Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
654 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 261 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#430 | C2 | semidirect |
| 3. #2911 | C2 | semidirect |
| 5. #1814 | C2 | semidirect |
| 6. C2×#344 | C2 | semidirect |
| 9. #568 | C4 | semidirect |
| 19. C2×#108 | C22 | semidirect |
| 21. #427 | C22 | semidirect |
| 25. #344 | C22 | semidirect |
| 41. #131 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#568 | 2 |
| C2×#430 | 2 |
| #2911 | 2 |
| C2×#421 | 2 |
| #1814 | 2 |
| C2×#344 | 2 |