Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×C22⋊Q8 | C2 |
| prime radical | C2×C22⋊D7 | C22 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | D4×Q8 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D7 | C2 | semidirect |
| 2. C22×Dic14 | C2 | semidirect |
| 3. C7×C22⋊Q8 | C2 | semidirect |
| 4. #6507 | C2 | semidirect |
| 6. #6460 | C2 | semidirect |
| 7. #6459 | C2 | semidirect |
| 8. #6451 | C2 | semidirect |
| 9. #6450 | C2 | semidirect |
| 11. #6441 | C2 | semidirect |
| 46. C22⋊D7 | Q8 | semidirect |
| 58. Dic14 | D4 | semidirect |
| 66. C22⋊Q8 | D7 | semidirect |
| 72. C22×C7 | C2×Q8 | semidirect |
| 76. C2×D7 | C2×Q8 | semidirect |
| 85. Dic7 | C2×Q8 | semidirect |
| 137. C7 | D4×Q8 | semidirect |
| 143. C22 | Q8×D7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D7 | 2 |
| C22×Dic14 | 2 |
| C7×C22⋊Q8 | 2 |
| #6507 | 2 |
| C4×C22⋊D7 | 2 |
| #6460 | 2 |
| #6459 | 2 |
| #6451 | 2 |
| #6450 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| #6441 | 2 |
| D4×Q8 | 7 |