Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
816 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C251 | D4 |
| commutator | C4 | C23×C4×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#103×C251 | C2 | direct (N × H) |
| 7. C22×Q8×C251 | C4 | semidirect |
| 16. #103×C251 | C22 | direct (N × H) |
| 21. C2×Q8×C251 | C2×C4 | semidirect |
| 41. Q8×C251 | C22×C4 | semidirect |
| 52. C22×C251 | #103 | direct (N × H) |
| 62. C2×C251 | C2×#103 | direct (N × H) |
| 66. C251 | C22×#103 | direct (N × H) |
| 67. C22×#103 | C251 | direct (N × H) |
| 71. C2×#103 | C2×C251 | direct (N × H) |
| 74. C22×Q8 | C4×C251 | semidirect |
| 83. #103 | C22×C251 | direct (N × H) |
| 88. C2×Q8 | C2×C4×C251 | semidirect |
| 108. Q8 | C22×C4×C251 | semidirect |
| 119. C22 | #103×C251 | direct (N × H) |
| 129. C2 | C2×#103×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C251 | 2 |
| C23×C8×C251 | 2 |
| C22×C4⋊C4×C251 | 2 |
| C2×#103×C251 | 2 |
| C22×#103 | 251 |