Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
404 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C251 |
| prime radical | C2×D8×C251 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C4×Q8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×Q8×C251 | C2 | semidirect |
| 2. #500×C251 | C2 | semidirect |
| 3. #474×C251 | C2 | semidirect |
| 4. C8×Q8×C251 | C2 | semidirect |
| 22. D8×C251 | Q8 | direct (N × H) |
| 32. Q8×C251 | D8 | direct (N × H) |
| 34. D4×C251 | C2×Q8 | semidirect |
| 41. C8×C251 | C2×Q8 | semidirect |
| 55. C251 | Q8×D8 | direct (N × H) |
| 56. Q8×D8 | C251 | direct (N × H) |
| 57. D4×Q8 | C2×C251 | semidirect |
| 58. #500 | C2×C251 | semidirect |
| 59. #474 | C2×C251 | semidirect |
| 60. C8×Q8 | C2×C251 | semidirect |
| 78. D8 | Q8×C251 | direct (N × H) |
| 88. Q8 | D8×C251 | direct (N × H) |
| 90. D4 | C2×Q8×C251 | semidirect |
| 97. C8 | C2×Q8×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×Q8×C251 | 2 |
| #500×C251 | 2 |
| #474×C251 | 2 |
| C8×Q8×C251 | 2 |
| C4×D8×C251 | 2 |
| Q8×D8 | 251 |