Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
624 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×D4 | C2 |
| hypercenter | C4×D4 | D21 |
| hypocenter | C3×C7 | #550 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66728 | C2 | semidirect |
| 2. C3×C4×C7×D4 | C2 | semidirect |
| 3. #10621423 | C2 | semidirect |
| 4. #10621413 | C2 | semidirect |
| 5. #10621376 | C2 | semidirect |
| 6. #10621367 | C2 | semidirect |
| 7. #10621357 | C2 | semidirect |
| 8. C4×Dic42 | C2 | semidirect |
| 28. C4×C7×D4 | D3 | semidirect |
| 52. Dic42 | D4 | semidirect |
| 71. C3×C4×D4 | D7 | semidirect |
| 88. C3×C4×C7 | #41 | semidirect |
| 138. C4×D4 | D21 | semidirect |
| 148. C4×C7 | #235 | semidirect |
| 169. C3×C7 | #550 | semidirect |
| 201. C3×C4 | #1157 | semidirect |
| 222. C7 | #5553 | semidirect |
| 239. C4 | #66728 | semidirect |
| 241. C3 | #90244 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66728 | 2 |
| C3×C4×C7×D4 | 2 |
| #10621423 | 2 |
| #10621413 | 2 |
| #10621376 | 2 |
| #10621367 | 2 |
| #10621357 | 2 |
| C4×Dic42 | 2 |
| #90244 | 3 |
| #5553 | 7 |