We generalize the mixture integer-valued ARCH model (MINARCH) introduced by Zhu et al. in 2010 to a mixture integer-valued GARCH (MINGARCH) for modeling time series of counts. This model includes the ability to take into account the moving average component of the series. We give the necessary and sufficient first and second order stationarity conditions. The estimation is done via the EM algorithm. The model selection problem is studied by using three information criterions. We also study the performance of the method via simulations and include a real data application.