This paper investigates automatic speech recognition system using context-dependent graphemes as subword units based on the conventional HMM/GMM system as well as TANDEM system. Experimental studies conducted on two different continuous speech recognition tasks show that systems using only context-dependent graphemes can yield competitive performance when compared to state-of-the-art context-dependent phoneme-based automatic speech recognition system. We further demonstrate that a system using both context-dependent phoneme and grapheme subword units can out perform either of these systems alone.