Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
350 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting | C22×C7×#41 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊(C7⋊SL(2,𝔽3)) | C2 | semidirect |
| 2. C22×C7×#41 | C3 | semidirect |
| 17. C7×#41 | A4 | semidirect |
| 27. C22×#41 | ΓL(1,𝔽8) | semidirect |
| 54. C22×C7 | #41⋊C3 | semidirect |
| 65. #41 | C22⋊ΓL(1,𝔽8) | semidirect |
| 91. C7 | #5948 | semidirect |
| 98. C22 | #41⋊ΓL(1,𝔽8) | 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 |
| C22×C7×#41 | 3 |
| #41⋊ΓL(1,𝔽8) | 4 |
| C4×C22⋊ΓL(1,𝔽8) | 4 |
| #5948 | 7 |