Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C2×C7×#41 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C2×C4 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. #10621541 | C2 | direct (N × H) |
| 3. C2×D7⋊SL(2,𝔽3) | C2 | semidirect |
| 4. C2×#6554 | C3 | semidirect |
| 5. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 7. D7⋊SL(2,𝔽3) | C22 | semidirect |
| 9. C2×C7×#41 | C2×C3 | semidirect |
| 11. #6554 | C2×C3 | semidirect |
| 12. C2×Q8×D7 | C2×C3 | semidirect |
| 22. C7×#41 | C22×C3 | semidirect |
| 29. Q8×D7 | C22×C3 | semidirect |
| 78. C2×#41 | AGL(1,𝔽7) | semidirect |
| 100. #41 | C2×AGL(1,𝔽7) | semidirect |
| 115. C2×C7 | 2−1+4⋊C3 | semidirect |
| 141. C7 | C2×2−1+4⋊C3 | semidirect |
| 157. C2 | #10621541 | direct (N × H) |
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 |
| #10621541 | 2 |
| C2×D7⋊SL(2,𝔽3) | 2 |
| C2×#6554 | 3 |
| C2×C4⋊AGL(1,𝔽7) | 4 |
| C2×2−1+4⋊C3 | 7 |