Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
284 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #120⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C24×C7 | C22×C3 |
| Frattini | C22 | C22×C22⋊ΓL(1,𝔽8) |
| Fitting | C2×C7×#120 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊(C22⋊ΓL(1,𝔽8)) | C2 | semidirect |
| 2. #120⋊ΓL(1,𝔽8) | C2 | direct (N × H) |
| 3. C2×C7×#120 | C3 | semidirect |
| 4. C22⋊(C22⋊ΓL(1,𝔽8)) | C22 | semidirect |
| 6. C25×C7 | C2×C3 | semidirect |
| 9. C7×#120 | C2×C3 | semidirect |
| 19. C24×C7 | A4 | semidirect |
| 20. C24×C7 | C22×C3 | semidirect |
| 33. C2×#120 | ΓL(1,𝔽8) | semidirect |
| 51. C23×C7 | C2×A4 | semidirect |
| 61. C25 | C2×ΓL(1,𝔽8) | semidirect |
| 64. #120 | C2×ΓL(1,𝔽8) | semidirect |
| 93. C24 | C22×ΓL(1,𝔽8) | semidirect |
| 94. C24 | C22⋊ΓL(1,𝔽8) | semidirect |
| 107. C2×C7 | #120⋊C3 | semidirect |
| 128. C23 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 134. C7 | C2×#120⋊C3 | semidirect |
| 162. C2 | #120⋊ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊(C22⋊ΓL(1,𝔽8)) | 2 |
| #120⋊ΓL(1,𝔽8) | 2 |
| C2×C7×#120 | 3 |
| C22×C22⋊ΓL(1,𝔽8) | 4 |
| C2×#120⋊C3 | 7 |