Does finance commission work under planning commission?

Since there is no Central Planning Commission or Finance Commission in the US I assume you ask this question in reference to India. The answer in the Indian context is No. The Finance Commission is a constitutional body while the Planning Commission is not statutory and merely an advisory body.

While some of their functions overlap, by and large the Finance Commission focuses on non-plan revenues (administrative costs of government etc) and expenditure while the Planning Commission focuses on plan expenditure (social schemes and infrastructure).