Let \(kG\) be the group algebra of a finite group \(G\) over an algebraically closed field \(k\) of characteristic \(p>0\). The author introduces the Dade group \(D(G)\) of \(G\), a generalization of the well-known Dade group of a finite \(p\)-group. The elements of \(D(G)\) are equivalence classes of strongly capped endo-\(p\)-permutation \(kG\)-modules; an endo-\(p\)-permutation \(kG\)-module \(M\) is called strongly capped if \(\text{End}_k(M)\) is isomorphic to \(k\oplus N\) where \(N\) is a \(p\)-permutation \(kG\)-module all of whose indecomposable direct summands have vertices strictly contained in a Sylow \(p\)-subgroup \(P\) of \(G\). In this case \(M\) has a unique indecomposable direct summand \(\text{Cap}(M)\) with vertex \(P\), up to isomorphism, and two strongly capped endo-\(p\)-permutation \(kG\)-modules \(M,N\) are called equivalent if \(\text{Cap}(M)\) is isomorphic to \(\text{Cap}(N)\). The tensor product makes \(D(G)\) into a finitely generated Abelian group. Moreover, \(D(G)\) has a subgroup \(\Gamma(X(N_G(P)))\) coming from Green correspondents of one-dimensional \(kN_G(P)\)-modules, and a subgroup \(D^\Omega(G)\) generated by certain relative syzygies. The author proves that, in certain cases, one has \(D(G)=D^\Omega(G)+\Gamma(X(N_G(P)))\).