Which class of objects in Solar System carries more mass?
The sun by far occupies the largest part of the solar system's mass. The mass of the sun happens to be approximately $(1.98855±0.00025)×10^{30}$ kg. Which is immensely huge. By comparison the inter-planetary medium though may occupy a large volume is nowhere dense enough to even compare to the mass of the sun which is approximately 99% of the solar system's mass.

The mass of all planets combined by comparison happens to be approximately $2.6634×10^{27}$kg.
As far as interplanetary matter is concerned, its density is to low and its volume is pretty hard to define so mass remains a highly variable quantity.

Moons and asteroids account only for a fraction of the mass of the solar system though they happen to be present in large numbers.