Finite Group DatabasebetaFinite Group DatabasebetaOrder 704.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D11 |
| commutator, socle, cosocle | C22×C11 | C24 |
| Fitting | C4×Q8×C11 | C2 |
| prime radical | #66948 | C2 |
| hypercenter | C4×Q8 | D11 |
| hypocenter | C11 | #542 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×Q8×C11 | C2 | semidirect |
| 2. #67000 | C2 | semidirect |
| 3. #66999 | C2 | semidirect |
| 4. #66949 | C2 | semidirect |
| 6. #66947 | C2 | semidirect |
| 7. #66941 | C2 | semidirect |
| 8. #66929 | C2 | semidirect |
| 9. #66928 | C2 | semidirect |
| 10. C4×D44 | C2 | semidirect |
| 11. #66925 | C2 | semidirect |
| 77. C4×Q8 | D11 | semidirect |
| 108. C11 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×Q8×C11 | 2 |
| #67000 | 2 |
| #66999 | 2 |
| #66949 | 2 |
| #66948 | 2 |
| #66947 | 2 |
| #66941 | 2 |
| #66929 | 2 |
| #66928 | 2 |
| C4×D44 | 2 |
| #66925 | 2 |
| #542 | 11 |