Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
180 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 62 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×C127 | #99 |
| commutator | C2×C4 | C2×C16×C127 |
| Frattini, cosocle | C2×C4⋊C8 | C22×C127 |
| socle | C22×C127 | #324 |
| prime radical | C23×C127 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4⋊C8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. C42×C127 | C16 | semidirect |
| 68. #7062 | C127 | direct (N × H) |
| 76. C127 | #7062 | direct (N × H) |
| 108. C42 | C16×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#1831 | 2 |
| C127×#1322 | 2 |
| #7062 | 127 |