One fundamental type of query for graph databases is subgraph isomorphism queries (a.k.a subgraph queries). Due to the computational hardness of subgraph queries coupled with the cost of managing massive graph data, outsourcing the query computation to a third-party service provider has been an economical and scalable approach. However, confidentiality is known to be an important attribute of Quality of Service (QoS) in Query as a Service (QaaS). In this paper, we propose the first practical private approach for subgraph query services, asymmetric structure-preserving subgraph query processing, where the data graph is publicly known and the query structure/topology is kept secret. Unlike other previous methods for subgraph queries, this paper proposes a series of novel optimizations that only exploit graph structures, not the queries. Further, we propose a robust query encoding and adopt the novel cyclic group based encryption so that query processing is transformed into a series of private matrix operations. Our experiments confirm that our techniques are efficient and the optimizations are effective.