Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
210 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C2×D7⋊A4 |
| commutator, cosocle | C2×C7×Q8 | C22×C3 |
| Fitting | C22×C7×Q8 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C23 | D7⋊A4 |
| hypocenter | C7×Q8 | C3×D4 |
| LCS | G ▷ C2×C7×Q8 ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ 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 | semidirect |
| 3. Dic7⋊SL(2,𝔽3) | C2 | semidirect |
| 4. Q8×C22⋊D7 | C3 | semidirect |
| 9. C22×C7×Q8 | C2×C3 | semidirect |
| 10. C2×Q8×D7 | C2×C3 | semidirect |
| 12. Q8×Dic7 | C2×C3 | semidirect |
| 19. C7⋊SL(2,𝔽3) | D4 | semidirect |
| 55. C7×Q8 | C3×D4 | semidirect |
| 60. C22⋊D7 | SL(2,𝔽3) | semidirect |
| 78. C22×Q8 | AGL(1,𝔽7) | semidirect |
| 86. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 90. C2×D7 | C2×SL(2,𝔽3) | semidirect |
| 96. Dic7 | C2×SL(2,𝔽3) | semidirect |
| 139. Q8 | (C22⋊D7)⋊C3 | semidirect |
| 150. C7 | D4×SL(2,𝔽3) | semidirect |
| 161. C22 | D7⋊SL(2,𝔽3) | semidirect |
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 |
| Dic7⋊SL(2,𝔽3) | 2 |
| Q8×C22⋊D7 | 3 |
| C2×(C22⋊D7)⋊C3 | 4 |
| D4×SL(2,𝔽3) | 7 |