Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D127 |
| commutator, socle, cosocle | C22×C127 | C25 |
| Frattini | C22 | C24×D127 |
| Fitting | C22×C42×C127 | C2 |
| hypercenter | C22×C42 | D127 |
| hypocenter | C127 | #3444 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22550433 | C2 | semidirect |
| 2. #22550426 | C2 | semidirect |
| 3. #22550410 | C2 | semidirect |
| 4. #22550387 | C2 | semidirect |
| 5. C4×#14523248 | C2 | semidirect |
| 6. C22×C42×C127 | C2 | semidirect |
| 8. #16861209 | C22 | semidirect |
| 9. #16861206 | C22 | semidirect |
| 10. #16861203 | C22 | semidirect |
| 11. #16861201 | C22 | semidirect |
| 12. #16861197 | C22 | semidirect |
| 17. C4×Dic254 | C22 | semidirect |
| 18. C4×D508 | C22 | semidirect |
| 21. #16861166 | C22 | semidirect |
| 22. C42×D127 | C22 | semidirect |
| 60. C2×C4×C127 | #41 | semidirect |
| 77. C127 | #3444 | semidirect |
| 83. C22×C42 | D127 | semidirect |
| 137. C2×C4 | #14523248 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22550433 | 2 |
| #22550426 | 2 |
| #22550410 | 2 |
| #22550387 | 2 |
| C4×#14523248 | 2 |
| C22×C42×C127 | 2 |
| #3444 | 127 |