Lavender is evergreen . I've never tried to divide it and i'm fairly certain that isn't the usual approach to propagation.

Propagation is by semi ripe cuttings taken about now.

I agree, the main stems become very woody, you can trim it back when flowering is finished, they say not into the old wood, but I have and it just takes longer to recover, so a bit at the time where needed is best.

Cuttings are easy and reliable. Although they are evergreen, mine goes a dull grey in the winter.

Some supermarket and garden centre Lavender plants are made up of several plants. You can normally separate them. You can tell if there are several stems in the pot. It is cheaper for them to produce a sell-able pot that way as it is quicker to add lots of plants in one pot to make it bushy looking than growing the plant.

Semi-ripe cuttings do work, sandy well drained soil works best as they rot before rooting if the soil is too damp as outlined above.

English Lavender works best in Scotland - Hidcote and Munstead varieties are reliable. I also have Vera which grows larger and can get woody. It smells great though. Adding lots of grit to make the soil drain well and cutting regularly is the secret to keeping it going.