Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
298 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 126 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 | C22×C127 | C2×D16 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×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 |
|---|---|---|
| 2. C22×C32×C127 | C2 | semidirect |
| 3. C127×#2158 | C2 | semidirect |
| 4. C127×#1416 | C2 | semidirect |
| 5. C127×#1415 | C2 | semidirect |
| 8. #508×C127 | C4 | semidirect |
| 87. #13615 | C127 | direct (N × H) |
| 93. C22×C32 | C2×C127 | semidirect |
| 94. #2158 | C2×C127 | semidirect |
| 95. #1416 | C2×C127 | semidirect |
| 96. #1415 | C2×C127 | semidirect |
| 97. C127 | #13615 | direct (N × H) |
| 100. #508 | C4×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#508×C127 | 2 |
| C22×C32×C127 | 2 |
| C127×#2158 | 2 |
| C127×#1416 | 2 |
| C127×#1415 | 2 |
| #13615 | 127 |