The Square Network
The simplest one. The square network is isotropic and optimal for conductivity but not iosotropic for elasticity.
The Hexagonal Network
Hexagonal networks are isotropic and optimal for conductivity, isotropic for elasticity but not optimal.
The Triangular Network
Triangles are isotropic and optimal both for conductivity and elasticity. This is has been known for quite a long time.
The Kagome Network
The Kagome network is actually very interesting, because one can prove that an infinite network is isotropic and optimal, both for conductivity and elasticity. Any finite instance of this network is bound to be suboptimal from the elastic point of view though, because its degree (the number of beams at each node) is strictly smaller than four, which is a necessary conditions coming from our equations (and linked to Maxwell criteria). Note that numerically it is still possible to find optimal moduli for them under some circumstances.