Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
322 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| prime radical | C24×C7 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×Q8 | C2 | semidirect |
| 3. C7×C22⋊Q8 | C2 | direct (N × H) |
| 4. C2×C7×C4⋊C4 | C2 | semidirect |
| 5. C2×C7×C4⋊C4 | C2 | semidirect |
| 9. C2×C7×Q8 | C22 | semidirect |
| 16. C7×C4⋊C4 | C22 | semidirect |
| 17. C7×C4⋊C4 | C22 | semidirect |
| 19. C2×C22⋊Q8 | C7 | direct (N × H) |
| 22. C23×C7 | Q8 | semidirect |
| 35. C22×Q8 | C2×C7 | semidirect |
| 37. C22⋊Q8 | C2×C7 | direct (N × H) |
| 38. C2×C4⋊C4 | C2×C7 | semidirect |
| 39. C2×C4⋊C4 | C2×C7 | semidirect |
| 46. C22×C7 | C2×Q8 | semidirect |
| 55. C2×Q8 | C22×C7 | semidirect |
| 62. C4⋊C4 | C22×C7 | semidirect |
| 63. C4⋊C4 | C22×C7 | semidirect |
| 66. C2×C7 | C22⋊Q8 | direct (N × H) |
| 72. C23 | C7×Q8 | semidirect |
| 85. C7 | C2×C22⋊Q8 | direct (N × H) |
| 91. C22 | C2×C7×Q8 | semidirect |
| 99. C2 | C7×C22⋊Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×Q8 | 2 |
| C23×C4×C7 | 2 |
| C7×C22⋊Q8 | 2 |
| C2×C7×C4⋊C4 | 2 |
| C2×C7×C22⋊C4 | 2 |
| C2×C22⋊Q8 | 7 |