With a triangular prism, we have the 2 equilateral on the end. The symmetries of the an equilateral triangle is 6. With this we deduce the 3D solid has $2\times6=12$ symmetries.
For the reflective symmetries we have four that reflects on planes that go through the end of the prism, and one that is perpendicular to the rectangular faces
For the rotational symmetries, we can rotate through the triangular end by $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$. We can also rotate around an axis that goes through the midpoint each rectangular edge that joins to another rectangular edge and through the opposing centre of the rectangular face, e.g.
So, that gives 4 reflection symmetries, and 5 rotational symmetries, and 1 identity. What am I missing with the two other symmetries?