Finite Group DatabasebetaFinite Group DatabasebetaOrder 8224.
62 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16861628 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #16861626 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16861619 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #14523346 | C22 | 1 | 1 | yes | characteristic |
| 5 | #14523346 | C22 | 1 | 3 | yes | — |
| 6 | D1028 | C4 | 1 | 3 | yes | — |
| 7 | C4×D257 | C22 | 1 | 3 | yes | — |
| 8 | Q8×C257 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×D257 | C2×C4 | 1 | 3 | yes | — |
| 10 | C4×C257 | C2×C4 | 1 | 3 | yes | — |
| 11 | Dic257 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C257 | C22×C4 | 1 | 1 | yes | characteristic |
| 13 | D257 | — | 2 | 3 | no | — |
| 14 | C257 | #131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | #131 | — | 257 | 1 | no | maximal, Sylow |
| 16 | #41 | — | 257 | 1 | no | — |
| 17 | M16 | — | 257 | 3 | no | — |
| 18 | C2×C8 | — | 257 | 3 | no | — |
| 19 | Q8 | #11759967 | 1 | 1 | yes | characteristic |
| 20 | D4 | — | 257 | 3 | no | — |
| 21 | C2×C4 | — | 257 | 3 | no | — |
| 22 | C8 | — | 257 | 1 | no | — |
| 23 | C8 | — | 257 | 3 | no | — |
| 24 | C22 | — | 257 | 3 | no | — |
| 25 | C4 | — | 257 | 1 | no | — |
| 26 | C4 | C2×#11759967 | 1 | 3 | yes | — |
| 27 | C2 | — | 514 | 3 | no | — |
| 28 | C2 | C22×#11759967 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#11759967 |
| commutator, socle | C2×C257 | C22×C4 |
| Fitting | Q8×C257 | C4 |
| hypercenter | Q8 | #11759967 |
| hypocenter | C257 | #131 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16861628 | C2 | semidirect |
| 2. #16861626 | C2 | semidirect |
| 14. C257 | #131 | semidirect |