ◁ G | |
| LECS | G ▷ C8192 ▷ C4096 ▷ C2048 ▷ C1024 ▷ C512 ▷ C256 ▷ C128 ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q16384 | 2 |
| C16384 | 2 |