Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
472 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 225 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×#555 | C3 |
| hypercenter | #555 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#555 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #555 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×C4⋊Q8 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#124 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 4. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#117 | C2 | semidirect |
| 6. C7×#555 | C3 | semidirect |
| 17. C7×C4⋊Q8 | C2×C3 | semidirect |
| 18. C7×#124 | C2×C3 | semidirect |
| 19. C7×#121 | C2×C3 | semidirect |
| 20. C4×C7×D4 | C2×C3 | semidirect |
| 21. C7×#117 | C2×C3 | semidirect |
| 55. #555 | ΓL(1,𝔽8) | direct (N × H) |
| 78. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 79. #124 | C2×ΓL(1,𝔽8) | semidirect |
| 80. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 81. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 82. #117 | C2×ΓL(1,𝔽8) | semidirect |
| 100. ΓL(1,𝔽8) | #555 | direct (N × H) |
| 131. C7 | C3×#555 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| ΓL(1,𝔽8)×#124 | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| C7×#555 | 3 |
| C3×#555 | 7 |