Finite Group DatabasebetaFinite Group DatabasebetaOrder 8224.
68 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16861627 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 2 | C2×#14523346 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C22×Dic257 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #14523346 | C22 | 1 | 4 | yes | complemented |
| 5 | C2×Dic257 | C4 | 1 | 2 | yes | — |
| 6 | C2×Dic257 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×Dic257 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×Dic257 | C22 | 1 | 2 | yes | — |
| 9 | C23×C257 | C4 | 1 | 1 | yes | characteristic |
| 10 | C22×C257 | — | 2 | 2 | no | — |
| 11 | C22×C257 | C2×C4 | 1 | 2 | yes | — |
| 12 | C22×C257 | C2×C4 | 1 | 1 | yes | characteristic |
| 13 | Dic257 | C2×C4 | 1 | 1 | yes | characteristic |
| 14 | Dic257 | C2×C4 | 1 | 2 | yes | — |
| 15 | Dic257 | C23 | 1 | 1 | yes | characteristic |
| 16 | C2×C257 | — | 2 | 2 | no | — |
| 17 | C2×C257 | M16 | 1 | 2 | yes | complemented |
| 18 | C2×C257 | C22×C4 | 1 | 1 | yes | characteristic |
| 19 | C257 | C2×M16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | C2×M16 | — | 257 | 1 | no | maximal, Sylow |
| 21 | C22×C4 | — | 257 | 1 | no | — |
| 22 | M16 | — | 257 | 4 | no | — |
| 23 | C2×C8 | — | 257 | 2 | no | — |
| 24 | C23 | #11759967 | 1 | 1 | yes | characteristic |
| 25 | C2×C4 | — | 257 | 2 | no | — |
| 26 | C2×C4 | — | 257 | 2 | no | — |
| 27 | C2×C4 | — | 257 | 1 | no | — |
| 28 | C2×C4 | — | 257 | 1 | no | — |
| 29 | C8 | — | 257 | 4 | no | — |
| 30 | C22 | — | 2 | 2 | no | — |
| 31 | C22 | C2×#11759967 | 1 | 2 | yes | — |
| 32 | C22 | C2×#11759967 | 1 | 1 | yes | characteristic |
| 33 | C4 | — | 257 | 1 | no | — |
| 34 | C4 | — | 257 | 2 | no | — |
| 35 | C4 | — | 257 | 1 | no | — |
| 36 | C2 | — | 2 | 2 | no | — |
| 37 | C2 | #16861627 | 1 | 2 | yes | direct product factor |
| 38 | C2 | C22×#11759967 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#11759967 |
| commutator | C2×C257 | C22×C4 |
| Frattini | C2 | C22×#11759967 |
| Fitting | C23×C257 | C4 |
| socle | C22×C257 | C2×C4 |
| hypercenter | C23 | #11759967 |
| hypocenter | C257 | C2×M16 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16861627 | C2 | direct (N × H) |
| 2. C2×#14523346 | C2 | semidirect |
| 4. #14523346 | C22 | semidirect |
| 17. C2×C257 | M16 | semidirect |
| 19. C257 | C2×M16 | semidirect |
| 37. C2 | #16861627 | direct (N × H) |