Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 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 | C2×2−1+4×C127 | C2 |
| hypercenter | C2×2−1+4 | D127 |
| hypocenter | C127 | #3487 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14523249 | C2 | semidirect |
| 2. #22550545 | C2 | semidirect |
| 3. #22550537 | C2 | semidirect |
| 4. #22550440 | C2 | semidirect |
| 5. #22550383 | C2 | semidirect |
| 6. #22550354 | C2 | semidirect |
| 7. C2×2−1+4×C127 | C2 | semidirect |
| 12. #16861205 | C22 | semidirect |
| 13. #16861198 | C22 | semidirect |
| 15. #16861185 | C22 | semidirect |
| 16. #16861183 | C22 | semidirect |
| 17. #16861180 | C22 | semidirect |
| 19. Q8×Dic127 | C22 | semidirect |
| 20. C4×C22⋊D127 | C22 | semidirect |
| 24. C2×#41×C127 | C22 | semidirect |
| 26. C22×Q8×C127 | C22 | semidirect |
| 44. C2×Q8×C127 | D4 | semidirect |
| 88. C127 | #3487 | semidirect |
| 89. C2×2−1+4 | D127 | semidirect |
| 100. C2×#41 | C2×D127 | semidirect |
| 102. C22×Q8 | C2×D127 | semidirect |
| 120. C2×Q8 | C22⋊D127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14523249 | 2 |
| #22550545 | 2 |
| #22550537 | 2 |
| #22550440 | 2 |
| #22550383 | 2 |
| #22550354 | 2 |
| C2×2−1+4×C127 | 2 |
| #3487 | 127 |