Understanding nonlinear manifolds of scientific data extracted via autoencoder is important to propel practical uses of non-intrusive reduced-order modeling in the community. We here tackle this matter by visualizing nonlinear autoencoder modes with the aid of mode-decomposing convolutional neural network autoencoder (MD-CNN-AE). The MD-CNN-AE has a customization in the decoder part, which enable us to visualize individual modes extracted through the encoder part. The present demonstration is performed with a three-dimensional flow around a square cylinder at Re_D=300, which possesses complex nonlinear vortical phenomena associated with strong nonlinearities. The results are compared with a conventional linear model order reduction method, i.e, principal component analysis (PCA). The reconstructed fields with MD-CNN-AE hold more energetic information than that with PCA, despite the same number of latent variables. The present results indicate the strong capability of MD-CNN-AE for efficient low-dimensionalization and data compression of three-dimensional flow fields in an interpretable manner.