Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
430 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×Q8 | C2×C3 |
| hypercenter | Q8 | D7⋊A4 |
| hypocenter | C22×C7 | C3×#41 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. C4⋊(D7⋊A4) | C2 | semidirect |
| 3. C4×D7⋊A4 | C2 | semidirect |
| 4. C22×#1161 | C3 | semidirect |
| 9. C22×C7×Q8 | C2×C3 | semidirect |
| 14. C22×D28 | C2×C3 | semidirect |
| 15. C22×C4×D7 | C2×C3 | semidirect |
| 33. #1161 | A4 | semidirect |
| 44. C22⋊ΓL(1,𝔽8) | #41 | semidirect |
| 56. C7×Q8 | C2×A4 | semidirect |
| 64. D28 | C2×A4 | semidirect |
| 69. C4×D7 | C2×A4 | semidirect |
| 80. C22×Q8 | AGL(1,𝔽7) | semidirect |
| 85. C22×C7 | C3×#41 | semidirect |
| 137. Q8 | D7⋊A4 | semidirect |
| 151. C7 | A4×#41 | semidirect |
| 162. C22 | #66659 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C22⋊ΓL(1,𝔽8) | 2 |
| C4⋊(D7⋊A4) | 2 |
| C4×D7⋊A4 | 2 |
| C22×#1161 | 3 |
| #66659 | 4 |
| A4×#41 | 7 |