Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic1952 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21443638 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C61×D64 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic976 | — | 2 | 1 | no | — |
| 5 | C61×D32 | — | 2 | 1 | no | — |
| 6 | C61×C64 | C22 | 1 | 1 | yes | characteristic |
| 7 | Dic488 | — | 4 | 1 | no | — |
| 8 | D16×C61 | — | 4 | 1 | no | — |
| 9 | C32×C61 | D4 | 1 | 1 | yes | characteristic |
| 10 | D8×C61 | — | 8 | 1 | no | — |
| 11 | Dic244 | — | 8 | 1 | no | — |
| 12 | C16×C61 | D8 | 1 | 1 | yes | characteristic |
| 13 | D4×C61 | — | 16 | 1 | no | — |
| 14 | Dic122 | — | 16 | 1 | no | — |
| 15 | C8×C61 | D16 | 1 | 1 | yes | characteristic |
| 16 | SD256 | — | 61 | 1 | no | maximal, Sylow |
| 17 | C22×C61 | — | 32 | 1 | no | — |
| 18 | C4×C61 | D32 | 1 | 1 | yes | characteristic |
| 19 | Dic61 | — | 32 | 1 | no | — |
| 20 | Q128 | — | 61 | 1 | no | — |
| 21 | D64 | D61 | 1 | 1 | yes | characteristic |
| 22 | C128 | — | 61 | 1 | no | — |
| 23 | C2×C61 | — | 64 | 1 | no | — |
| 24 | C2×C61 | D64 | 1 | 1 | yes | characteristic |
| 25 | Q64 | — | 122 | 1 | no | — |
| 26 | D32 | — | 2 | 1 | no | — |
| 27 | C64 | C2×D61 | 1 | 1 | yes | characteristic |
| 28 | C61 | SD256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 29 | Q32 | — | 244 | 1 | no | — |
| 30 | D16 | — | 4 | 1 | no | — |
| 31 | C32 | C22⋊D61 | 1 | 1 | yes | characteristic |
| 32 | Q16 | — | 488 | 1 | no | — |
| 33 | D8 | — | 8 | 1 | no | — |
| 34 | C16 | #11758199 | 1 | 1 | yes | characteristic |
| 35 | Q8 | — | 976 | 1 | no | — |
| 36 | D4 | — | 16 | 1 | no | — |
| 37 | C8 | #14519718 | 1 | 1 | yes | characteristic |
| 38 | C22 | — | 32 | 1 | no | — |
| 39 | C4 | — | 1952 | 1 | no | — |
| 40 | C4 | #16839597 | 1 | 1 | yes | characteristic |
| 41 | C2 | — | 64 | 1 | no | — |
| 42 | C2 | #21444031 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #21444031 |
| commutator, cosocle | C61×C64 | C22 |
| Frattini | C64 | C2×D61 |
| Fitting, prime radical | C61×D64 | C2 |
| socle | C2×C61 | D64 |
| hypercenter | D64 | D61 |
| hypocenter | C61 | SD256 |
| LCS | G ▷ C61×C64 ▷ C32×C61 ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ D64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic1952 | C2 | semidirect |
| 2. #21443638 | C2 | semidirect |
| 28. C61 | SD256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic1952 | 2 |
| #21443638 | 2 |
| C61×D64 | 2 |
| SD256 | 61 |