Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C251 | D8 |
| commutator | C8 | C22×C4×C251 |
| Frattini, cosocle | C2×C8 | C23×C251 |
| socle | C22×C251 | C2×D8 |
| prime radical | C2×D8×C251 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ C42×C251 ◁ C4×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×Q16×C251 | C2 | semidirect |
| 4. #367×C251 | C2 | semidirect |
| 5. #358×C251 | C2 | semidirect |
| 7. C4×C16×C251 | C2 | semidirect |
| 12. SD32×C251 | C4 | direct (N × H) |
| 24. Q16×C251 | C2×C4 | semidirect |
| 36. C16×C251 | C2×C4 | semidirect |
| 60. C4×C251 | SD32 | direct (N × H) |
| 67. C251 | C4×SD32 | direct (N × H) |
| 68. C4×SD32 | C251 | direct (N × H) |
| 70. C4×Q16 | C2×C251 | semidirect |
| 72. #367 | C2×C251 | semidirect |
| 73. #358 | C2×C251 | semidirect |
| 75. C4×C16 | C2×C251 | semidirect |
| 80. SD32 | C4×C251 | direct (N × H) |
| 92. Q16 | C2×C4×C251 | semidirect |
| 104. C16 | C2×C4×C251 | semidirect |
| 128. C4 | SD32×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD32×C251 | 2 |
| C4×Q16×C251 | 2 |
| C4×D8×C251 | 2 |
| #367×C251 | 2 |
| #358×C251 | 2 |
| #357×C251 | 2 |
| C4×C16×C251 | 2 |
| C4×SD32 | 251 |