Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
499 conjugacy classes of subgroups; only the 291 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 291 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 | C2×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2875 | C2 | semidirect |
| 4. C4×#108 | C2 | semidirect |
| 5. #1772 | C2 | semidirect |
| 6. #1765 | C2 | semidirect |
| 7. #1759 | C2 | semidirect |
| 8. #1726 | C2 | semidirect |
| 12. #443 | C4 | semidirect |
| 13. C4×Q16 | C4 | semidirect |
| 14. C4×SD16 | C4 | semidirect |
| 15. C4×D8 | C4 | semidirect |
| 38. #131 | C2×C4 | semidirect |
| 53. #108 | C2×C4 | semidirect |
| 56. #104 | C2×C4 | semidirect |
| 69. 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×#443 | 2 |
| C4×#135 | 2 |
| #2875 | 2 |
| C4×#108 | 2 |
| #1772 | 2 |
| #1765 | 2 |
| #1759 | 2 |
| #1726 | 2 |