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Heatmap regression (HR) has become one of the mainstream approaches for face alignment and has obtained promising results under constrained environments. However, when a face image suffers from large pose variations, heavy occlusions and complicated illuminations, the performances of HR methods degrade greatly due to the low resolutions of the generated landmark heatmaps and the exclusion of important high-order information that can be used to learn more discriminative features. To address the alignment problem for faces with extremely large poses and heavy occlusions, this paper proposes a heatmap subpixel regression (HSR) method and a multi-order cross geometry-aware (MCG) model, which are seamlessly integrated into a novel multi-order high-precision hourglass network (MHHN). The HSR method is proposed to achieve high-precision landmark detection by a well-designed subpixel detection loss (SDL) and subpixel detection technology (SDT). At the same time, the MCG model is able to use the proposed multi-order cross information to learn more discriminative representations for enhancing facial geometric constraints and context information. To the best of our knowledge, this is the first study to explore heatmap subpixel regression for robust and high-precision face alignment. The experimental results from challenging benchmark datasets demonstrate that our approach outperforms state-of-the-art methods in the literature.