Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
416 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C3×C7 |
| Frattini | C2 | C23×C7×A4 |
| Fitting | C2×C7×H(5,𝔽2) | C3 |
| hypercenter | C2×C7×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#994 | C2 | direct (N × H) |
| 2. C2×C7×#41⋊C3 | C2 | semidirect |
| 3. C2×C7×H(5,𝔽2) | C3 | semidirect |
| 5. C7×#41⋊C3 | C22 | semidirect |
| 6. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 8. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 12. C2×#994 | C7 | direct (N × H) |
| 15. C7×SL(2,𝔽3) | C23 | semidirect |
| 28. #994 | C2×C7 | direct (N × H) |
| 29. C2×#41⋊C3 | C2×C7 | semidirect |
| 32. C2×H(5,𝔽2) | C3×C7 | semidirect |
| 47. #41⋊C3 | C22×C7 | semidirect |
| 48. C2×SL(2,𝔽3) | C22×C7 | semidirect |
| 52. H(5,𝔽2) | C2×C3×C7 | semidirect |
| 65. SL(2,𝔽3) | C23×C7 | semidirect |
| 80. C2×C7 | #994 | direct (N × H) |
| 97. C7 | C2×#994 | direct (N × H) |
| 109. C2 | C7×#994 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#994 | 2 |
| C2×C7×#41⋊C3 | 2 |
| C2×C7×H(5,𝔽2) | 3 |
| C2×C3×C7×Q8 | 4 |
| C2×#994 | 7 |