Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
1266 conjugacy classes of subgroups; only the 406 normal subgroups are shown.
This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 406 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C127 | C23 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C23×C127 | C23×C4 |
| prime radical | C127×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C8×D4×C127 | C2 | semidirect |
| 3. C127×#1955 | C2 | semidirect |
| 4. C127×#1843 | C2 | semidirect |
| 5. C127×#1726 | C2 | semidirect |
| 30. C4×C8×C127 | D4 | semidirect |
| 73. #21683 | C127 | direct (N × H) |
| 77. C2×C8×D4 | C2×C127 | semidirect |
| 79. #1955 | C2×C127 | semidirect |
| 80. #1843 | C2×C127 | semidirect |
| 81. #1726 | C2×C127 | semidirect |
| 82. C127 | #21683 | direct (N × H) |
| 107. C4×C8 | D4×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D4×C127 | 2 |
| C127×#2397 | 2 |
| C127×#1955 | 2 |
| C127×#1843 | 2 |
| C127×#1726 | 2 |
| #21683 | 127 |