Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×D8 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | C2×D8 | D3 |
| hypocenter | C3 | #576 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #1004 | C2 | semidirect |
| 2. C2×C3×D8 | C2 | semidirect |
| 3. #932 | C2 | semidirect |
| 4. #912 | C2 | semidirect |
| 5. #911 | C2 | semidirect |
| 6. D3×D8 | C2 | semidirect |
| 7. #904 | C2 | semidirect |
| 8. #901 | C2 | semidirect |
| 10. C2×C3×D4 | C22 | semidirect |
| 13. #237 | C22 | semidirect |
| 15. D3×D4 | C22 | semidirect |
| 17. #235 | C22 | semidirect |
| 18. C3×D8 | C22 | semidirect |
| 19. C3×D8 | C22 | semidirect |
| 20. C2×C3×C8 | C22 | semidirect |
| 21. #214 | C22 | semidirect |
| 22. #213 | C22 | semidirect |
| 23. #208 | C22 | semidirect |
| 24. Dic12 | C22 | semidirect |
| 25. D24 | C22 | semidirect |
| 26. #204 | C22 | semidirect |
| 27. ΓL(1,𝔽25) | C22 | semidirect |
| 28. D3×C8 | C22 | semidirect |
| 35. C2×D8 | D3 | semidirect |
| 63. C2×D4 | C2×D3 | semidirect |
| 69. D8 | C2×D3 | semidirect |
| 70. D8 | C2×D3 | semidirect |
| 74. C2×C8 | C2×D3 | semidirect |
| 119. C3 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1004 | 2 |
| C2×C3×D8 | 2 |
| #932 | 2 |
| #912 | 2 |
| #911 | 2 |
| D3×D8 | 2 |
| #904 | 2 |
| #901 | 2 |
| #576 | 3 |