Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
390 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | #41 | C22⋊ΓL(1,𝔽8) |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. #10621615 | C2 | semidirect |
| 3. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 4. C7×#585 | C3 | semidirect |
| 5. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 6. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 7. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 10. C7×2−1+4 | C2×C3 | semidirect |
| 12. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 15. C2×C7×#41 | C2×C3 | semidirect |
| 46. #585 | ΓL(1,𝔽8) | semidirect |
| 73. 2−1+4 | C2×ΓL(1,𝔽8) | semidirect |
| 75. H(5,𝔽2) | C2×ΓL(1,𝔽8) | semidirect |
| 78. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 138. C7 | #5945 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊ΓL(1,𝔽8) | 2 |
| #10621615 | 2 |
| C2×#41⋊ΓL(1,𝔽8) | 2 |
| C7×#585 | 3 |
| #41×ΓL(1,𝔽8) | 4 |
| #5945 | 7 |