Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1123 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 1123 conjugacy classes of subgroups — too many to draw. Only its 407 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×M16 | C2 | semidirect |
| 4. #1749 | C2 | semidirect |
| 5. C4×C22⋊C8 | C2 | semidirect |
| 6. #1728 | C2 | semidirect |
| 17. #407 | C4 | direct (N × H) |
| 20. C2×#97 | C22 | semidirect |
| 21. C2×C4×C8 | C22 | semidirect |
| 22. #336 | C22 | semidirect |
| 39. C2×M16 | C2×C4 | semidirect |
| 54. C22⋊C8 | C2×C4 | semidirect |
| 82. C22×C4 | M16 | semidirect |
| 96. C2×C8 | C22×C4 | semidirect |
| 152. C22 | C4×M16 | semidirect |
| 161. C4 | #407 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C42 | 2 |
| C2×#407 | 2 |
| C2×C4×M16 | 2 |
| #1749 | 2 |
| C4×C22⋊C8 | 2 |
| #1728 | 2 |