Phononic bandgap metamaterials, which consist of periodic cellular structures, are capable of absorbing energy within a certain frequency range. Designing metamaterials that trap waves across a wide wave frequency range is still a challenging task. In this paper, we present a deep feature learning-based design framework for both unsupervised generative design and supervised learning-based exploitative optimization. The Gaussian mixture beta variational autoencoder (GM-βVAE) is used to extract latent features as design variables. Gaussian process (GP) regression models are trained to predict the relationship between latent features and properties for property-driven optimization. The optimal structural designs are reconstructed by mapping the optimized latent feature values to the original image space. Compared with the regular variational autoencoder (VAE), we demonstrate that GM-βVAE has a better learning capability and is able to generate a more diversified design set in unsupervised generative design. Furthermore, we propose an iterative GM-βVAE model updating-based design framework. In each iteration, the optimal designs found property-driven optimization is used to update the training dataset. The GM-βVAE model is re-trained with the updated dataset for the optimization search in the next iteration. The effectiveness of the iterative design framework is demonstrated by comparing the proposed designs with the designs found by the traditional single-loop design method and the topologically optimized designs reported in literatures. The caveats to designing phonic bandgap metamaterials are summarized.