Context. Geminids are the most active annual meteor shower observed on Earth. Their parent is an active asteroid, (3200) Phaethon, which is a target of the planned DESTINY+ mission of the Japan Aerospace Exploration Agency (JAXA). The exact physical nature of (3200) Phaethon and Geminids is still debated. Aims. This paper is devoted to fragmentation modeling of bright Geminid fireballs, which should reveal information about the structure of centimeter-sized Geminid meteoroids. These fireballs were observed by the European Fireball Network (EN) over the past few years. We aim to describe their disintegration cascade in the atmosphere and their mechanical properties, and to derive their precise initial masses and velocities. Methods. We used a semi-empirical fragmentation model that employs an automatic procedure based on parallel genetic algorithms to determine the aerodynamic pressures at which a meteoroid and its parts fragment. This serves as a proxy for the mechanical strength of the body and its subsequent fragments. It enabled us to derive the minimum, median, and maximum mechanical strength and the strength distribution inside the meteoroid and reveal its internal structure. Results. We find that the Geminids begin to crumble at pressures 1–100 kPa, with the strongest parts reaching pressures of between 0.4 and 1.55 MPa before fragmenting. Knowing the spectral type of (3200) Phaethon (a B-type asteroid, part of the C complex), we conclude that the Geminids are made of compact and coherent carbonaceous material. We also find that the minimum aerodynamic pressure that causes the fragmentation of Geminids increases with increasing entry mass of Geminids. In contrast, the median aerodynamic pressure decreases as their entry mass increases. The spectra of all the observed Geminid fireballs show normal content and little variation in terms of sodium.