Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 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 | Q82×C127 | C2 |
| prime radical | C2×#14523249 | C22 |
| hypercenter | Q82 | D127 |
| hypocenter | C127 | Q8×#41 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22550605 | C2 | semidirect |
| 2. #22550587 | C2 | semidirect |
| 3. Q8×D508 | C2 | semidirect |
| 4. #22550431 | C2 | semidirect |
| 6. C4×Q8×D127 | C2 | semidirect |
| 7. Q82×C127 | C2 | semidirect |
| 27. #14523249 | Q8 | direct (N × H) |
| 50. D508 | C2×Q8 | semidirect |
| 56. C4×D127 | C2×Q8 | semidirect |
| 58. Q8×C127 | C2×Q8 | semidirect |
| 59. Q8×C127 | #41 | semidirect |
| 78. C127 | Q8×#41 | semidirect |
| 79. Q82 | D127 | semidirect |
| 129. Q8 | Q8×D127 | semidirect |
| 130. Q8 | #14523249 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22550605 | 2 |
| #22550587 | 2 |
| Q8×D508 | 2 |
| #22550431 | 2 |
| C4×#14523249 | 2 |
| C4×Q8×D127 | 2 |
| Q82×C127 | 2 |
| Q8×#41 | 127 |