Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
218 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23×C127 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 39. Q16×C127 | Q16 | semidirect |
| 79. #12446 | C127 | direct (N × H) |
| 88. C127 | #12446 | direct (N × H) |
| 122. Q16 | Q16×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2225 | 2 |
| C127×#1713 | 2 |
| C8×Q16×C127 | 2 |
| C127×#1370 | 2 |
| C127×#1364 | 2 |
| #12446 | 127 |