Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C7×#392 | C3 |
| hypercenter | #392 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#392 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×C22⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#95 | C2 | semidirect |
| 4. C7×#392 | C3 | semidirect |
| 11. C22×C7×D4 | C2×C3 | semidirect |
| 12. C2×C7×C22⋊C4 | C2×C3 | semidirect |
| 13. C7×#95 | C2×C3 | semidirect |
| 19. C23×ΓL(1,𝔽8) | D4 | semidirect |
| 24. C2×C4×ΓL(1,𝔽8) | D4 | semidirect |
| 44. #392 | ΓL(1,𝔽8) | direct (N × H) |
| 49. C23×C7 | C3×D4 | semidirect |
| 54. C2×C4×C7 | C3×D4 | semidirect |
| 65. C22×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 66. C2×C22⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 67. #95 | C2×ΓL(1,𝔽8) | semidirect |
| 88. ΓL(1,𝔽8) | #392 | direct (N × H) |
| 113. C23 | D4×ΓL(1,𝔽8) | semidirect |
| 118. C2×C4 | D4×ΓL(1,𝔽8) | semidirect |
| 119. C7 | C3×#392 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×ΓL(1,𝔽8) | 2 |
| C2×C22⋊C4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#95 | 2 |
| C7×#392 | 3 |
| C3×#392 | 7 |