The title is badly worded, so let me expand. Suppose I have a finite field $\mathbb{F}_p$, and I'd like to know which extension of this field contains all elements of a certain order $n$. For example, take $p = 7$ and $n = 24$. If I am correct, all elements of order $24$ is contained in $\mathbb{F}_{7^2}$. The fact that $\mathbb{F}_{49}$ contains elements of order 24 is obvious as $24$ divides $7^2-1$ (at least, I believe that that's obvious?). But why shouldn't there be an element of order $24$ in a field $\mathbb{F}_{7^{2l}}$ that isn't contained in the subfield $\mathbb{F}_{7^2}$?
Is it generally true that if I have a finite field with one element of order $m$ then all elements of order $m$ will be in that finite field?