Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
588 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 243 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×#537 | C3 |
| hypercenter | #537 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#537 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #537 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×ΓL(1,𝔽8) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×C22⋊Q8 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 5. C2×C4⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×#537 | C3 | semidirect |
| 18. C2×C7×#41 | C2×C3 | semidirect |
| 19. C7×C22⋊Q8 | C2×C3 | semidirect |
| 20. C7×#121 | C2×C3 | semidirect |
| 21. C7×#121 | C2×C3 | semidirect |
| 22. C2×C7×C4⋊C4 | C2×C3 | semidirect |
| 32. C2×C4×ΓL(1,𝔽8) | D4 | semidirect |
| 62. #537 | ΓL(1,𝔽8) | direct (N × H) |
| 71. C2×C4×C7 | C3×D4 | semidirect |
| 91. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 92. C22⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 93. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 94. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 95. C2×C4⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 118. ΓL(1,𝔽8) | #537 | direct (N × H) |
| 153. C2×C4 | D4×ΓL(1,𝔽8) | semidirect |
| 157. C7 | C3×#537 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C7×#537 | 3 |
| C3×#537 | 7 |