Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
560 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C22×C7×Q8 | C2×C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7⋊SL(2,𝔽3) | C2 | semidirect |
| 2. C2×D7⋊SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C22×Q8×D7 | C3 | semidirect |
| 5. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 6. D7⋊SL(2,𝔽3) | C22 | direct (N × H) |
| 7. C22×C7×Q8 | C2×C3 | semidirect |
| 9. C2×Q8×D7 | C2×C3 | semidirect |
| 16. C7⋊SL(2,𝔽3) | C23 | semidirect |
| 19. C2×C7×Q8 | C22×C3 | semidirect |
| 24. Q8×D7 | C22×C3 | semidirect |
| 37. C22×D7 | SL(2,𝔽3) | semidirect |
| 41. C7×Q8 | C23×C3 | semidirect |
| 58. C22×Q8 | AGL(1,𝔽7) | semidirect |
| 60. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 62. C2×D7 | C2×SL(2,𝔽3) | semidirect |
| 79. C2×Q8 | C2×AGL(1,𝔽7) | semidirect |
| 84. C2×C7 | C22×SL(2,𝔽3) | semidirect |
| 86. D7 | C22×SL(2,𝔽3) | semidirect |
| 98. Q8 | C22×AGL(1,𝔽7) | semidirect |
| 104. C7 | C23×SL(2,𝔽3) | semidirect |
| 110. C22 | D7⋊SL(2,𝔽3) | direct (N × H) |
| 117. C2 | C2×D7⋊SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7⋊SL(2,𝔽3) | 2 |
| C2×D7⋊SL(2,𝔽3) | 2 |
| C22×Q8×D7 | 3 |
| C23×AGL(1,𝔽7) | 4 |
| C23×SL(2,𝔽3) | 7 |