  We present the pressure-temperature phase diagram La$_5$Co$_2$Ge$_3$ up to
$\sim$ 5\,GPa, which was constructed from magnetization, resistivity and
specific heat measurements. At ambient pressure, La$_5$Co$_2$Ge$_3$ is an
itinerant ferromagnet with a Curie temperature $T_\textrm C\sim$ 4\,K. Upon
increasing pressure up to $\sim$ 1.7\,GPa, $T_\textrm C$ is suppressed down to
$\sim$ 3\,K. Upon further increasing pressure, our results suggest that
La$_5$Co$_2$Ge$_3$ enters a different low-temperature ground state. The
corresponding transition temperature, $T^*$, has a nonmonotonic pressure
dependence up to $\sim$ 5\,GPa. Our results demonstrate that the ferromagnetic
quantum critical point in La$_5$Co$_2$Ge$_3$ is avoided by the appearance of a
different, likely magnetically ordered state that has an antiferromagnetic
component.
