Finite Group DatabasebetaFinite Group DatabasebetaOrder 32896.
408 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C257 | C23 |
| commutator | C23 | C22×C4×C257 |
| Frattini, cosocle | C22×C4 | C23×C257 |
| socle | C23×C257 | C22×C4 |
| prime radical | C22×D4×C257 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C257 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×M16×C257 | C2 | semidirect |
| 3. C2×C22⋊C8×C257 | C2 | semidirect |
| 4. C2×C22⋊C8×C257 | C2 | semidirect |
| 5. #336×C257 | C2 | semidirect |
| 45. C2×C8×C257 | D4 | semidirect |
| 60. C23×C257 | M16 | semidirect |
| 77. C2×C4×C257 | M16 | semidirect |
| 113. C257 | #1870 | direct (N × H) |
| 114. #1870 | C257 | direct (N × H) |
| 115. C22×M16 | C2×C257 | semidirect |
| 117. C2×C22⋊C8 | C2×C257 | semidirect |
| 118. C2×C22⋊C8 | C2×C257 | semidirect |
| 119. #336 | C2×C257 | semidirect |
| 159. C2×C8 | D4×C257 | semidirect |
| 174. C23 | M16×C257 | semidirect |
| 191. C2×C4 | M16×C257 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C257 | 2 |
| C2×C4×D4×C257 | 2 |
| C2×C22⋊C8×C257 | 2 |
| #336×C257 | 2 |
| #1870 | 257 |