Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
555 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 555 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#441 | C2 | semidirect |
| 4. C2×C4×SD16 | C2 | semidirect |
| 5. #2326 | C2 | semidirect |
| 7. #1971 | C2 | semidirect |
| 8. #1968 | C2 | semidirect |
| 9. #1959 | C2 | semidirect |
| 10. #1844 | C2 | semidirect |
| 11. #1726 | C2 | semidirect |
| 18. #495 | C4 | semidirect |
| 46. C2×Q16 | C2×C4 | semidirect |
| 49. C2×SD16 | C2×C4 | semidirect |
| 58. #124 | C2×C4 | semidirect |
| 85. 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×#495 | 2 |
| C2×#442 | 2 |
| C2×#441 | 2 |
| C2×C4×SD16 | 2 |
| #2326 | 2 |
| #1972 | 2 |
| #1971 | 2 |
| #1968 | 2 |
| #1959 | 2 |
| #1844 | 2 |
| #1726 | 2 |