Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
480 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 168 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 | C7×#127 | C2×C3 |
| hypercenter | #127 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#540 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#127 | C2 | semidirect |
| 2. #6502⋊C3 | C2 | semidirect |
| 3. #6501⋊C3 | C2 | semidirect |
| 4. #6498⋊C3 | C2 | semidirect |
| 5. D4×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 6. #6435⋊C3 | C2 | semidirect |
| 7. #6433⋊C3 | C2 | semidirect |
| 8. #90410 | C3 | semidirect |
| 22. C7×#127 | C2×C3 | semidirect |
| 23. #6502 | C2×C3 | semidirect |
| 24. #6501 | C2×C3 | semidirect |
| 25. #6498 | C2×C3 | semidirect |
| 26. D4×Dic7 | C2×C3 | semidirect |
| 27. #6435 | C2×C3 | semidirect |
| 28. #6433 | C2×C3 | semidirect |
| 103. #127 | AGL(1,𝔽7) | semidirect |
| 133. ΓL(1,𝔽8) | #540 | semidirect |
| 174. C7 | C3×#540 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#127 | 2 |
| #6502⋊C3 | 2 |
| #6501⋊C3 | 2 |
| #6498⋊C3 | 2 |
| D4×ΓL(1,𝔽8)⋊C4 | 2 |
| #6435⋊C3 | 2 |
| #6433⋊C3 | 2 |
| #90410 | 3 |
| C3×#540 | 7 |