Finite Group DatabasebetaFinite Group DatabasebetaOrder 8128.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic1016 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C127⋊C32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D16×C127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic508 | — | 2 | 1 | no | — |
| 5 | D8×C127 | — | 2 | 1 | no | — |
| 6 | C16×C127 | C22 | 1 | 1 | yes | characteristic |
| 7 | Dic254 | — | 4 | 1 | no | — |
| 8 | D4×C127 | — | 4 | 1 | no | — |
| 9 | C8×C127 | D4 | 1 | 1 | yes | characteristic |
| 10 | C22×C127 | — | 8 | 1 | no | — |
| 11 | C4×C127 | D8 | 1 | 1 | yes | characteristic |
| 12 | Dic127 | — | 8 | 1 | no | — |
| 13 | C2×C127 | — | 16 | 1 | no | — |
| 14 | C2×C127 | D16 | 1 | 1 | yes | characteristic |
| 15 | C127 | SD64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | SD64 | — | 127 | 1 | no | maximal, Sylow |
| 17 | Q32 | — | 127 | 1 | no | — |
| 18 | D16 | D127 | 1 | 1 | yes | characteristic |
| 19 | C32 | — | 127 | 1 | no | — |
| 20 | Q16 | — | 254 | 1 | no | — |
| 21 | D8 | — | 2 | 1 | no | — |
| 22 | C16 | C2×D127 | 1 | 1 | yes | characteristic |
| 23 | Q8 | — | 508 | 1 | no | — |
| 24 | D4 | — | 4 | 1 | no | — |
| 25 | C8 | C22⋊D127 | 1 | 1 | yes | characteristic |
| 26 | C22 | — | 8 | 1 | no | — |
| 27 | C4 | — | 1016 | 1 | no | — |
| 28 | C4 | #14523233 | 1 | 1 | yes | characteristic |
| 29 | C2 | — | 16 | 1 | no | — |
| 30 | C2 | #16861150 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16861150 |
| commutator | C16×C127 | C22 |
| Frattini | C16 | C2×D127 |
| Fitting | D16×C127 | C2 |
| socle | C2×C127 | D16 |
| hypercenter | D16 | D127 |
| hypocenter | C127 | SD64 |
| LCS | G ▷ C16×C127 ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic1016 | C2 | semidirect |
| 2. C127⋊C32 | C2 | semidirect |
| 15. C127 | SD64 | semidirect |