Finite Group DatabasebetaFinite Group DatabasebetaValid at every prime. The subgroup lattice of this shape is one object: each count below is a polynomial in p, and substituting a prime gives the lattice of that group.
| Order | p4 |
| Subgroups | p4 + 3p3 + 4p2 + 3p + 5 |
| Classes of subgroups | 3 |
| Automorphisms | p16 − p15 − p14 + 2p11 − p8 − p7 + p6 |
| Classes with a complement | 3 |
Each circle is one class of subgroups, labeled by the exponents of its shape — 3,2 is Cp3 × Cp2, and 2²,1 is (Cp2)2 × Cp. The badge is how many subgroups the class holds. Hover a row of the table to light its circle, or click a circle to jump to its row.
| Subgroup | Quotient | Subgroups |
|---|---|---|
| Cp (direct factor) | (Cp)3 | p3 + p2 + p + 1 |
| (Cp)2 (direct factor) | (Cp)2 | p4 + p3 + 2p2 + p + 1 |
| (Cp)3 (direct factor) | Cp | p3 + p2 + p + 1 |