Sequential patterns exist in data such as DNA string databases, occurrences of recurrent illness, etc. In this study, we present an algorithm, SEQUEST, to mine frequent subsequences from sequential patterns. The challenges of mining a very large database of sequences is computationally expensive and require large memory space. SEQUEST uses a Direct Memory Access Strips (DMA-Strips) structure to efficiently generate candidate subsequences. DMA-Strips structure provides direct access to each item to be manipulated and thus is optimized for speed and space performance. In addition, the proposed technique uses a hybrid principle of frequency counting by the vertical join approach and candidate generation by structure guided method. The structure guided method is adapted from the TMG approach used for enumerating subtrees in our previous work [8]. Experiments utilizing very large databases of sequences which compare our technique with the existing technique, PLWAP [4], demonstrate the effectiveness of our proposed technique.