Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
521 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #1773 | C2 | semidirect |
| 6. #1772 | C2 | semidirect |
| 13. #430 | C4 | semidirect |
| 15. #428 | C4 | semidirect |
| 20. C4×M16 | C4 | semidirect |
| 52. #108 | C2×C4 | semidirect |
| 53. #107 | C2×C4 | semidirect |
| 54. #106 | C2×C4 | semidirect |
| 56. #97 | C2×C4 | semidirect |
| 58. C4×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#430 | 2 |
| #2911 | 2 |
| #2873 | 2 |
| C2×C4×M16 | 2 |
| #1773 | 2 |
| #1772 | 2 |
| C2×#339 | 2 |