Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
416 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10621615 | C2 | direct (N × H) |
| 2. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×C7×H(5,𝔽2) | C3 | semidirect |
| 4. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 5. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 8. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 13. C7⋊SL(2,𝔽3) | C23 | semidirect |
| 32. C2×H(5,𝔽2) | ΓL(1,𝔽8) | semidirect |
| 52. H(5,𝔽2) | C2×ΓL(1,𝔽8) | semidirect |
| 80. C2×C7 | #994 | semidirect |
| 97. C7 | C2×#994 | semidirect |
| 109. C2 | #10621615 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621615 | 2 |
| C2×#41⋊ΓL(1,𝔽8) | 2 |
| C2×C7×H(5,𝔽2) | 3 |
| C2×Q8×ΓL(1,𝔽8) | 4 |
| C2×#994 | 7 |