Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
7039 conjugacy classes of subgroups; only the 3695 normal subgroups are shown.
This group has 7039 conjugacy classes of subgroups — too many to draw. Only its 3695 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C23 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | semidirect |
| 3. C22×C22⋊Q8 | C2 | direct (N × H) |
| 4. C23×C4⋊C4 | C2 | semidirect |
| 5. C23×C4⋊C4 | C2 | semidirect |
| 9. C23×Q8 | C22 | semidirect |
| 16. C2×C22⋊Q8 | C22 | direct (N × H) |
| 17. C22×C4⋊C4 | C22 | semidirect |
| 18. C22×C4⋊C4 | C22 | semidirect |
| 22. C25 | Q8 | semidirect |
| 26. C22×Q8 | C23 | semidirect |
| 37. C22⋊Q8 | C23 | direct (N × H) |
| 38. C2×C4⋊C4 | C23 | semidirect |
| 39. C2×C4⋊C4 | C23 | semidirect |
| 46. C24 | C2×Q8 | semidirect |
| 51. C2×Q8 | C24 | semidirect |
| 62. C4⋊C4 | C24 | semidirect |
| 63. C4⋊C4 | C24 | semidirect |
| 68. C23 | C22⋊Q8 | direct (N × H) |
| 71. C23 | C22×Q8 | semidirect |
| 88. C22 | C2×C22⋊Q8 | direct (N × H) |
| 91. C22 | C23×Q8 | semidirect |
| 99. C2 | C22×C22⋊Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C25×C4 | 2 |
| C22×C22⋊Q8 | 2 |
| C23×C4⋊C4 | 2 |
| C23×C22⋊C4 | 2 |