Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2231 conjugacy classes of subgroups; only the 687 normal subgroups are shown.
This group has 2231 conjugacy classes of subgroups — too many to draw. Only its 687 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 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊Q8 | C2 | semidirect |
| 3. C22×#124 | C2 | semidirect |
| 4. #2858 | C2 | direct (N × H) |
| 5. C2×#390 | C2 | semidirect |
| 6. C2×C43 | C2 | semidirect |
| 11. C2×C4⋊Q8 | C22 | semidirect |
| 15. C2×#124 | C22 | semidirect |
| 21. #390 | C22 | semidirect |
| 22. C43 | C22 | semidirect |
| 46. C2×C42 | D4 | semidirect |
| 80. C42 | C2×D4 | semidirect |
| 99. C2×C4 | #124 | semidirect |
| 118. C4 | C2×#124 | semidirect |
| 121. C2 | #2858 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊Q8 | 2 |
| C22×#127 | 2 |
| C22×#124 | 2 |
| #2858 | 2 |
| C2×#390 | 2 |
| C2×C43 | 2 |