Finite Group DatabasebetaFinite Group DatabasebetaOrder 32512.
212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | C2×D8 |
| commutator | C8 | C22×C8×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C2×C127 | C8×D8 |
| prime radical | #508×C127 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C2×C16×C127 ◁ C4×C16×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C127×#2170 | C2 | semidirect |
| 3. C127×#1373 | C2 | semidirect |
| 4. C127×#1335 | C2 | semidirect |
| 5. C127×#1311 | C2 | semidirect |
| 18. Q32×C127 | C8 | semidirect |
| 19. SD32×C127 | C8 | semidirect |
| 20. D16×C127 | C8 | semidirect |
| 67. #12047 | C127 | direct (N × H) |
| 72. #2170 | C2×C127 | semidirect |
| 73. #1373 | C2×C127 | semidirect |
| 74. #1335 | C2×C127 | semidirect |
| 75. #1311 | C2×C127 | semidirect |
| 76. C127 | #12047 | direct (N × H) |
| 89. Q32 | C8×C127 | semidirect |
| 90. SD32 | C8×C127 | semidirect |
| 91. D16 | C8×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127×#2178 | 2 |
| C127×#2170 | 2 |
| C127×#1373 | 2 |
| C127×#1335 | 2 |
| C127×#1311 | 2 |
| #12047 | 127 |