Finite Group DatabasebetaFinite Group DatabasebetaOrder 8064.
1890 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1890 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214620 |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C23×C3 |
| Frattini | C22 | C22×A4⋊AGL(1,𝔽7) |
| Fitting | C2×C7×#120 | C3×D3 |
| socle | C23×C7 | C2×C3×S4 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#13057164 | 3 |
| C2×ΓL(1,𝔽8)×#120⋊C3 | 2 |
| C2×#89804⋊C3 | 3 |
| C2×#72214032 | 2 |
| C22×A4⋊AGL(1,𝔽7) | 4 |
| #72214620 | 2 |
| C2×C3×#1020⋊C2 | 7 |