Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
706 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×C22⋊C4 | C2 | semidirect |
| 3. #1964 | C2 | semidirect |
| 4. #1799 | C2 | semidirect |
| 5. C4×#99 | C2 | semidirect |
| 6. C22×#41 | C4 | semidirect |
| 10. C2×#124 | C4 | semidirect |
| 11. C2×C22⋊Q8 | C4 | semidirect |
| 14. C2×C4×Q8 | C4 | semidirect |
| 102. C22⋊C4 | #41 | semidirect |
| 121. Q8 | #99 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#548 | 2 |
| Q8×C22⋊C4 | 2 |
| #1964 | 2 |
| #1799 | 2 |
| C4×#99 | 2 |