Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
580 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 196 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 | C2×C7×#41 | C2×C3 |
| hypercenter | C2×#41 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#548 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×Q8×AGL(1,𝔽7) | C2 | semidirect |
| 3. Q8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. #6501⋊C3 | C2 | semidirect |
| 5. #6499⋊C3 | C2 | semidirect |
| 6. C4×(C22⋊D7)⋊C3 | C2 | semidirect |
| 7. #6487⋊C3 | C2 | semidirect |
| 8. #90522 | C3 | semidirect |
| 23. C2×C7×#41 | C2×C3 | semidirect |
| 24. C2×Q8×D7 | C2×C3 | semidirect |
| 25. Q8×Dic7 | C2×C3 | semidirect |
| 26. #6501 | C2×C3 | semidirect |
| 27. #6499 | C2×C3 | semidirect |
| 28. C4×C22⋊D7 | C2×C3 | semidirect |
| 29. #6487 | C2×C3 | semidirect |
| 33. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 71. C2×AGL(1,𝔽7) | #41 | semidirect |
| 80. C7×Q8 | C3×D4 | semidirect |
| 110. C2×#41 | AGL(1,𝔽7) | semidirect |
| 123. C2×D7 | C3×#41 | semidirect |
| 142. ΓL(1,𝔽8) | #548 | semidirect |
| 177. Q8 | (C22⋊D7)⋊C3 | semidirect |
| 187. C7 | C3×#548 | 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 |
| C2×Q8×AGL(1,𝔽7) | 2 |
| Q8×ΓL(1,𝔽8)⋊C4 | 2 |
| #6501⋊C3 | 2 |
| #6499⋊C3 | 2 |
| C4×(C22⋊D7)⋊C3 | 2 |
| #6487⋊C3 | 2 |
| #90522 | 3 |
| C3×#548 | 7 |