Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
548 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×M16 | C2 | semidirect |
| 3. C42×Q8 | C2 | semidirect |
| 4. #1994 | C2 | semidirect |
| 5. #1986 | C2 | semidirect |
| 8. Q82 | C4 | semidirect |
| 9. #557 | C4 | semidirect |
| 11. #554 | C4 | semidirect |
| 13. D4×Q8 | C4 | semidirect |
| 77. #104 | Q8 | direct (N × H) |
| 105. M16 | C2×Q8 | semidirect |
| 135. C42 | C2×Q8 | semidirect |
| 142. Q8 | #104 | direct (N × H) |
| 143. Q8 | C4×Q8 | semidirect |
| 147. D4 | C4×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#41 | 2 |
| Q8×M16 | 2 |
| C42×Q8 | 2 |
| #1994 | 2 |
| #1986 | 2 |
| #1806 | 2 |
| C4×#104 | 2 |