Plane instance segmentation from RGB-D data is a crucial research topic for many downstream tasks. However, most existing deep-learning-based methods utilize only information within the RGB bands, neglecting the important role of the depth band in plane instance segmentation. Based on EfficientSAM, a fast version of SAM, we propose a plane instance segmentation network called PlaneSAM, which can fully integrate the information of the RGB bands (spectral bands) and the D band (geometric band), thereby improving the effectiveness of plane instance segmentation in a multimodal manner. Specifically, we use a dual-complexity backbone, with primarily the simpler branch learning D-band features and primarily the more complex branch learning RGB-band features. Consequently, the backbone can effectively learn D-band feature representations even when D-band training data is limited in scale, retain the powerful RGB-band feature representations of EfficientSAM, and allow the original backbone branch to be fine-tuned for the current task. To enhance the adaptability of our PlaneSAM to the RGB-D domain, we pretrain our dual-complexity backbone using the segment anything task on large-scale RGB-D data through a self-supervised pretraining strategy based on imperfect pseudo-labels. To support the segmentation of large planes, we optimize the loss function combination ratio of EfficientSAM. In addition, Faster R-CNN is used as a plane detector, and its predicted bounding boxes are fed into our dual-complexity network as prompts, thereby enabling fully automatic plane instance segmentation. Experimental results show that the proposed PlaneSAM sets a new SOTA performance on the ScanNet dataset, and outperforms previous SOTA approaches in zero-shot transfer on the 2D-3D-S, Matterport3D, and ICL-NUIM RGB-D datasets, while only incurring a 10% increase in computational overhead compared to EfficientSAM.