You need to beat Attack Mode (Story) before Dire Subjugations appear. (Not DLC/Epilogue)
You can check the requirements in "Special Mission" in the pause menu when playing solo.
You can also see your number of subjugations, and side missions completed there.
Check Combat Status Log for Mysterious Titan, and Pyxis's plan subjugations. (Tells you every 5)
When going for Dire Subjugation you don't need S-Rank, and can skip cutscenes to save time.

Kind of stopped working on this guide. May finish it some day.

The 1004th Cadet Corps:
S-Rank requirements:
Dire Subjugation requirements:



Do the level at moderate speed till the Dire Subjugation Titan appears. Subjugate it. Then finish.
Dire Subjugation Titan appears at the bottom were you started from.
While you're striking limbs in the final training session. (Before all Training Titans are destroyed.)

• Mission Reward(s):
  Iron Scraps
  Iron Bamboo Fragment
  Iceburst Stone Fragment
  
• S-Rank Reward(s):
  Low-Quality Rare Metal
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

Invasion:
S-Rank requirements:
Dire Subjugation requirements:



Play through the level till the Dire Subjugation Titan appears. Then subjugate it,
and continue as normal. Dire Subjugation Titan appears at the top right of the map.
After healing, and subjugating the 2nd Titan. (Requirements can't be missed)

• Mission Reward(s):
  Low-Quality Rare Metal
  Iron Bamboo Fragment
  Wood
  
• S-Rank Reward(s):
  Low-Quality Rare Metal
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

First Battle:
S-Rank requirements:
Dire Subjugation requirements:


Dire Subjugation Titan appears at the bottom of the map shortly after subjugating 20 Titans
("Side-missions" aren't miss-able here, and there won't be 20 Titan's before the 3 side are done)
Subjugate the Dire Subjugation Titan before you subjugate the Final Mission Titan.

• Mission Reward(s):
  Low-Quality Rare Metal
  Iron Scraps
  Iceburst Stone Fragment
  
• S-Rank Reward(s):
  Leather
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

The World the Girl Saw:
S-Rank requirements:
Dire Subjugation requirements:

Subjugate the Titans attacking the citizens quickly, and beat the side-missions as they come up.
Dire Subjugation Titan appears at the bottom of the map after all 5 side-missions have been done.
Subjugate the Dire Subjugation Titan before you subjugate the Final Mission Titans.

• Mission Reward(s):
  Generic Statement of Deployment
  Iron Scraps
  Iron Bamboo Fragment
  
• S-Rank Reward(s):
  Generic Statement of Deployment
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

The Mysterious Titan:
S-Rank requirements:
Dire Subjugation requirements:

Can't do S-Rank and the Dire Subjugation on the same run since there's not enough Titans.
Don't subjugate Titans in the Mysterious Titan's way. Let it get 15 subjugations itself.
The Dire Subjugation Titan appears after the Mysterious Titan defends Trost District base.

• Mission Reward(s):
  Leather
  Low-Quality Rare Metal
  Iron Scraps
  
• S-Rank Reward(s):
  Iron Scraps
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

Roar:
S-Rank requirements:
Dire Subjugation requirements:


Recommended you play as Eren to control the Titan. Make sure to not let any Cadets die.
Subjugate all Titans, but the Final Mission Titan before 3 minutes.
Dire Subjugation Titan appears on the left of the Trost District base, after subjugating all Titans.

• Mission Reward(s):
  Leather
  Low-Quality Rare Metal
  Iron Bamboo Fragment
  
• S-Rank Reward(s):
  Iron Bamboo Fragment
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment

Idol:
S-Rank requirements:
Dire Subjugation requirements:

Lure Titans next to the wall till you get the message "Pyxis's plan has subjugated 10 Titans!"
After escorting Annie, and she leaves the battlefield look for Reiner at the bottom left.
After adding Reiner to your squad go to the top right to find, and add Bertholdt.
Dire Subjugation Titan should appear at the bottom of the map when the final mission shows up.

• Mission Reward(s):
  Leather
  Low-Quality Rare Metal
  Iceburst Stone Fragment
  
• S-Rank Reward(s):
  Iceburst Stone Fragment
  
• Dire Subjugation Reward(s):
  Ultra-hardened Crystal Fragment