Following the idea that hydrogen-rich compounds might be high-Tc superconductors at high pressures, and the very recent breakthrough in predicting and synthesizing hydrogen sulfide with record-high Tc=203K, an ab initio evolutionary algorithm for crystal structure prediction was employed to find stable germanium hydrides. In addition to the earlier structure of germane with space group Ama2, we propose a C2/m structure, which is energetically more favorable at pressures above 278 GPa (with inclusion of zero-point energy). Our calculations indicate that the C2/m phase of germane is a superconductor with Tc=67K at 280 GPa. Germane is found to become thermodynamically unstable to decomposition to hydrogen and the compound Ge3H11 at pressures above 300 GPa. Ge3H11 with space group I4m2 is found to become stable at above 285 GPa with Tc=43K. We find that the pressure-induced phase stability of germanium hydrides is distinct from analogous isoelectronic systems, e.g., Si hydrides and Sn hydrides. Superconductivity stems from large electron-phonon coupling associated with the wagging, bending, and stretching intermediate-frequency modes derived mainly from hydrogen.