Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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 | D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle, prime radical | C22×C127 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. Q64×C127 | C4 | direct (N × H) |
| 51. C4×C127 | Q64 | direct (N × H) |
| 54. C4×Q64 | C127 | direct (N × H) |
| 63. C127 | C4×Q64 | direct (N × H) |
| 66. Q64 | C4×C127 | direct (N × H) |
| 109. C4 | Q64×C127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q64×C127 | 2 |
| C4×Q32×C127 | 2 |
| C127×#1423 | 2 |
| C127×#1416 | 2 |
| C4×C32×C127 | 2 |
| C4×Q64 | 127 |