Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
222 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C127 | C22⋊C4 |
| commutator | C22 | C4×C16×C127 |
| Frattini, cosocle | C2×C4×C8 | C22×C127 |
| socle | C22×C127 | #336 |
| prime radical | C23×C127 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 30. C2×C8×C127 | C16 | semidirect |
| 64. #7195 | C127 | direct (N × H) |
| 72. C127 | #7195 | direct (N × H) |
| 100. C2×C8 | C16×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2162 | 2 |
| C2×#322×C127 | 2 |
| #7195 | 127 |