Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
342 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 342 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 | C23×C127 | D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C22×D16×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×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. C2×#366×C127 | C2 | semidirect |
| 4. C127×#1415 | C2 | direct (N × H) |
| 8. C2×D16×C127 | C4 | semidirect |
| 13. C2×C32×C127 | C22 | semidirect |
| 15. #366×C127 | C22 | semidirect |
| 25. D16×C127 | C2×C4 | semidirect |
| 81. C2×#1415 | C127 | direct (N × H) |
| 83. C2×C127 | #1415 | direct (N × H) |
| 88. C22×C32 | C2×C127 | semidirect |
| 89. C2×#366 | C2×C127 | semidirect |
| 90. #1415 | C2×C127 | direct (N × H) |
| 91. C127 | C2×#1415 | direct (N × H) |
| 95. C2×D16 | C4×C127 | semidirect |
| 100. C2×C32 | C22×C127 | semidirect |
| 102. #366 | C22×C127 | semidirect |
| 112. D16 | C2×C4×C127 | semidirect |
| 169. C2 | C127×#1415 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D16×C127 | 2 |
| C22×C32×C127 | 2 |
| C2×#366×C127 | 2 |
| C127×#1415 | 2 |
| C2×#1415 | 127 |