Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
306 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | C127×#2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C127×#2252 | C2 | semidirect |
| 3. C127×#2240 | C2 | semidirect |
| 4. C127×#1706 | C2 | semidirect |
| 5. C8×C16×C127 | C2 | semidirect |
| 45. C16×C127 | SD16 | semidirect |
| 57. C8×C127 | D16 | semidirect |
| 69. #12503 | C127 | direct (N × H) |
| 74. #2252 | C2×C127 | semidirect |
| 76. #2240 | C2×C127 | semidirect |
| 77. #1706 | C2×C127 | semidirect |
| 78. C8×C16 | C2×C127 | semidirect |
| 79. C127 | #12503 | direct (N × H) |
| 119. C16 | SD16×C127 | semidirect |
| 131. C8 | D16×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2252 | 2 |
| C127×#2246 | 2 |
| C127×#2240 | 2 |
| C127×#1706 | 2 |
| C8×C16×C127 | 2 |
| #12503 | 127 |