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