Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C23×C127 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q64×C127 | C2 | semidirect |
| 4. C127×#1423 | C2 | semidirect |
| 5. C127×#1416 | C2 | semidirect |
| 56. C22×C127 | Q64 | semidirect |
| 63. #13695 | C127 | direct (N × H) |
| 68. C2×Q64 | C2×C127 | semidirect |
| 71. #1423 | C2×C127 | semidirect |
| 72. #1416 | C2×C127 | semidirect |
| 73. C127 | #13695 | direct (N × H) |
| 124. C22 | Q64×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q64×C127 | 2 |
| C22×C32×C127 | 2 |
| C127×#2216 | 2 |
| C127×#1423 | 2 |
| C127×#1416 | 2 |
| #13695 | 127 |