Finite Group DatabasebetaFinite Group DatabasebetaOrder 8768.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D137 |
| commutator, socle | C22×C137 | C24 |
| Fitting | C22⋊Q8×C137 | C2 |
| hypercenter | C22⋊Q8 | D137 |
| hypocenter | C137 | #536 |
| LCS | G ▷ C22×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q8×D137 | C2 | semidirect |
| 3. #17984087 | C2 | semidirect |
| 4. #17984074 | C2 | semidirect |
| 5. #17984073 | C2 | semidirect |
| 6. #17984070 | C2 | semidirect |
| 7. #17984068 | C2 | semidirect |
| 8. #17984067 | C2 | semidirect |
| 9. #17984064 | C2 | semidirect |
| 10. #17984046 | C2 | semidirect |
| 11. C22⋊Q8×C137 | C2 | semidirect |
| 83. C137 | #536 | semidirect |
| 94. C22⋊Q8 | D137 | semidirect |