Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
540 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 84 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. ΓL(1,𝔽8)×H(3,𝔽4) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×B(4,𝔽2) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#455 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#354 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×AΣL(1,𝔽16) | C2 | semidirect |
| 6. C7×#2199 | C3 | semidirect |
| 20. C7×H(3,𝔽4) | C2×C3 | semidirect |
| 21. C7×B(4,𝔽2) | C2×C3 | semidirect |
| 22. C7×#455 | C2×C3 | semidirect |
| 23. C7×#354 | C2×C3 | semidirect |
| 24. C7×AΣL(1,𝔽16) | C2×C3 | semidirect |
| 26. C24×ΓL(1,𝔽8) | D4 | semidirect |
| 30. C2×Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 47. C22⋊C4×ΓL(1,𝔽8) | D4 | semidirect |
| 49. C42×ΓL(1,𝔽8) | D4 | semidirect |
| 92. #2199 | ΓL(1,𝔽8) | direct (N × H) |
| 93. C24×C7 | C3×D4 | semidirect |
| 97. C2×C7×Q8 | C3×D4 | semidirect |
| 114. C7×C22⋊C4 | C3×D4 | semidirect |
| 116. C42×C7 | C3×D4 | semidirect |
| 146. H(3,𝔽4) | C2×ΓL(1,𝔽8) | semidirect |
| 147. B(4,𝔽2) | C2×ΓL(1,𝔽8) | semidirect |
| 148. #455 | C2×ΓL(1,𝔽8) | semidirect |
| 149. #354 | C2×ΓL(1,𝔽8) | semidirect |
| 150. AΣL(1,𝔽16) | C2×ΓL(1,𝔽8) | semidirect |
| 257. ΓL(1,𝔽8) | #2199 | direct (N × H) |
| 258. C24 | D4×ΓL(1,𝔽8) | semidirect |
| 262. C2×Q8 | D4×ΓL(1,𝔽8) | semidirect |
| 279. C22⋊C4 | D4×ΓL(1,𝔽8) | semidirect |
| 281. C42 | D4×ΓL(1,𝔽8) | semidirect |
| 327. C7 | C3×#2199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×H(3,𝔽4) | 2 |
| ΓL(1,𝔽8)×B(4,𝔽2) | 2 |
| ΓL(1,𝔽8)×#455 | 2 |
| ΓL(1,𝔽8)×#354 | 2 |
| ΓL(1,𝔽8)×AΣL(1,𝔽16) | 2 |
| C7×#2199 | 3 |
| C3×#2199 | 7 |