Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
210 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 62 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 | C2×Q8 | C22×C3×C7 |
| Frattini | C22 | C22×C7×A4 |
| Fitting | C7×D4×Q8 | C3 |
| hypercenter | C2×C7×D4 | A4 |
| hypocenter | Q8 | C3×C7×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 2. C4×C7×SL(2,𝔽3) | C2 | semidirect |
| 3. C7×D4×Q8 | C3 | semidirect |
| 7. C22×C7×Q8 | C2×C3 | semidirect |
| 10. C4×C7×Q8 | C2×C3 | semidirect |
| 12. D4×SL(2,𝔽3) | C7 | direct (N × H) |
| 16. C7×SL(2,𝔽3) | D4 | direct (N × H) |
| 28. C22×SL(2,𝔽3) | C2×C7 | semidirect |
| 29. C4×SL(2,𝔽3) | C2×C7 | semidirect |
| 35. D4×Q8 | C3×C7 | semidirect |
| 40. C7×Q8 | C3×D4 | semidirect |
| 41. C7×D4 | SL(2,𝔽3) | direct (N × H) |
| 55. C22×Q8 | C2×C3×C7 | semidirect |
| 58. C4×Q8 | C2×C3×C7 | semidirect |
| 62. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 68. C4×C7 | C2×SL(2,𝔽3) | semidirect |
| 72. SL(2,𝔽3) | C7×D4 | direct (N × H) |
| 98. Q8 | C3×C7×D4 | semidirect |
| 99. D4 | C7×SL(2,𝔽3) | direct (N × H) |
| 106. C7 | D4×SL(2,𝔽3) | direct (N × H) |
| 113. C22 | C2×C7×SL(2,𝔽3) | semidirect |
| 119. C4 | C2×C7×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×SL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| C7×D4×Q8 | 3 |
| C2×C3×C7×D4 | 4 |
| D4×SL(2,𝔽3) | 7 |