Molar heat capacity and heat capacity
The symbols, which are standardized in ISO/IEC 80000 Quantities and units (14 parts) and recommended by IUPAC in Quantities, Units and Symbols in Physical Chemistry (Green Book), distinguish the various related quantities:

$$\small
\begin{array}{llll}
\hline
\text{Quantity} &amp; \text{Quantity} &amp; \text{Unit} &amp; \text{Unit} \\
\text{name} &amp; \text{symbol} &amp; \text{name} &amp; \text{symbol} \\
\hline
\text{heat capacity} &amp; C &amp; \text{joule per kelvin} &amp; \mathrm{J/K} \\
\text{heat capacity at constant pressure} &amp; C_p &amp; \text{joule per kelvin} &amp; \mathrm{J/K} \\
\text{heat capacity at constant volume} &amp; C_V &amp; \text{joule per kelvin} &amp; \mathrm{J/K} \\
\text{specific heat capacity} &amp; c &amp; \text{joule per kilogram kelvin} &amp; \mathrm{J/( kg \cdot K )} \\
\text{specific heat capacity at constant pressure} &amp; c_p &amp; \text{joule per kilogram kelvin} &amp; \mathrm{J/( kg \cdot K )} \\
\text{specific heat capacity at constant volume} &amp; c_V &amp; \text{joule per kilogram kelvin} &amp; \mathrm{J/( kg \cdot K )} \\
\text{molar heat capacity} &amp; C_\mathrm m &amp; \text{joule per mole kelvin} &amp; \mathrm{J/( mol \cdot K )} \\
\text{molar heat capacity at constant pressure} &amp; C_{\mathrm m, p} &amp; \text{joule per mole kelvin} &amp; \mathrm{J/( mol \cdot K )} \\
\text{molar heat capacity at constant volume} &amp; C_{\mathrm m, V} &amp; \text{joule per mole kelvin} &amp; \mathrm{J/( mol \cdot K )} \\
\hline
\end{array}$$

However, in contrast to symbols for units, symbols for quantities are only recommendations. Unfortunately, quantity symbols are used inconsistently in the literature, sometimes in a sloppy way. In principle, a given symbol can indicate different quantities.