Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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×Q8 | C2 |
| prime radical | C2×D28 | C22 |
| hypercenter | C4×Q8 | D7 |
| hypocenter | C7 | #554 |
| 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. C4×C7×Q8 | C2 | semidirect |
| 2. #6509 | C2 | semidirect |
| 3. #6459 | C2 | semidirect |
| 4. D7×C4⋊C4 | C2 | semidirect |
| 7. #6433 | C2 | semidirect |
| 8. #6432 | C2 | semidirect |
| 35. D28 | Q8 | semidirect |
| 43. C4×Q8 | D7 | semidirect |
| 91. C7 | #554 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×Q8 | 2 |
| #6509 | 2 |
| #6459 | 2 |
| D7×C4⋊C4 | 2 |
| C4×D28 | 2 |
| #6433 | 2 |
| #6432 | 2 |
| #554 | 7 |