Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
494 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #331 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | C2×#493×C127 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#364×C127 | C2 | semidirect |
| 3. C127×#1357 | C2 | direct (N × H) |
| 12. #364×C127 | C22 | semidirect |
| 88. C2×#1357 | C127 | direct (N × H) |
| 92. C2×C127 | #1357 | direct (N × H) |
| 95. C2×#364 | C2×C127 | semidirect |
| 96. #1357 | C2×C127 | direct (N × H) |
| 97. C127 | C2×#1357 | direct (N × H) |
| 106. #364 | C22×C127 | semidirect |
| 185. C2 | C127×#1357 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#493×C127 | 2 |
| C2×#364×C127 | 2 |
| C127×#1357 | 2 |
| C2×#1357 | 127 |