Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3563 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 3563 conjugacy classes of subgroups — too many to draw. Only its 647 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×SD16 | C2 | semidirect |
| 3. C22×C22⋊Q8 | C2 | semidirect |
| 4. C2×#450 | C2 | direct (N × H) |
| 5. C22×#102 | C2 | semidirect |
| 6. C22×C22⋊C8 | C2 | semidirect |
| 8. C23×Q8 | C22 | semidirect |
| 16. C22×SD16 | C22 | semidirect |
| 17. C23×C8 | C22 | semidirect |
| 19. C2×C22⋊Q8 | C22 | semidirect |
| 21. C22×C4⋊C4 | C22 | semidirect |
| 23. #450 | C22 | direct (N × H) |
| 24. C2×#102 | C22 | semidirect |
| 25. C2×C22⋊C8 | C22 | semidirect |
| 34. C22×Q8 | C23 | semidirect |
| 53. C2×SD16 | C23 | semidirect |
| 55. C22×C8 | C23 | semidirect |
| 57. C22⋊Q8 | C23 | semidirect |
| 60. C2×C4⋊C4 | C23 | semidirect |
| 62. #102 | C23 | semidirect |
| 63. C22⋊C8 | C23 | semidirect |
| 75. C24 | SD16 | semidirect |
| 81. C2×Q8 | C24 | semidirect |
| 106. C2×C8 | C24 | semidirect |
| 109. C4⋊C4 | C24 | semidirect |
| 125. C23 | C2×SD16 | semidirect |
| 156. C22 | #450 | direct (N × H) |
| 159. C22 | C22×SD16 | semidirect |
| 169. C2 | C2×#450 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D4 | 2 |
| C23×SD16 | 2 |
| C22×C22⋊Q8 | 2 |
| C2×#450 | 2 |
| C22×#102 | 2 |
| C22×C22⋊C8 | 2 |