Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #127⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C42×C7 | C22×C3 |
| Frattini | C22 | C22×C22⋊ΓL(1,𝔽8) |
| Fitting | C2×C7×#127 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #127⋊ΓL(1,𝔽8) | C2 | direct (N × H) |
| 2. C2×C7⋊(C42⋊C3) | C2 | semidirect |
| 3. C2×C7×#127 | C3 | semidirect |
| 5. C7⋊(C42⋊C3) | C22 | semidirect |
| 9. C7×#127 | C2×C3 | semidirect |
| 10. C2×C42×C7 | C2×C3 | semidirect |
| 26. C42×C7 | C22×C3 | semidirect |
| 31. C2×#127 | ΓL(1,𝔽8) | semidirect |
| 56. #127 | C2×ΓL(1,𝔽8) | semidirect |
| 57. C2×C42 | C2×ΓL(1,𝔽8) | semidirect |
| 86. C42 | C22×ΓL(1,𝔽8) | semidirect |
| 91. C2×C7 | #127⋊C3 | semidirect |
| 111. C7 | C2×#127⋊C3 | semidirect |
| 132. C2 | #127⋊ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #127⋊ΓL(1,𝔽8) | 2 |
| C2×C7⋊(C42⋊C3) | 2 |
| C2×C7×#127 | 3 |
| C22×C22⋊ΓL(1,𝔽8) | 4 |
| C2×#127⋊C3 | 7 |