The modular group of order $27$ has exponent $9$ and exactly three cyclic subgroups of order $9$

Given: The objects and hypotheses in the Example.

The modular group of order $p^3$ is the external semidirect product $C_{p^2}{⋊}_{\alpha }{C}_p$ for the order-$p$ automorphism $a\mapsto a^{1+p}$ (The modular group of order $p^3$ as a semidirect product $C_{p^2}⋊C_p$).

For every prime $p$ the map $a\mapsto a^{1+p}$ is an automorphism of order $p$ of a cyclic group of order $p^2$ (Raising to the power $1+p$ is an automorphism of order $p$ of a cyclic group of order $p^2$).

The modular group of order $p^3$ is extraspecial, and its exponent is $p^2$ (The modular group of order $p^3$ is extraspecial, of exponent $p^2$ when $p$ is odd).

For a finite group $G$, its exponent is $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}\mathrm{.}$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

$\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).