Dianthus can be propagated by taking cuttings from the mother plant or by sowing seeds

From cuttings

Take a cutting: In mid-summer, cut a 2–3 inch segment from a non-flowering shoot of the mother plant. 

Prepare the cutting: Remove the lower leaves of the cutting. You can also dip the cut end in rooting hormone. 

Plant the cutting: Insert the cutting into a pot of moist potting soil mixed with vermiculite. Make sure at least one node is below the soil. 

Place in a warm, sunny location: Keep the soil moist but not soggy. 

Wait for new growth: After about a month, new leaves should appear. 

Replant: Once the cutting has new growth, you can transplant it to its permanent location. 

From seeds

Annual dianthus varieties are usually propagated by seed. 

Dianthus can spread by self-seeding, about a foot from the parent plant. 

Dianthus prefer well-drained, neutral to slightly alkaline soil. They can tolerate clay, silt, or sandy soil, but over-watering can cause root rot and yellow leaves.