Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1112 conjugacy classes of subgroups; only the 621 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 621 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D7 |
| commutator, cosocle | C22×C7 | C25 |
| Frattini | C22 | C24×D7 |
| Fitting | C2×C4×C7×Q8 | C2 |
| socle, prime radical | C23×C7 | C24 |
| hypercenter | C2×C4×Q8 | D7 |
| hypocenter | C7 | C2×Q82 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. Q8×Dic14 | C2 | direct (N × H) |
| 22. C2×C7×Q8 | Q8 | semidirect |
| 30. C2×Dic14 | Q8 | direct (N × H) |
| 46. C7×Q8 | C2×Q8 | semidirect |
| 58. Dic14 | C2×Q8 | direct (N × H) |
| 86. C2×Q8 | Dic14 | direct (N × H) |
| 99. C2×C7 | Q82 | semidirect |
| 106. Q8 | C2×Dic14 | direct (N × H) |
| 117. C7 | C2×Q82 | semidirect |
| 127. C2 | Q8×Dic14 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×Q8 | 2 |
| C2×Q8×Dic7 | 2 |
| Q8×Dic14 | 2 |
| C2×#6451 | 2 |
| C2×#6432 | 2 |
| C2×C4×Dic14 | 2 |
| C2×Q82 | 7 |