Is the free product contained in amalgamated product?

Question

I'm trying to understand the subgroups of finite index of amalgamated products in some particular cases and it arises the following question:

Suppose that $G = A \ast_C B$ where $A$ and $B$ are finite groups. If $A_1 \leq A$ and $B_1 \leq B$, then is true that $A_1 \ast B_1 \leq A \ast_C B$?

I'm not sure if this is true. Because perhaps we can be $G$ being soluble and $A_1 \ast B_1$ not soluble, but this is only intuition.

Answer 1

Of course not, take $A_1=A$, $B_1=B$ and recall that the pushout (aka amalgam) is a quotient of the coproduct (aka free product) of groups. It doesn't make sense to regard the pushout as an subobject of the coproduct.