We present the pressure-temperature phase diagram La5Co2Ge3 up to ∼5 GPa, which was constructed from magnetization, resistivity, and specific heat measurements. At ambient pressure, La5Co2Ge3 is an itinerant ferromagnet with a Curie temperature TC∼ 4 K. Upon increasing pressure up to ∼1.7 GPa, TC is suppressed down to ∼3 K. Upon further increasing pressure, our results suggest that La5Co2Ge3 enters a different low-temperature ground state. The corresponding transition temperature T∗ has a nonmonotonic pressure dependence up to ∼5 GPa. Our results demonstrate that the ferromagnetic quantum critical point in La5Co2Ge3 is avoided by the appearance of a different, likely magnetically ordered, state that has an antiferromagnetic component.