Modularity was introduced by Newman and Girvan in 2004 and is used as a measure of community structure of networks represented by graphs. In our work we study modularity of the random intersection graph model first considered by Karoński, Scheinerman, and Singer--Cohen in 1999. Since their introduction, random intersection graphs has attracted much attention, mostly due to their application as networks models. In our work we determine the range of parameters in which modularity detects well the community structure of the random intersection graphs, as well as give a range of parameters for which there is a community structure present but not revealed by modularity. We also relate modularity of the random intersection graph to the modularity of other known random graph models.