Do price subsidies raise prices?

Since public health care in Finland is a disaster zone, I try to use private health care when I can. But private health care receives hefty price subsidies here. For example, my recent visit to a private dentist for a basic check up cost me 125 € ($160). Out of this, the government automatically paid 58,80 € ($76) and I had to pay 66,20 € ($86). The whole thing took approx. 20 minutes.

I know everything costs an arm and a leg here, that taxes rob you blind and employment costs will bankrupt you, but I just don't see how the 125 € price could be a market price. So does the price subsidy have a role to play?