Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C4×C7×D4 | C2 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | D4×Q8 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Dic14 | C2 | semidirect |
| 4. #6487 | C2 | semidirect |
| 5. #6451 | C2 | semidirect |
| 6. #6441 | C2 | semidirect |
| 7. #6432 | C2 | semidirect |
| 8. C4×Dic14 | C2 | semidirect |
| 28. C7×D4 | Q8 | semidirect |
| 40. Dic14 | D4 | direct (N × H) |
| 51. C22×C7 | C2×Q8 | semidirect |
| 56. C4×C7 | C2×Q8 | semidirect |
| 85. D4 | Dic14 | direct (N × H) |
| 95. C7 | D4×Q8 | semidirect |
| 98. C22 | C2×Dic14 | semidirect |
| 105. C4 | C2×Dic14 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Dic14 | 2 |
| C4×C7×D4 | 2 |
| D4×Dic7 | 2 |
| #6487 | 2 |
| #6451 | 2 |
| #6441 | 2 |
| #6432 | 2 |
| C4×Dic14 | 2 |
| D4×Q8 | 7 |