The secure hash algorithm 2 (SHA-2) family, including the SHA-224/256/384/512 hash functions, is widely adopted in many modern domains, ranging from Internet of Things devices to cryptocurrency. SHA-2 functions are often implemented on hardware to optimize performance and power. In addition to the high-performance and low-cost requirements, the hardware for SHA-2 must be highly flexible for many applications. This paper proposes an SHA-2 hardware architecture named the multimode SHA-2 accelerator (MSA), which has high performance and flexibility at the system-on-chip level. To achieve high performance and flexibility, our accelerator applies three optimal techniques. First, a multimode processing element architecture is proposed to enable the accelerator to compute various SHA-2 functions for many applications. Second, a three-stage arithmetic logic unit pipeline architecture is proposed to reduce the critical paths and hardware resources. Finally, nonce generator and nonce validator architectures are proposed to reduce memory access and maximize the performance of the proposed MSA for blockchain mining applications. The MSA accuracy is tested on a real hardware platform (the Xilinx Alveo U280 FPGA). The experimental results on the field programmable gate array (FPGA) prove that the proposed MSA achieves significantly better performance, hardware efficiency, and flexibility than previous works. The evaluation results for energy efficiency show that the proposed MSA achieves up to 38.05 Mhps/W, which is 543.6 and 29 times better than the state-of-the-art Intel i9-10940X CPU and RTX 3090 GPU, respectively.