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 | C4×C7×Q8 | C2×C3 |
| hypercenter | C4×Q8 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#548 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66659 | C2 | semidirect |
| 2. C2×Q8×AGL(1,𝔽7) | C2 | semidirect |
| 3. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. #6460⋊C3 | C2 | semidirect |
| 5. #6458⋊C3 | C2 | semidirect |
| 6. #6438⋊C3 | C2 | semidirect |
| 7. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 8. #90267 | C3 | semidirect |
| 23. C2×#1161 | C2×C3 | semidirect |
| 24. C2×Q8×D7 | C2×C3 | semidirect |
| 25. C4×C7×Q8 | C2×C3 | semidirect |
| 26. #6460 | C2×C3 | semidirect |
| 27. #6458 | C2×C3 | semidirect |
| 28. #6438 | C2×C3 | semidirect |
| 29. C4×D28 | C2×C3 | semidirect |
| 33. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 71. C2×AGL(1,𝔽7) | #41 | semidirect |
| 80. C7×Q8 | C3×D4 | semidirect |
| 115. C4×Q8 | AGL(1,𝔽7) | semidirect |
| 123. C2×D7 | C3×#41 | semidirect |
| 142. ΓL(1,𝔽8) | #548 | semidirect |
| 177. Q8 | C4⋊AGL(1,𝔽7) | 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×#66659 | 2 |
| C2×Q8×AGL(1,𝔽7) | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| #6460⋊C3 | 2 |
| #6458⋊C3 | 2 |
| #6438⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #90267 | 3 |
| C3×#548 | 7 |