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Design Pattern Detection is a part of re-engineering process and thus gives significant information to the designer. Detection of design patterns is helpful for improving the software characteristics. Therefore, a reliable design pattern discovery is required. The problem of finding an isomorphic subgraph is used to solve design pattern detection in past. It is noticed that ordering of vertices of the design pattern saves the time of process. In this paper we are doing ordering of vertices for few design patterns proposed by Gamma, Helm, Johnson, and Vlissides (1995) using an algorithm “GreatestConstraintFirst” proposed by Bonnici, Giugno, Pulvirenti, Shasha, and Ferro (2013). After getting this ordering, we use a matching algorithm that uses subgraph isomorphism conditions to check whether a particular design pattern exists in the system design or not (Bonnici et al., 2013). We redefine subgraph isomorphism conditions in the context of the problem of mining design patterns from the system design.