Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
288 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C2×C7×#41 | D3 |
| socle | C2×C7 | C22×S4 |
| hypercenter | C2×C4×C7 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41⋊C3 | C2 | semidirect |
| 2. C7×#986 | C2 | semidirect |
| 3. C7×#985 | C2 | semidirect |
| 4. C7×#984 | C2 | semidirect |
| 5. C7×#983 | C2 | semidirect |
| 9. C7×#41⋊C3 | C22 | semidirect |
| 10. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 11. C7×GL(2,𝔽3) | C22 | semidirect |
| 12. C7×#226 | C22 | semidirect |
| 14. C2×C7×#41 | D3 | semidirect |
| 24. #5923 | C7 | direct (N × H) |
| 58. C2×#41⋊C3 | C2×C7 | semidirect |
| 59. #986 | C2×C7 | semidirect |
| 60. #985 | C2×C7 | semidirect |
| 61. #984 | C2×C7 | semidirect |
| 62. #983 | C2×C7 | semidirect |
| 95. #41⋊C3 | C22×C7 | semidirect |
| 96. C2×SL(2,𝔽3) | C22×C7 | semidirect |
| 97. GL(2,𝔽3) | C22×C7 | semidirect |
| 98. #226 | C22×C7 | semidirect |
| 103. C2×#41 | D3×C7 | semidirect |
| 192. C7 | #5923 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41⋊C3 | 2 |
| C7×#986 | 2 |
| C7×#985 | 2 |
| C7×#984 | 2 |
| C7×#983 | 2 |
| C7×#575 | 3 |
| C7×#235 | 4 |
| #5923 | 7 |