Mathematica comes with LatticeData, a database of many lattices. This Demonstration lets you examine many of these lattices at the same time, and gives tooltip popups when you hover the mouse over the appropriate name.
There are two parts. At the top, components of a selected lattice can be examined. At the bottom, a group of lattices of a selected dimension and class can be examined.
Use the "lattice type" control to choose the type of lattice to look at. Use "lattice property" to select the requisite information. The second row of controls queries the LatticeData and returns names of lattices with the dimension chosen on the "dim" radio buttons. The third table row is similar in that it becomes populated with members of the database that meet the class criteria of the radio buttons along the bottom of the frame.
Items listed in the second and third rows are tied to a Tooltip. This toggle sets either the "image" or "dbase".
The normal toggle applies "normal" to data so that sparse arrays may be expanded in both the Tooltip and the third cell of the first row. Some sparse arrays may be too large. In these cases, select appropriate class or dimension data to see the entire array (as for the Leech lattice in particular).