Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
800 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×H(5,𝔽2) | C2 | semidirect |
| 2. C3×#534 | C2 | semidirect |
| 3. C3×#458 | C2 | semidirect |
| 4. C3×B(4,𝔽2) | C2 | semidirect |
| 5. C2×C3×#99 | C2 | semidirect |
| 6. #3026 | C3 | direct (N × H) |
| 12. C22×C3×D4 | C22 | semidirect |
| 14. C3×#123 | C22 | semidirect |
| 17. C3×#120 | C22 | semidirect |
| 18. C2×C3×C22⋊C4 | C22 | semidirect |
| 19. C3×#99 | C22 | semidirect |
| 20. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 21. #534 | C2×C3 | semidirect |
| 22. #458 | C2×C3 | semidirect |
| 23. B(4,𝔽2) | C2×C3 | semidirect |
| 24. C2×#99 | C2×C3 | semidirect |
| 26. C24×C3 | D4 | semidirect |
| 28. C24×C3 | C23 | semidirect |
| 40. C2×C3×D4 | D4 | semidirect |
| 44. C22×C3×C4 | D4 | semidirect |
| 45. C22×C3×C4 | C23 | semidirect |
| 50. C3×C22⋊C4 | C23 | semidirect |
| 56. C22×D4 | C22×C3 | semidirect |
| 58. #123 | C22×C3 | semidirect |
| 61. #120 | C22×C3 | semidirect |
| 62. C2×C22⋊C4 | C22×C3 | semidirect |
| 63. #99 | C22×C3 | semidirect |
| 91. C24 | C3×D4 | semidirect |
| 93. C24 | C23×C3 | semidirect |
| 105. C2×D4 | C3×D4 | semidirect |
| 109. C22×C4 | C3×D4 | semidirect |
| 110. C22×C4 | C23×C3 | semidirect |
| 115. C22⋊C4 | C23×C3 | semidirect |
| 180. C3 | #3026 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×H(5,𝔽2) | 2 |
| C3×#534 | 2 |
| C3×#458 | 2 |
| C3×B(4,𝔽2) | 2 |
| C2×C3×#99 | 2 |
| #3026 | 3 |