A new method for computing the conjugacy classes of subgroups of a finite permutation group \(G\) is described. The method requires a preliminary calculation and storage of appropriate data of all trivial-Fitting groups (i.e. groups with no nontrivial solvable normal subgroups) of order less than some fixed \(n\); at the time of writing of the paper, this was done for \(n=216000\), producing a database of \(154\) groups. The algorithm consists of two stages. At first, one has to find an appropriate ascending series of normal subgroups of \(G\), \(1=N_0\triangleleft N_1\triangleleft\cdots\triangleleft N_r=L\triangleleft G\), such that \(N_i/N_{i-1}\) is elementary Abelian (for \(i=1,\dots,r\)) and \(G/L\) is trivial-Fitting. Then, \(G/L\) must be explicitly identified with one of the trivial-Fitting groups in the database, obtaining representatives of all the conjugacy classes of subgroups of \(G/L\). The second stage is the successive lifting of such class representatives through the layers, finding class representatives of subgroups of \(G/N_{i-1}\) from those of \(G/N_i\). The paper ends with several examples and performance results of the algorithm as implemented in the algebra system MAGMA.