Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 316 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 |
|---|---|---|
| 1. C22×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 6. C7×#448 | C3 | semidirect |
| 18. C22×C7×Q8 | C2×C3 | semidirect |
| 19. C2×C7×SD16 | C2×C3 | semidirect |
| 21. C7×#103 | C2×C3 | semidirect |
| 22. C7×C22⋊C8 | C2×C3 | semidirect |
| 28. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 59. C22×ΓL(1,𝔽8) | SD16 | semidirect |
| 65. #448 | ΓL(1,𝔽8) | direct (N × H) |
| 70. C7×Q8 | C3×D4 | semidirect |
| 96. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 97. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 99. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 100. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 106. C22×C7 | C3×SD16 | semidirect |
| 126. ΓL(1,𝔽8) | #448 | direct (N × H) |
| 158. Q8 | D4×ΓL(1,𝔽8) | semidirect |
| 168. C7 | C3×#448 | semidirect |
| 179. C22 | SD16×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×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#448 | 3 |
| C3×#448 | 7 |