Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D3 |
| commutator, cosocle | C2×C3 | C25 |
| Frattini | C2 | C24×D3 |
| Fitting | C22×C3×D4 | C2 |
| socle | C22×C3 | C24 |
| hypercenter | C22×D4 | D3 |
| hypocenter | C3 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D4 | C2 | semidirect |
| 2. C22×#67 | C2 | semidirect |
| 3. #1004 | C2 | direct (N × H) |
| 4. C2×#237 | C2 | semidirect |
| 5. C2×D3×D4 | C2 | semidirect |
| 6. C2×#235 | C2 | semidirect |
| 8. C24×C3 | C22 | semidirect |
| 9. C23×D3 | C22 | semidirect |
| 10. C2×C3×D4 | C22 | semidirect |
| 11. C2×C3×D4 | C22 | semidirect |
| 12. C22×C3×C4 | C22 | semidirect |
| 13. C2×#67 | C22 | semidirect |
| 14. C2×#67 | C22 | semidirect |
| 15. C2×#67 | C22 | semidirect |
| 16. C22×Dic3 | C22 | semidirect |
| 17. #237 | C22 | semidirect |
| 18. D3×D4 | C22 | semidirect |
| 19. #235 | C22 | semidirect |
| 20. C2×D12 | C22 | semidirect |
| 21. C2×C4×D3 | C22 | semidirect |
| 22. C2×Dic6 | C22 | semidirect |
| 28. C22×D4 | D3 | semidirect |
| 32. C23×C3 | C23 | semidirect |
| 35. C22×D3 | C23 | semidirect |
| 36. C3×D4 | C23 | semidirect |
| 38. C2×C3×C4 | C23 | semidirect |
| 39. #67 | C23 | semidirect |
| 40. #67 | C23 | semidirect |
| 42. C2×Dic3 | C23 | semidirect |
| 43. D12 | C23 | semidirect |
| 44. C4×D3 | C23 | semidirect |
| 45. Dic6 | C23 | semidirect |
| 47. C24 | C2×D3 | semidirect |
| 56. C2×D4 | C2×D3 | semidirect |
| 57. C2×D4 | C2×D3 | semidirect |
| 60. C22×C4 | C2×D3 | semidirect |
| 78. C23 | C22×D3 | semidirect |
| 83. D4 | C22×D3 | semidirect |
| 88. C2×C4 | C22×D3 | semidirect |
| 91. C2×C3 | H(5,𝔽2) | semidirect |
| 105. C3 | C2×H(5,𝔽2) | semidirect |
| 109. C2 | #1004 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D4 | 2 |
| C22×#67 | 2 |
| #1004 | 2 |
| C2×#237 | 2 |
| C2×D3×D4 | 2 |
| C2×#235 | 2 |
| C2×H(5,𝔽2) | 3 |