When the powers of an element of a Finite Field generate a subfield?

Question

As it is known, all the elements on a Finite Field GF($q$) of $q$ elements have an order $n \le q−1$ in such a way that being $\beta$ a given element: $β^n=1$. The set $F^∗={β0,\beta^1,⋯,\beta^{n−1}}$ is a commutative group under multiplication. The element whose order is $q−1$ is named the primitive element, $\alpha$. Every Finite Field has a primitive element and its powers generate the Field. Sometimes, $F^* \cup \{0\}$ is a subfield of GF($q$) as for $\beta = \alpha^5$ in GF($2^4$). What conditions should be accomplished in order to get a subfield from the successive powers of a given element, is that enough that the order is a prime or the power of a prime number?.