Lufthansa provides extensive Information about traveling with Pets on their Airline on their own Website here and here.
This is just a Summary of Conditions and Charges. Please confirm Current Prices with Lufthansa.

Pets in the cabin
You can take your dog or cat with you in the cabin on your Lufthansa flight. We’ll check the availability and make the booking for you no later than 72 hours before departure. We shall inform you by email if it is not possible to make the booking for your pet. If your booking is successful, you must pay the fees incurred when checking in at the airport.

From 1 January 2020, restrictions will be imposed on the transport of dogs and cats belonging to snub-nosed (brachycephalic) breeds.
To stub-nosed breeds
Dogs and cats that weigh more than 8 kg including the Carrier Box.

Prices for the carriage of animals as excess baggage in the cargo hold

The price for the carriage of animals as excess baggage in the cargo hold depends both on the route and the size of the carrying container. The price applies per flight direction, so also with a Lufthansa connecting flight, and is charged only once (e.g. on a flight from Hamburg via Frankfurt to Madrid). The price is charged again for return flights and stop-overs (travel interrupted by more than 24 hours).

Payment for taking your pet with you should be made at the check-in counter or at the Lufthansa service counter at the airport. In addition to your pet and its carrying container, of course you can also take with you the free baggage allowance included with your ticket.

Note: For animals in the hold, a surcharge of EUR 150 / CHF 170 / USD 170 / CAD 220 is charged for transfers via Brussels, Geneva, Frankfurt, Vienna and Zurich.

The information contained is believed to be true, but we have made no attempt to verify it.Please check with the relevant Organization.