As @Benjamin said in comments above, this is a very difficult question. However if one is willing to add conditions, then there are some relevant results. Firstly:

Chein, Orin, * Moufang Loops of Small Order*, Trans. Amer. Math. Soc. 188 (1974), 31-51.

Chein completely classifies all non-associative Moufang loops of order $\leq 31$. There are 13 such loops - one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28.

In fact Chein extends this classification to give a full classification up to order $\leq 63$ in:

Chein, Orin, Moufang loops of small order, Mem. Amer. Math. Soc. 13 (1978), no. 197, iv+131 pp.

You may also be interested in this webpage which gives a full list of all Bol loops of order $\leq 31$.

As @Benjamin said in comments above, this is a very difficult question. However if one is willing to add conditions, then there are some relevant results. Firstly:

Chein, Orin, * Moufang Loops of Small Order*, Trans. Amer. Math. Soc. 188 (1974), 31-51.

Chein completely classifies all non-associative Moufang loops of order $\leq 31$. There are 13 such loops - one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28.

In fact Chein extends this classification to give a full classification up to order $\leq 63$ in:

Chein, Orin, Moufang loops of small order, Mem. Amer. Math. Soc. 13 (1978), no. 197, iv+131 pp.

You may also be interested in this webpage which gives a full list of all Bol loops of order $\leq 31$.