Finite Group DatabasebetaFinite Group DatabasebetaOrder 8048.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D2012 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C8×C503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×D503 | — | 2 | 2 | no | — |
| 4 | C4×C503 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×C503 | D4 | 1 | 1 | yes | characteristic |
| 6 | D503 | — | 4 | 2 | no | — |
| 7 | C503 | D8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | D8 | — | 503 | 1 | no | maximal, Sylow |
| 9 | D4 | — | 503 | 2 | no | — |
| 10 | C8 | D503 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22 | — | 1006 | 2 | no | — |
| 12 | C4 | C2×D503 | 1 | 1 | yes | characteristic |
| 13 | C2 | — | 2012 | 2 | no | — |
| 14 | C2 | D2012 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2012 |
| commutator | C4×C503 | C22 |
| Frattini | C4 | C2×D503 |
| Fitting | C8×C503 | C2 |
| socle | C2×C503 | D4 |
| hypercenter | C8 | D503 |
| hypocenter | C503 | D8 |
| LCS | G ▷ C4×C503 ▷ C2×C503 ▷ C503 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D2012 | C2 | semidirect |
| 2. C8×C503 | C2 | semidirect |
| 7. C503 | D8 | semidirect |
| 10. C8 | D503 | semidirect |