Finite Group DatabasebetaFinite Group DatabasebetaOrder 16064.
290 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 105 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×D251 |
| commutator, socle, cosocle | C22×C251 | C24 |
| Fitting | C22⋊Q8×C251 | C2 |
| hypercenter | C22⋊Q8 | D251 |
| hypocenter | C251 | #548 |
| LCS | G ▷ C22×C251 ▷ C251 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16860810 | C2 | semidirect |
| 2. C2×Q8×D251 | C2 | semidirect |
| 3. #22548704 | C2 | semidirect |
| 4. #22548701 | C2 | semidirect |
| 5. #22548693 | C2 | semidirect |
| 6. #22548690 | C2 | semidirect |
| 7. #22548688 | C2 | semidirect |
| 8. #22548680 | C2 | semidirect |
| 9. #22548677 | C2 | semidirect |
| 10. C4×C22⋊D251 | C2 | semidirect |
| 11. C22⋊Q8×C251 | C2 | semidirect |
| 40. Dic502 | D4 | semidirect |
| 68. C2×D251 | #41 | semidirect |
| 83. C251 | #548 | semidirect |
| 90. C22⋊Q8 | D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16860810 | 2 |
| C2×Q8×D251 | 2 |
| #22548704 | 2 |
| #22548701 | 2 |
| #22548693 | 2 |
| #22548690 | 2 |
| #22548688 | 2 |
| #22548680 | 2 |
| #22548677 | 2 |
| C4×C22⋊D251 | 2 |
| C22⋊Q8×C251 | 2 |
| #548 | 251 |