Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
164 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 164 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, prime radical | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×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 |
|---|---|---|
| 19. Q32×C127 | Q8 | semidirect |
| 63. #13704 | C127 | direct (N × H) |
| 72. C127 | #13704 | direct (N × H) |
| 86. Q32 | Q8×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2252 | 2 |
| C4×Q32×C127 | 2 |
| C127×#1424 | 2 |
| C127×C4⋊C32 | 2 |
| C127×#1416 | 2 |
| #13704 | 127 |