Most administrative institutions are basically under the management of both local party and government agencies and higher-level departments.
But in different units, some have greater horizontal management power, and some have stronger vertical power.
Even if the same unit is in different regions and at different times, it is subject to different constraints from the same level or superior departments.
For example, the Finance Bureau, in developed coastal cities such as Zhengzhou, does not listen to the Provincial Finance Department at all. Because the superiors want to ask them for money. But like other places, such as Zhumadian, Zhoukou and other places, they have to ask the department for money, so they have to listen to the Department of Finance.
I am not very familiar with trade unions, but in terms of their functions, they should be better managed by government departments at the same level. In theory, superiors only play a "guiding" role in their work.