Charge q is divided in two parts q₁ and q₂ and these are placed at a distance r. For maximum repulsion between these parts

Here one charge is q₁ then ${\mathrm{q}}_2=\left(\mathrm{q}-{\mathrm{q}}_1\right)$
Force between these two charges is given by $\mathrm{F}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\times \frac{{\mathrm{q}}_1\left(\mathrm{q}-{\mathrm{q}}_1\right)}{{\mathrm{r}}^2}$
For maximum repulsion $\frac{\mathrm{dF}}{{\mathrm{dq}}_1}=0=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\times \frac{\left(\mathrm{q}-2{\mathrm{q}}_1\right)}{{\mathrm{r}}^2}$
or   $\mathrm{q}-2{\mathrm{q}}_1=0\text{ or }{\mathrm{q}}_1=\mathrm{q}/2$ and  ${\mathrm{q}}_2=\mathrm{q}/2$.