You can do it -- you'll need a subpanel loadcenter in a NEMA 3R enclosure, though, as well as watertight conduit/raceway and fittings for the outdoor runs, which will be done using THWN in the conduits.

You will want to put a 30A 2pole for the dryer and a 20A GFCI for the laundry outlets, as well as a spare 20A breaker (if you want a lighting circuit there, or an extra laundry outlet circuit) in your subpanel, which does not need a main breaker of its own, so it can be a four-slot NEMA 3R loadcenter, such as an Eaton BR48L125RP with a GBK10 ground bar in it, as the neutral and ground will be on separate, isolated bars in the subpanel.

As to the feeder breaker, the NEC calls for 1500VA for the laundry outlet circuit (220.52(B)) and a minimum of 5000VA (a 20A dryer) or the wattage of the dryer (whichever is higher) for an electric dryer load (220.54). Assuming your dryer pulls the full 30A (7500VA) and you have 1500VA of laundry circuit load, this yields 9000VA or 37.5A, which requires a 40A feeder breaker in the main panel, and 8AWG THWN for the feeder wires. In reality, your dryer can't sit there and pull 30A continuously without tripping your breaker, so this is a conservative calculation.

It's difficult to tell from the photo if there's even a grounding conductor run along with the two ungrounded (hot) conductors, so you'll have to verify that before you go shopping for supplies. You'll also have to verify that the ungrounded (hot) conductors are the proper size. You said the panel is fed through a 60 ampere breaker, so you'll need at least 6 AWG copper conductors (solid or stranded, it makes no difference).

Shopping

Here's what you're going to need.

• A grounding bus bar.
• 6 AWG copper THWN wire with white or gray insulation, long enough to reach from the main grounded (neutral) bar to second panel grounded (neutral) bar.
• 6 AWG bare or green copper conductor, long enough to reach from the main grounding bar to the second panel grounding bar (which you'll install).

If the current ungrounded (hot) conductors are undersized, you have a couple choices. You can replace them, with appropriate conductors. Or you can install a smaller feeder breaker in the main panel, and down size the grounded and grounding conductors accordingly.

Installation

Once you have the supplies, here's what you're going to do.

• Remove the brass bonding screw from the existing bus bar in the second panel.
• Install the grounding bus bar (the one you purchased), in the panel.
• Move all the grounding conductors from the old grounding bus, to the new grounding bus bar.
• Pull the required conductors between the panels.
• Connect the grounded (neutral) conductor to the grounded (neutral) bar in the main panel.
• Connect the grounding conductor to the grounding bar in the main panel.
• Connect the grounded (neutral) conductor to the grounded (neutral) bar in the second panel.
• Connect the grounding conductor to the new grounding bar in the second panel.
• Connect the grounded (neutral) pigtail from the GFCI breaker, to the grounded (neutral) bar in the second panel.

Now you should be all set.

NOTE: Depending on where the second panel is installed. You may need to install a grounding electrode, and bond the panel to it. Check with your local building department, to determine if this is required.