Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
634 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×D4 | C2×C3 |
| hypercenter | D4 | D7⋊A4 |
| hypocenter | C22×C7 | C2×C3×D4 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D7⋊A4 | C2 | semidirect |
| 2. D4×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. (C22⋊D7)⋊A4 | C2 | semidirect |
| 4. C4⋊(D7⋊A4) | C2 | semidirect |
| 5. C4×D7⋊A4 | C2 | semidirect |
| 6. C22×D4×D7 | C3 | semidirect |
| 7. C22×C22⋊ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×D7⋊A4 | C22 | semidirect |
| 11. C4×C22⋊ΓL(1,𝔽8) | C22 | semidirect |
| 12. Dic7⋊A4 | C22 | semidirect |
| 14. C24×D7 | C2×C3 | semidirect |
| 15. C22×C7×D4 | C2×C3 | semidirect |
| 16. C22×C22⋊D7 | C2×C3 | semidirect |
| 23. C22×D28 | C2×C3 | semidirect |
| 24. C22×C4×D7 | C2×C3 | semidirect |
| 29. D7⋊A4 | D4 | direct (N × H) |
| 35. C24×C7 | C22×C3 | semidirect |
| 43. C23×D7 | C22×C3 | semidirect |
| 47. C22×C4×C7 | C22×C3 | semidirect |
| 52. C22×Dic7 | C22×C3 | semidirect |
| 63. D4×D7 | A4 | semidirect |
| 76. C22⋊ΓL(1,𝔽8) | C2×D4 | semidirect |
| 108. C22×D7 | C3×D4 | semidirect |
| 109. C22×D7 | C2×A4 | semidirect |
| 113. C7×D4 | C2×A4 | semidirect |
| 120. C22⋊D7 | C2×A4 | semidirect |
| 126. D28 | C2×A4 | semidirect |
| 131. C4×D7 | C2×A4 | semidirect |
| 152. C22×D4 | AGL(1,𝔽7) | semidirect |
| 161. C22×C7 | C2×C3×D4 | semidirect |
| 162. C22×C7 | C22×A4 | semidirect |
| 179. C2×D7 | C22×A4 | semidirect |
| 181. C4×C7 | C22×A4 | semidirect |
| 183. Dic7 | C22×A4 | semidirect |
| 202. C24 | C2×AGL(1,𝔽7) | semidirect |
| 229. C22×C4 | C2×AGL(1,𝔽7) | semidirect |
| 238. D7 | D4×A4 | semidirect |
| 284. D4 | D7⋊A4 | direct (N × H) |
| 294. C7 | C2×D4×A4 | semidirect |
| 323. C22 | D4×AGL(1,𝔽7) | semidirect |
| 324. C22 | C2×D7⋊A4 | semidirect |
| 328. C4 | C2×D7⋊A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D7⋊A4 | 2 |
| D4×C22⋊ΓL(1,𝔽8) | 2 |
| (C22⋊D7)⋊A4 | 2 |
| C4⋊(D7⋊A4) | 2 |
| C4×D7⋊A4 | 2 |
| C22×D4×D7 | 3 |
| D4×AGL(1,𝔽7) | 4 |
| C2×D4×A4 | 7 |