Standard ASR systems typically use phoneme as the subword units. Preliminary studies have shown that the performance of the ASR system could be improved by using grapheme as additional subword units. In this paper, we investigate such a system where the word models are defined in terms of two different subword units, i.e., phoneme and grapheme. During training, models for both the subword units are trained, and then during recognition either both or just one subword unit is used. We have studied this system for a continuous speech recognition task in American English language. Our studies show that grapheme information used along with phoneme information improves the performance of ASR.