I know that every analytic $C_0$-semigroup is differentiable and the every differentiable semigroup is norm continuous.
I wanted to know where uniform continuity fits in the above picture.
My intuition is that since the generator of a uniformly continuous semigroup is bounded, it is of the form $(e^{tA})_{t\geq 0}$ for some bounded operator $A$, and being the generalization of "exponential" it should certainly be analytic (or at least differentiable). On the other hand, the implication "uniform continuity implies analyticity" seems questionable.
A uniformly continuous semigroup is given by the standard exponential of a bounded operator which is the generator. The exponential operator function is analytic on the right half plane $\mathbb{C}^+$ and satisfies the semigroup low. Thus the semigroup is analytic of angle $\pi/2$.