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 | p7 |
| Subgroups | 2p6 + 6p5 + 10p4 + 12p3 + 10p2 + 6p + 8 |
| Classes of subgroups | 18 |
| Automorphisms | p25 − 2p24 + p22 + p21 − 2p19 + p18 |
| Classes with a complement | 6 |
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) | (Cp2)3 | p3 |
| Cp | (Cp2)2 × (Cp)2 | p2 + p + 1 |
| Cp2 (direct factor) | (Cp2)2 × Cp | p5 + p4 + p3 |
| (Cp)2 | (Cp2)2 × Cp | p4 + p3 + p2 |
| (Cp)2 | Cp2 × (Cp)3 | p2 + p + 1 |
| Cp2 × Cp (direct factor) | (Cp2)2 | p6 + p5 + p4 |
| Cp2 × Cp | Cp2 × (Cp)2 | p5 + 2p4 + 2p3 + p2 |
| (Cp)3 | Cp2 × (Cp)2 | p3 + p2 + p |
| (Cp)3 (characteristic) | (Cp)4 | 1 |
| (Cp2)2 (direct factor) | Cp2 × Cp | p6 + p5 + p4 |
| Cp2 × (Cp)2 | Cp2 × Cp | p5 + 2p4 + 2p3 + p2 |
| Cp2 × (Cp)2 | (Cp)3 | p3 + p2 + p |
| (Cp)4 (characteristic) | (Cp)3 | 1 |
| (Cp2)2 × Cp (direct factor) | Cp2 | p5 + p4 + p3 |
| (Cp2)2 × Cp | (Cp)2 | p4 + p3 + p2 |
| Cp2 × (Cp)3 | (Cp)2 | p2 + p + 1 |
| (Cp2)3 (direct factor) | Cp | p3 |
| (Cp2)2 × (Cp)2 | Cp | p2 + p + 1 |