Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
634 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D7⋊A4 |
| commutator, cosocle | C23×C7 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C25×C7 | C2×C3 |
| socle | C24×C7 | C22×C3 |
| hypercenter | C23 | D7⋊A4 |
| hypocenter | C22×C7 | C2×C3×D4 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×D7⋊A4 | C2 | semidirect |
| 3. (C22⋊D7)⋊A4 | C2 | direct (N × H) |
| 4. C2×Dic7⋊A4 | C2 | semidirect |
| 5. C23×C22⋊D7 | C3 | semidirect |
| 8. C22×C22⋊ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×D7⋊A4 | C22 | semidirect |
| 11. Dic7⋊A4 | C22 | semidirect |
| 13. C25×C7 | C2×C3 | semidirect |
| 14. C24×D7 | C2×C3 | semidirect |
| 20. C22×C22⋊D7 | C2×C3 | semidirect |
| 21. C23×Dic7 | C2×C3 | semidirect |
| 24. C2×C22⋊ΓL(1,𝔽8) | D4 | semidirect |
| 37. C24×C7 | C22×C3 | semidirect |
| 43. C23×D7 | C22×C3 | semidirect |
| 56. C2×C22⋊D7 | A4 | semidirect |
| 60. C22×Dic7 | C22×C3 | semidirect |
| 65. C22⋊ΓL(1,𝔽8) | C2×D4 | semidirect |
| 89. C23×C7 | C3×D4 | semidirect |
| 90. C23×C7 | C2×A4 | semidirect |
| 105. C22×D7 | C2×A4 | semidirect |
| 110. C22⋊D7 | C2×A4 | semidirect |
| 116. C2×Dic7 | C2×A4 | semidirect |
| 129. C25 | AGL(1,𝔽7) | semidirect |
| 153. C22×C7 | C2×C3×D4 | semidirect |
| 155. C22×C7 | C22×A4 | semidirect |
| 165. C2×D7 | C22×A4 | semidirect |
| 167. Dic7 | C22×A4 | semidirect |
| 189. C24 | C2×AGL(1,𝔽7) | semidirect |
| 212. C2×C7 | D4×A4 | semidirect |
| 253. C23 | (C22⋊D7)⋊C3 | semidirect |
| 255. C23 | D7⋊A4 | semidirect |
| 267. C7 | C2×D4×A4 | semidirect |
| 298. C22 | C2×(C22⋊D7)⋊C3 | semidirect |
| 300. C22 | C2×D7⋊A4 | semidirect |
| 311. C2 | (C22⋊D7)⋊A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊ΓL(1,𝔽8) | 2 |
| C22×D7⋊A4 | 2 |
| (C22⋊D7)⋊A4 | 2 |
| C2×Dic7⋊A4 | 2 |
| C23×C22⋊D7 | 3 |
| C2×(C22⋊D7)⋊C3 | 4 |
| C2×D4×A4 | 7 |