Hello  aspirant,
Here's the information about IIST exam and colleges :

IIST Admission:

• Undergraduate (BTech):
  • JEE Main
  • JEE Advanced
    
    
• Postgraduate (MTech, MSc):
  • GATE
    
    
• PhD:
  • Online screening test (based on GATE score)
  • Interview

Other Top Colleges for Space Technology:

• Indian Institute of Technology (IITs):
  • IIT Bombay, IIT Madras, IIT Kanpur, IIT Delhi, IIT Kharagpur, IIT Roorkee, IIT Guwahati, IIT Hyderabad, IIT Bhubaneswar, IIT Ropar, IIT Gandhinagar, IIT Patna, IIT Jodhpur, IIT Mandi, IIT Dharwad, IIT Jammu, IIT Palakkad, IIT Bhilai, IIT Tirupati
• National Institute of Technology (NITs):
  • NIT Trichy, NIT Warangal, NIT Surathkal, NIT Calicut, NIT Kurukshetra, NIT Jaipur, NIT Hamirpur, NIT Rourkela, NIT Silchar, NIT Durgapur, NIT Nagpur, NIT Raipur, NIT Bhopal, NIT Patna, NIT Jalandhar, NIT Kurukshetra, NIT Jamshedpur, NIT Goa, NIT Puducherry, NIT Agartala, NIT Arunachal Pradesh, NIT Uttarakhand, NIT Meghalaya, NIT Nagaland, NIT Sikkim, NIT Mizoram, NIT Manipur, NIT Andhra Pradesh, NIT Telangana, NIT Karnataka, NIT Kerala, NIT Tamil Nadu, NIT Odisha, NIT West Bengal
• Other Government Colleges:
  • Birla Institute of Technology and Science (BITS) Pilani
  • University of Hyderabad
  • Anna University
  • University of Kerala
  • University of Mysore
  • University of Pune
  • Indian Institute of Astrophysics (IIA)
  • Aryabhatta Research Institute of Observational Sciences (ARIES)

• Top Colleges for Physics Research:

  • Indian Institute of Science (IISc) Bangalore
  • Tata Institute of Fundamental Research (TIFR) Mumbai
  • Indian Institute of Technology (IITs):
    • IIT Bombay, IIT Madras, IIT Kanpur, IIT Delhi, IIT Kharagpur, IIT Roorkee, IIT Guwahati, IIT Hyderabad, IIT Bhubaneswar, IIT Ropar, IIT Gandhinagar, IIT Patna, IIT Jodhpur, IIT Mandi, IIT Dharwad, IIT Jammu, IIT Palakkad, IIT Bhilai, IIT Tirupati
      
      
• National Institute of Technology (NITs):
  • NIT Trichy, NIT Warangal, NIT Surathkal, NIT Calicut, NIT Kurukshetra, NIT Jaipur, NIT Hamirpur, NIT Rourkela, NIT Silchar, NIT Durgapur, NIT Nagpur, NIT Raipur, NIT Bhopal, NIT Patna, NIT Jalandhar, NIT Kurukshetra, NIT Jamshedpur, NIT Goa, NIT Puducherry, NIT Agartala, NIT Arunachal Pradesh, NIT Uttarakhand, NIT Meghalaya, NIT Nagaland, NIT Sikkim, NIT Mizoram, NIT Manipur, NIT Andhra Pradesh, NIT Telangana, NIT Karnataka, NIT Kerala, NIT Tamil Nadu, NIT Odisha, NIT West Bengal
• Other Government Colleges:
  • University of Delhi
  • University of Mumbai
  • University of Calcutta
  • University of Hyderabad
  • University of Pune
  • Bose Institute
  • Saha Institute of Nuclear Physics (SINP)
  • Raman Research Institute (RRI)
    
    

I hope this information helps you.