While homophily -- the tendency to link with similar others -- may nurture a sense of belonging and shared values, it can also hinder diversity and widen inequalities. Here, we unravel this trade-off analytically, revealing homophily traps for minority groups: scenarios where increased homophilic interaction among minorities negatively affects their structural opportunities within a network. We demonstrate that homophily traps arise when minority size falls below 25% of a network, at which point homophily comes at the expense of lower structural visibility for the minority group. Our work reveals that social groups require a critical size to benefit from homophily without incurring structural costs, providing insights into core processes underlying the emergence of group inequality in networks.