Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C24×D127 |
| commutator, socle, cosocle | C22×C127 | C25 |
| Fitting | #519×C127 | C2 |
| hypercenter | #519 | D127 |
| hypocenter | C127 | #3486 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16861257 | C2 | semidirect |
| 2. C22×Q8×D127 | C2 | semidirect |
| 3. Q8×D508 | C2 | semidirect |
| 4. #22550538 | C2 | semidirect |
| 5. #22550452 | C2 | semidirect |
| 6. #22550388 | C2 | semidirect |
| 7. #519×C127 | C2 | semidirect |
| 9. C2×#14523249 | C22 | semidirect |
| 14. C2×Q8×D127 | C22 | semidirect |
| 16. #16861209 | C22 | semidirect |
| 17. #16861197 | C22 | semidirect |
| 20. #16861189 | C22 | semidirect |
| 25. C4×Q8×C127 | C22 | semidirect |
| 45. C2×Q8×C127 | D4 | semidirect |
| 78. C2×D127 | 2−1+4 | semidirect |
| 89. C127 | #3486 | semidirect |
| 96. #519 | D127 | semidirect |
| 113. C4×Q8 | C2×D127 | semidirect |
| 124. C2×Q8 | D508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16861257 | 2 |
| C22×Q8×D127 | 2 |
| Q8×D508 | 2 |
| #22550538 | 2 |
| #22550452 | 2 |
| #22550388 | 2 |
| #519×C127 | 2 |
| #3486 | 127 |