C

$\because$棱柱的上下两个底面边数之和是侧棱数的$2$倍，
$\therefore 24\div 3=8$，即侧棱有$8$条，
$\therefore$侧面有$8$个，底面有$2$个，
$\therefore$这个棱柱共有$10$个面.
故选：$C$.