Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
994 conjugacy classes of subgroups; only the 382 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 382 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 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| prime radical | C2×#127×C127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#498×C127 | C2 | semidirect |
| 3. C127×#1705 | C2 | direct (N × H) |
| 4. C2×C82×C127 | C2 | semidirect |
| 9. #498×C127 | C22 | semidirect |
| 14. C82×C127 | C22 | semidirect |
| 40. C2×C8×C127 | SD16 | semidirect |
| 58. C8×C127 | C2×SD16 | semidirect |
| 66. C2×#1705 | C127 | direct (N × H) |
| 68. C2×C127 | #1705 | direct (N × H) |
| 70. C2×#498 | C2×C127 | semidirect |
| 72. #1705 | C2×C127 | direct (N × H) |
| 73. C2×C82 | C2×C127 | semidirect |
| 74. C127 | C2×#1705 | direct (N × H) |
| 79. #498 | C22×C127 | semidirect |
| 84. C82 | C22×C127 | semidirect |
| 110. C2×C8 | SD16×C127 | semidirect |
| 128. C8 | C2×SD16×C127 | semidirect |
| 137. C2 | C127×#1705 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#498×C127 | 2 |
| C2×#492×C127 | 2 |
| C127×#1705 | 2 |
| C2×C82×C127 | 2 |
| C2×#1705 | 127 |