Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
560 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 187 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×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×Q8 | C2 |
| prime radical | C2×D84 | C22 |
| hypercenter | C4×Q8 | D21 |
| hypocenter | C3×C7 | D4×Q8 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D21 | C2 | semidirect |
| 2. C3×C4×C7×Q8 | C2 | semidirect |
| 3. #10621380 | C2 | semidirect |
| 5. #10621352 | C2 | semidirect |
| 18. C4×C7×Q8 | D3 | semidirect |
| 24. C3×C7×Q8 | D4 | semidirect |
| 29. D84 | Q8 | direct (N × H) |
| 43. C3×C4×Q8 | D7 | semidirect |
| 50. C2×D21 | C2×Q8 | semidirect |
| 53. C3×C4×C7 | C2×Q8 | semidirect |
| 57. C7×Q8 | D12 | semidirect |
| 82. C4×Q8 | D21 | semidirect |
| 90. C4×C7 | D3×Q8 | semidirect |
| 93. C3×Q8 | D28 | semidirect |
| 101. C3×C7 | D4×Q8 | semidirect |
| 122. C3×C4 | Q8×D7 | semidirect |
| 126. Q8 | D84 | direct (N × H) |
| 133. C7 | Q8×D12 | semidirect |
| 145. C4 | Q8×D21 | semidirect |
| 146. C3 | Q8×D28 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D21 | 2 |
| C3×C4×C7×Q8 | 2 |
| #10621380 | 2 |
| C4×D84 | 2 |
| #10621352 | 2 |
| Q8×D28 | 3 |
| Q8×D12 | 7 |