Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
440 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊ΓL(1,𝔽8) |
| Fitting | C22×C7×#41 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41⋊ΓL(1,𝔽8) | C2 | direct (N × H) |
| 2. C22×C7⋊SL(2,𝔽3) | C2 | semidirect |
| 3. C22×C7×#41 | C3 | semidirect |
| 4. #41⋊ΓL(1,𝔽8) | C22 | direct (N × H) |
| 5. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 8. C2×C7×#41 | C2×C3 | semidirect |
| 14. C7⋊SL(2,𝔽3) | C23 | semidirect |
| 19. C7×#41 | C22×C3 | semidirect |
| 33. C22×#41 | ΓL(1,𝔽8) | semidirect |
| 53. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 59. C22×C7 | #41⋊C3 | semidirect |
| 71. #41 | C22×ΓL(1,𝔽8) | semidirect |
| 81. C2×C7 | C2×#41⋊C3 | semidirect |
| 99. C7 | C22×#41⋊C3 | semidirect |
| 104. C22 | #41⋊ΓL(1,𝔽8) | direct (N × H) |
| 111. C2 | C2×#41⋊ΓL(1,𝔽8) | 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 |
| C22×C7⋊SL(2,𝔽3) | 2 |
| C22×C7×#41 | 3 |
| C22×C4×ΓL(1,𝔽8) | 4 |
| C22×#41⋊C3 | 7 |