Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1296 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 517 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 | #543×C127 | C2 |
| prime radical | C24×D127 | C22 |
| hypercenter | #543 | D127 |
| hypocenter | C127 | C2×#543 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22550502 | C2 | semidirect |
| 2. #22550501 | C2 | semidirect |
| 3. C22⋊Q8×D127 | C2 | semidirect |
| 5. #543×C127 | C2 | semidirect |
| 7. C2×Q8×D127 | C22 | semidirect |
| 9. #16861189 | C22 | semidirect |
| 10. #16861188 | C22 | semidirect |
| 11. #16861187 | C22 | semidirect |
| 12. #16861186 | C22 | semidirect |
| 13. #16861185 | C22 | semidirect |
| 14. C4⋊C4×D127 | C22 | semidirect |
| 19. C22⋊Q8×C127 | C22 | semidirect |
| 22. C23×D127 | Q8 | semidirect |
| 25. C2×Dic254 | C23 | semidirect |
| 31. C4⋊Dic127 | C23 | semidirect |
| 32. Dic127⋊C4 | C23 | semidirect |
| 36. C2×Q8×C127 | C23 | semidirect |
| 38. C4⋊C4×C127 | C23 | semidirect |
| 44. C22×D127 | C2×Q8 | semidirect |
| 53. C23×C127 | C2×Q8 | semidirect |
| 74. D127 | #543 | direct (N × H) |
| 76. C127 | C2×#543 | semidirect |
| 79. #543 | D127 | direct (N × H) |
| 90. C22⋊Q8 | C2×D127 | semidirect |
| 104. C2×Q8 | C22×D127 | semidirect |
| 112. C4⋊C4 | C22×D127 | semidirect |
| 124. C23 | Q8×D127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22550502 | 2 |
| #22550501 | 2 |
| C22⋊Q8×D127 | 2 |
| C2×C22⋊C4×D127 | 2 |
| #543×C127 | 2 |
| C2×#543 | 127 |