Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
6439 conjugacy classes of subgroups; only the 5407 normal subgroups are shown.
This group has 6439 conjugacy classes of subgroups — too many to draw. Only its 5407 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| socle | C24 | C24 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×#41 | C2 | direct (N × H) |
| 3. C22×C4×Q8 | C2 | semidirect |
| 4. C22×C4×D4 | C2 | semidirect |
| 5. C22×#117 | C2 | semidirect |
| 6. C23×C42 | C2 | semidirect |
| 7. C22×#41 | C4 | direct (N × H) |
| 13. C4×#41 | C22 | direct (N × H) |
| 14. C2×C4×Q8 | C22 | semidirect |
| 15. C2×C4×D4 | C22 | semidirect |
| 16. C2×#117 | C22 | semidirect |
| 22. C22×C42 | C22 | semidirect |
| 24. C2×#41 | C2×C4 | direct (N × H) |
| 26. C22×Q8 | C2×C4 | semidirect |
| 28. C22×D4 | C2×C4 | semidirect |
| 31. C23×C4 | C2×C4 | semidirect |
| 38. C4×Q8 | C23 | semidirect |
| 39. C4×D4 | C23 | semidirect |
| 40. #117 | C23 | semidirect |
| 46. C2×C42 | C23 | semidirect |
| 51. #41 | C22×C4 | direct (N × H) |
| 52. C2×Q8 | C22×C4 | semidirect |
| 54. C2×D4 | C22×C4 | semidirect |
| 60. C22×C4 | C22×C4 | semidirect |
| 61. C22×C4 | #41 | direct (N × H) |
| 80. Q8 | C23×C4 | semidirect |
| 81. D4 | C23×C4 | semidirect |
| 86. C2×C4 | C23×C4 | semidirect |
| 87. C2×C4 | C2×#41 | direct (N × H) |
| 94. C22 | C4×#41 | direct (N × H) |
| 102. C4 | C22×#41 | direct (N × H) |
| 104. C2 | C2×C4×#41 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C2×C4×#41 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |
| C22×#117 | 2 |
| C23×C42 | 2 |