 Any of the three proper subgroups of formula_40 (all of which have order 2) is normal in formula_40; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in formula_40, but none of these three subgroups of formula_40 is a conjugate of any other one, because formula_40 is Abelian.The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product.Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as such formula_17 and the quaternions.