Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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 |
|---|---|---|
| 2. C2×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. C7×#451 | C3 | semidirect |
| 19. C2×C7×Q16 | C2×C3 | semidirect |
| 21. C7×#103 | C2×C3 | semidirect |
| 55. C22×ΓL(1,𝔽8) | Q16 | semidirect |
| 62. #451 | ΓL(1,𝔽8) | direct (N × H) |
| 92. C2×Q16 | C2×ΓL(1,𝔽8) | semidirect |
| 94. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 98. C22×C7 | C3×Q16 | semidirect |
| 118. ΓL(1,𝔽8) | #451 | direct (N × H) |
| 156. C7 | C3×#451 | semidirect |
| 163. C22 | Q16×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×Q16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#451 | 3 |
| C3×#451 | 7 |