Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
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×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C4⋊Q8 | C2 |
| prime radical | #6458 | C2 |
| hypercenter | C4⋊Q8 | D7 |
| hypocenter | C7 | #542 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C4⋊Q8 | C2 | semidirect |
| 2. #6510 | C2 | semidirect |
| 3. #6509 | C2 | semidirect |
| 4. #6461 | C2 | semidirect |
| 5. #6460 | C2 | semidirect |
| 7. #6457 | C2 | semidirect |
| 8. #6456 | C2 | semidirect |
| 9. #6450 | C2 | semidirect |
| 10. #6438 | C2 | semidirect |
| 11. #6435 | C2 | semidirect |
| 53. C4⋊Q8 | D7 | semidirect |
| 125. C7 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C4⋊Q8 | 2 |
| #6510 | 2 |
| #6509 | 2 |
| #6461 | 2 |
| #6460 | 2 |
| #6458 | 2 |
| #6457 | 2 |
| #6456 | 2 |
| #6450 | 2 |
| #6438 | 2 |
| #6435 | 2 |
| #542 | 7 |