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We propose SnCQA, a set of hardware-efficient variational circuits of equivariant quantum convolutional circuits respective to permutation symmetries and spatial lattice symmetries with the number of qubits $n$. By exploiting permutation symmetries of the system, such as lattice Hamiltonians common to many quantum many-body and quantum chemistry problems, Our quantum neural networks are suitable for solving machine learning problems where permutation symmetries are present, which could lead to significant savings of computational costs. Aside from its theoretical novelty, we find our simulations perform well in practical instances of learning ground states in quantum computational chemistry, where we could achieve comparable performances to traditional methods with few tens of parameters. Compared to other traditional variational quantum circuits, such as the pure hardware-efficient ansatz (pHEA), we show that SnCQA is more scalable, accurate, and noise resilient (with $20\times$ better performance on $3 \times 4$ square lattice and $200\% - 1000\%$ resource savings in various lattice sizes and key criterions such as the number of layers, parameters, and times to converge in our cases), suggesting a potentially favorable experiment on near-time quantum devices.