Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
634 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D7⋊A4 |
| commutator, hypocenter | C22×C7 | C22×C3×C4 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting, socle | C25×C7 | C2×C3 |
| prime radical | C23×C22⋊ΓL(1,𝔽8) | C2 |
| cosocle | C23×C7 | C23×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Dic7⋊A4 | C2 | direct (N × H) |
| 3. C24×Dic7 | C3 | semidirect |
| 4. C22×C22⋊ΓL(1,𝔽8) | C4 | semidirect |
| 6. Dic7⋊A4 | C22 | direct (N × H) |
| 12. C23×Dic7 | C2×C3 | semidirect |
| 14. C2×C22⋊ΓL(1,𝔽8) | C2×C4 | semidirect |
| 21. C24×C7 | C3×C4 | semidirect |
| 30. C22×Dic7 | A4 | semidirect |
| 31. C22×Dic7 | C22×C3 | semidirect |
| 34. C22⋊ΓL(1,𝔽8) | C22×C4 | semidirect |
| 48. C23×C7 | C2×C3×C4 | semidirect |
| 56. C2×Dic7 | C2×A4 | semidirect |
| 77. C22×C7 | C4×A4 | semidirect |
| 78. C22×C7 | C22×C3×C4 | semidirect |
| 81. Dic7 | C22×A4 | semidirect |
| 90. C24 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 104. C2×C7 | C2×C4×A4 | semidirect |
| 119. C23 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 128. C7 | C22×C4×A4 | semidirect |
| 140. C22 | C22×ΓL(1,𝔽8)⋊C4 | semidirect |
| 141. C22 | Dic7⋊A4 | direct (N × H) |
| 149. C2 | C2×Dic7⋊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 |
| C2×Dic7⋊A4 | 2 |
| C24×Dic7 | 3 |
| C22×ΓL(1,𝔽8)⋊C4 | 4 |
| C22×C4×A4 | 7 |