Here is the full list of all 45 Carrier Command: Gaea Mission achievements.
Capture over 250 islands
Destroy 1000 Walruses using a Manta
Capture 25 islands with a Deadly defense
Capture 25 islands with a Very Strong defense
Capture 50 islands with a Strong defense
Destroy 1000 Turrets from Direct Control
Destroy 1000 Mantas from Direct Control
Destroy 1000 Walruses from Direct Control
Mine over 1 million in resources
Capture 10 islands with a Very Strong defense
Capture 10 islands with a Deadly defense
Capture 10 islands with a Strong defense
Capture all enemy islands in the Dead Zone
Capture the enemy Stockpile
Capture 1 island with a Deadly defense
Produce over 1000 items
Capture 1 island with a Strong defense
Capture 5 islands without using load
Locate and acquire all available Blueprints
Capture 1 island with a Very Strong defense
Capture 3 islands without using load
Capture 1 island without using load
Execute Dr. Mao Shin
Shoot down Shin's Manta
Sink the enemy Carrier
Decommission the Geothermal power plants
Rescue Captain Aurora from the APA
Destroy 100 Mantas from Direct Control
Defeat the enemy Carrier in naval combat
Unleash the Lord of Death within you, eliminate Fulcrum's facilities
Reassemble the Hammerhead Nuclear Missile system
Locate all Green Laser research files
Run over 100 enemy infantry units
Destroy 100 Walruses using a Manta
Destroy 100 Turrets from Direct Control
Destroy 100 Walruses from Direct Control
Run over 50 enemy infantry units
Restore contact with Captain Aurora
Locate survivors from the Dead Zone drop
Destroy 1 Manta from Direct Control
Destroy 1 Walrus using a Manta
Destroy 1 Walrus from Direct Control
Run over 1 enemy infantry unit
Destroy 1 Turret from Direct Control
Successfully commandeer an enemy Carrier