Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3087 conjugacy classes of subgroups; only the 1375 normal subgroups are shown.
This group has 3087 conjugacy classes of subgroups — too many to draw. Only its 1375 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#391 | C2 | direct (N × H) |
| 8. C23×Q8 | C4 | semidirect |
| 18. #391 | C22 | direct (N × H) |
| 24. C22×Q8 | C2×C4 | semidirect |
| 36. C23×C4 | Q8 | semidirect |
| 55. C2×Q8 | C22×C4 | semidirect |
| 70. C22×C4 | C2×Q8 | semidirect |
| 103. C2×C4 | C22×Q8 | semidirect |
| 104. C22 | #391 | direct (N × H) |
| 119. C2 | C2×#391 | 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 |
| C23×C4⋊C4 | 2 |
| C23×C42 | 2 |
| C2×#391 | 2 |
| C22×#95 | 2 |