Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
472 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 174 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×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C4×C7×D4 | C2×C3 |
| hypercenter | C4×D4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#555 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 2. #6501⋊C3 | C2 | semidirect |
| 3. #6493⋊C3 | C2 | semidirect |
| 4. #6489⋊C3 | C2 | semidirect |
| 5. #6455⋊C3 | C2 | semidirect |
| 6. #6448⋊C3 | C2 | semidirect |
| 7. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 8. #6432⋊C3 | C2 | semidirect |
| 9. #90254 | C3 | semidirect |
| 26. C4×C7×D4 | C2×C3 | semidirect |
| 27. #6501 | C2×C3 | semidirect |
| 28. #6493 | C2×C3 | semidirect |
| 29. #6489 | C2×C3 | semidirect |
| 30. #6455 | C2×C3 | semidirect |
| 31. #6448 | C2×C3 | semidirect |
| 32. C4×D28 | C2×C3 | semidirect |
| 33. #6432 | C2×C3 | semidirect |
| 127. C4×D4 | AGL(1,𝔽7) | semidirect |
| 157. ΓL(1,𝔽8) | #555 | semidirect |
| 206. C7 | C3×#555 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D4×ΓL(1,𝔽8) | 2 |
| #6501⋊C3 | 2 |
| #6493⋊C3 | 2 |
| #6489⋊C3 | 2 |
| #6455⋊C3 | 2 |
| #6448⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #6432⋊C3 | 2 |
| #90254 | 3 |
| C3×#555 | 7 |