Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
202 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 202 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 | C22×C127 | C2×D16 |
| prime radical | C2×D16×C127 | C22 |
| 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 |
|---|---|---|
| 2. C4×Q32×C127 | C2 | semidirect |
| 4. C127×#1424 | C2 | semidirect |
| 5. C127×#1416 | C2 | semidirect |
| 7. C4×C32×C127 | C2 | semidirect |
| 12. SD64×C127 | C4 | direct (N × H) |
| 25. Q32×C127 | C2×C4 | semidirect |
| 36. C32×C127 | C2×C4 | semidirect |
| 78. C4×C127 | SD64 | direct (N × H) |
| 81. C4×SD64 | C127 | direct (N × H) |
| 87. C4×Q32 | C2×C127 | semidirect |
| 89. #1424 | C2×C127 | semidirect |
| 90. #1416 | C2×C127 | semidirect |
| 92. C4×C32 | C2×C127 | semidirect |
| 93. C127 | C4×SD64 | direct (N × H) |
| 98. SD64 | C4×C127 | direct (N × H) |
| 111. Q32 | C2×C4×C127 | semidirect |
| 122. C32 | C2×C4×C127 | semidirect |
| 164. C4 | SD64×C127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD64×C127 | 2 |
| C4×Q32×C127 | 2 |
| C4×D16×C127 | 2 |
| C127×#1424 | 2 |
| C127×#1416 | 2 |
| C127×#1415 | 2 |
| C4×C32×C127 | 2 |
| C4×SD64 | 127 |