Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
220 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 98 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 | C2×D8 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C22×C127 | C4×D8 |
| prime radical | C2×D8×C127 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ C82×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C127×#1591 | C2 | semidirect |
| 4. C127×#1371 | C2 | semidirect |
| 5. C127×#1332 | C2 | semidirect |
| 7. C127×#1311 | C2 | semidirect |
| 67. C16×C127 | M16 | semidirect |
| 101. #12084 | C127 | direct (N × H) |
| 107. #1591 | C2×C127 | semidirect |
| 109. #1371 | C2×C127 | semidirect |
| 110. #1332 | C2×C127 | semidirect |
| 112. #1311 | C2×C127 | semidirect |
| 113. C127 | #12084 | direct (N × H) |
| 173. C16 | M16×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×SD32×C127 | 2 |
| C127×#1591 | 2 |
| C127×#1589 | 2 |
| C127×#1371 | 2 |
| C127×#1332 | 2 |
| C127×#1331 | 2 |
| C127×#1311 | 2 |
| #12084 | 127 |