State-of-the-art Automatic Speech Recognition (ASR) systems typically use phoneme as the subword units. In this paper, we investigate a system where the word models are defined in-terms of two different subword units, i.e., phonemes and graphemes. We train models for both the subword units, and then perform decoding using either both or just one subword unit. We have studied this system for American English language where there is weak correspondence between the grapheme and phoneme. The results from our studies show that there is good potential in using grapheme as auxiliary subword units.