Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C2×C3×C7×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D21 |
| hypocenter | C3×C7 | #537 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66728 | C2 | semidirect |
| 2. C2×C3×C7×C4⋊C4 | C2 | semidirect |
| 3. #10621413 | C2 | semidirect |
| 4. #10621380 | C2 | semidirect |
| 5. #10621378 | C2 | semidirect |
| 19. C2×C7×C4⋊C4 | D3 | semidirect |
| 27. C2×C3×C4×C7 | D4 | semidirect |
| 48. C2×C3×C4⋊C4 | D7 | semidirect |
| 66. C2×C4×C7 | D12 | semidirect |
| 95. C2×C4⋊C4 | D21 | semidirect |
| 109. C2×C3×C4 | D28 | semidirect |
| 118. C3×C7 | #537 | semidirect |
| 154. C2×C4 | D84 | semidirect |
| 157. C7 | #5510 | semidirect |
| 173. C3 | #90201 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66728 | 2 |
| C2×C3×C7×C4⋊C4 | 2 |
| #10621413 | 2 |
| #10621380 | 2 |
| #10621378 | 2 |
| #90201 | 3 |
| #5510 | 7 |