Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
1588 conjugacy classes of subgroups; only the 587 normal subgroups are shown.
This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 587 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D21 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C24×D21 |
| Fitting | C22×C3×C7×Q8 | C2 |
| socle | C22×C3×C7 | C24 |
| hypercenter | C22×Q8 | D21 |
| hypocenter | C3×C7 | C2×2−1+4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C7×Q8 | C2 | semidirect |
| 2. #10621660 | C2 | direct (N × H) |
| 3. C2×#66732 | C2 | semidirect |
| 4. C2×Q8×D21 | C2 | semidirect |
| 5. C2×#66728 | C2 | semidirect |
| 8. C2×C3×C7×Q8 | C22 | semidirect |
| 11. #66732 | C22 | semidirect |
| 12. Q8×D21 | C22 | semidirect |
| 13. #66728 | C22 | semidirect |
| 17. C22×C7×Q8 | D3 | semidirect |
| 34. C2×C7×Q8 | C2×D3 | semidirect |
| 43. C22×C3×Q8 | D7 | semidirect |
| 67. C2×C3×Q8 | C2×D7 | semidirect |
| 77. C2×C3×C7 | 2−1+4 | semidirect |
| 84. C22×Q8 | D21 | semidirect |
| 102. C3×C7 | C2×2−1+4 | semidirect |
| 108. C2×Q8 | C2×D21 | semidirect |
| 114. C2×C7 | #1007 | semidirect |
| 134. C7 | C2×#1007 | semidirect |
| 136. C2×C3 | #6551 | semidirect |
| 146. C3 | C2×#6551 | semidirect |
| 149. C2 | #10621660 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C7×Q8 | 2 |
| #10621660 | 2 |
| C2×#66732 | 2 |
| C2×Q8×D21 | 2 |
| C2×#66728 | 2 |
| C2×#6551 | 3 |
| C2×#1007 | 7 |