Finite Group DatabasebetaFinite Group DatabasebetaOrder 8224.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#14523348 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×#11759967 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14523348 | — | 2 | 2 | no | — |
| 4 | C2×#11759967 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#11759967 | C22 | 1 | 1 | yes | characteristic |
| 6 | C4×D257 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×D257 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | #11759967 | — | 2 | 1 | no | — |
| 9 | #11759967 | D4 | 1 | 1 | yes | characteristic |
| 10 | #11759967 | Q8 | 1 | 1 | yes | characteristic |
| 11 | C4×C257 | C8 | 1 | 1 | yes | characteristic, complemented |
| 12 | Dic257 | C8 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×C257 | C2×C8 | 1 | 1 | yes | characteristic |
| 14 | D257 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 15 | D257 | M16 | 1 | 1 | yes | characteristic |
| 16 | C257 | C4⋊C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C4⋊C8 | — | 257 | 1 | no | maximal, Sylow |
| 18 | C2×C8 | — | 257 | 2 | no | — |
| 19 | C42 | — | 257 | 1 | no | — |
| 20 | C2×C4 | — | 257 | 1 | no | — |
| 21 | C2×C4 | — | 257 | 1 | no | — |
| 22 | C2×C4 | — | 257 | 1 | no | — |
| 23 | C8 | — | 514 | 2 | no | — |
| 24 | C22 | — | 257 | 1 | no | — |
| 25 | C4 | — | 257 | 1 | no | — |
| 26 | C4 | — | 257 | 1 | no | — |
| 27 | C4 | — | 257 | 1 | no | — |
| 28 | C4 | — | 514 | 1 | no | — |
| 29 | C4 | #14523348 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2 | — | 257 | 1 | no | — |
| 31 | C2 | — | 257 | 1 | no | — |
| 32 | C2 | C2×#14523348 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#14523348 |
| commutator, socle | C2×C257 | C2×C8 |
| Fitting | C4×C257 | C8 |
| hypercenter | C4 | #14523348 |
| hypocenter | C257 | C4⋊C8 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C4×C257 | C8 | semidirect |
| 12. Dic257 | C8 | semidirect |
| 16. C257 | C4⋊C8 | semidirect |
| 29. C4 | #14523348 | semidirect |