Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
557 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×#102 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#504 | C2 | semidirect |
| 3. #2916 | C2 | semidirect |
| 4. #2143 | C2 | semidirect |
| 5. C2×#359 | C2 | semidirect |
| 15. C2×M32 | C22 | semidirect |
| 16. C22×C16 | C22 | semidirect |
| 18. C2×#108 | C22 | semidirect |
| 19. #359 | C22 | semidirect |
| 20. #359 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3584 | 2 |
| C2×#504 | 2 |
| #2916 | 2 |
| #2143 | 2 |
| C2×#359 | 2 |