Automatic modulation classification can be a core component for intelligent spectrally efficient wireless communication networks, and deep learning techniques have recently been shown to deliver superior performance to conventional model-based strategies, particularly when distinguishing between a large number of modulation types. However, such deep learning techniques have also been recently shown to be vulnerable to gradient-based adversarial attacks that rely on subtle input perturbations, which would be particularly feasible in a wireless setting via jamming. One such potent attack is the one known as the Carlini-Wagner attack, which we consider in this work. We further consider a data-driven subsampling setting, where several recently introduced deep-learning-based algorithms are employed to select a subset of samples that lead to reducing the final classifier's training time with minimal loss in accuracy. In this setting, the attacker has to make an assumption about the employed subsampling strategy, in order to calculate the loss gradient. Based on state of the art techniques available to both the attacker and defender, we evaluate best strategies under various assumptions on the knowledge of the other party's strategy. Interestingly, in presence of knowledgeable attackers, we identify computational cost reduction opportunities for the defender with no or minimal loss in performance.