Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3998 conjugacy classes of subgroups; only the 651 normal subgroups are shown.
This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 651 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D7 |
| commutator | C23×C7 | C23×C4 |
| Fitting | C24×C7×Q8 | C2 |
| socle, cosocle | C24×C7 | C24 |
| prime radical | C25×C7 | C23 |
| hypercenter | C24×Q8 | D7 |
| hypocenter | C7 | #14336 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C24 ◁ C24×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #11729888 | C2 | semidirect |
| 5. C23×C7×Q8 | C4 | semidirect |
| 44. C23×Q8 | Dic7 | semidirect |
| 119. C7 | #14336 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7×Q8 | 2 |
| C2×#89983 | 2 |
| #11729888 | 2 |
| #14336 | 7 |