Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
452 conjugacy classes of subgroups; only the 268 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 268 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 | C24 |
| commutator | C22 | C23×C4×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| socle | C22×C251 | C23×C4 |
| prime radical | C22×D4×C251 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #436×C251 | C2 | semidirect |
| 3. #431×C251 | C2 | semidirect |
| 4. #407×C251 | C2 | semidirect |
| 7. C2×M16×C251 | C22 | semidirect |
| 13. C4⋊C8×C251 | C22 | semidirect |
| 15. C22⋊C8×C251 | C22 | semidirect |
| 16. C4×C8×C251 | C22 | semidirect |
| 44. C2×C4×C251 | M16 | semidirect |
| 69. C251 | #2975 | direct (N × H) |
| 70. #2975 | C251 | direct (N × H) |
| 72. #436 | C2×C251 | semidirect |
| 73. #431 | C2×C251 | semidirect |
| 74. #407 | C2×C251 | semidirect |
| 77. C2×M16 | C22×C251 | semidirect |
| 83. C4⋊C8 | C22×C251 | semidirect |
| 85. C22⋊C8 | C22×C251 | semidirect |
| 86. C4×C8 | C22×C251 | semidirect |
| 114. C2×C4 | M16×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D4×C251 | 2 |
| #436×C251 | 2 |
| #431×C251 | 2 |
| #407×C251 | 2 |
| #2975 | 251 |