Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
182 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×S4 |
| commutator, cosocle | C2×C7×SL(2,𝔽3) | C22 |
| Fitting | C2×C7×Q8 | C2×D3 |
| socle | C22×C7 | C2×S4 |
| hypocenter | C7×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7⋊GL(2,𝔽3) | C2 | semidirect |
| 2. C2×C7×GL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)×Dic7 | C2 | semidirect |
| 10. Q8×Dic7 | D3 | semidirect |
| 20. C7×SL(2,𝔽3) | D4 | semidirect |
| 36. C2×GL(2,𝔽3) | D7 | semidirect |
| 50. C7×Q8 | D12 | semidirect |
| 88. Dic7 | GL(2,𝔽3) | semidirect |
| 93. SL(2,𝔽3) | C22⋊D7 | semidirect |
| 124. Q8 | #4247 | semidirect |
| 136. C7 | C4⋊GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7⋊GL(2,𝔽3) | 2 |
| C2×C7×GL(2,𝔽3) | 2 |
| SL(2,𝔽3)×Dic7 | 2 |
| #89935 | 3 |
| C2×#4247 | 4 |
| C4⋊GL(2,𝔽3) | 7 |