Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1975 conjugacy classes of subgroups; only the 1631 normal subgroups are shown.
This group has 1975 conjugacy classes of subgroups — too many to draw. Only its 1631 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×M16 | C2 | direct (N × H) |
| 4. C22×#97 | C2 | semidirect |
| 5. C22×C4×C8 | C2 | semidirect |
| 8. C22×M16 | C4 | direct (N × H) |
| 16. C4×M16 | C22 | direct (N × H) |
| 17. C2×#97 | C22 | semidirect |
| 18. C2×C4×C8 | C22 | semidirect |
| 29. C2×M16 | C2×C4 | direct (N × H) |
| 31. C22×C8 | C2×C4 | semidirect |
| 40. #97 | C23 | semidirect |
| 41. C4×C8 | C23 | semidirect |
| 50. C22×C4 | M16 | direct (N × H) |
| 61. M16 | C22×C4 | direct (N × H) |
| 62. C2×C8 | C22×C4 | semidirect |
| 81. C2×C4 | C2×M16 | direct (N × H) |
| 89. C8 | C23×C4 | semidirect |
| 92. C22 | C4×M16 | direct (N × H) |
| 100. C4 | C22×M16 | direct (N × H) |
| 103. C2 | C2×C4×M16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C23×C42 | 2 |
| C2×C4×M16 | 2 |
| C22×#97 | 2 |
| C22×C4×C8 | 2 |