Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
480 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 219 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 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#540 | C3 |
| hypercenter | #540 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#540 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #540 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#127 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#125 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#123 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 6. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 7. ΓL(1,𝔽8)×#117 | C2 | semidirect |
| 8. C7×#540 | C3 | semidirect |
| 22. C7×#127 | C2×C3 | semidirect |
| 23. C7×#125 | C2×C3 | semidirect |
| 24. C7×#123 | C2×C3 | semidirect |
| 25. C7×#121 | C2×C3 | semidirect |
| 26. C7×#121 | C2×C3 | semidirect |
| 27. C4×C7×D4 | C2×C3 | semidirect |
| 28. C7×#117 | C2×C3 | semidirect |
| 73. #540 | ΓL(1,𝔽8) | direct (N × H) |
| 103. #127 | C2×ΓL(1,𝔽8) | semidirect |
| 104. #125 | C2×ΓL(1,𝔽8) | semidirect |
| 105. #123 | C2×ΓL(1,𝔽8) | semidirect |
| 106. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 107. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 108. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 109. #117 | C2×ΓL(1,𝔽8) | semidirect |
| 133. ΓL(1,𝔽8) | #540 | direct (N × H) |
| 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 |
| ΓL(1,𝔽8)×#125 | 2 |
| ΓL(1,𝔽8)×#123 | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| C7×#540 | 3 |
| C3×#540 | 7 |