SAMtools is a widely-used genomics application for post-processing high-throughput sequence alignment data. Such sequence alignment data are commonly sorted to make downstream analysis more efficient. However, this sorting process itself can be computationally- and I/O-intensive: high-throughput sequence alignment files in the de facto standard binary alignment/map (BAM) format can be many gigabytes in size, and may need to be decompressed before sorting and compressed afterwards. As a result, BAM-file sorting can be a bottleneck in genomics workflows. This paper describes a case study on the performance analysis and optimization of SAMtools for sorting large BAM files. OpenMP task parallelism and memory optimization techniques resulted in a speedup of 5.9X versus the upstream SAMtools 1.3.1 for an internal (in-memory) sort of 24.6 GiB of compressed BAM data (102.6 GiB uncompressed) with 32 processor cores, while a 1.98X speedup was achieved for an external (out-of-core) sort of a 271.4 GiB BAM file.