Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1316 conjugacy classes of subgroups; only the 672 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 672 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C33 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C33×D4 | C2 | semidirect |
| 2. C2×C33×D4 | C2 | semidirect |
| 3. C22×C33×C4 | C2 | semidirect |
| 4. C33×C4⋊C4 | C2 | semidirect |
| 5. C33×C22⋊C4 | C2 | semidirect |
| 6. C32×#121 | C3 | direct (N × H) |
| 15. C2×C32×D4 | C2×C3 | semidirect |
| 16. C2×C32×D4 | C2×C3 | semidirect |
| 17. C22×C32×C4 | C2×C3 | semidirect |
| 18. C32×C4⋊C4 | C2×C3 | semidirect |
| 19. C32×C22⋊C4 | C2×C3 | semidirect |
| 25. C22×C33 | D4 | semidirect |
| 29. C33×C4 | D4 | semidirect |
| 30. C3×#121 | C32 | direct (N × H) |
| 44. C2×C3×D4 | C2×C32 | semidirect |
| 45. C2×C3×D4 | C2×C32 | semidirect |
| 46. C22×C3×C4 | C2×C32 | semidirect |
| 47. C3×C4⋊C4 | C2×C32 | semidirect |
| 48. C3×C22⋊C4 | C2×C32 | semidirect |
| 54. C22×C32 | C3×D4 | semidirect |
| 58. C32×C4 | C3×D4 | semidirect |
| 59. #121 | C33 | direct (N × H) |
| 60. C33 | #121 | direct (N × H) |
| 74. C2×D4 | C2×C33 | semidirect |
| 75. C2×D4 | C2×C33 | semidirect |
| 76. C22×C4 | C2×C33 | semidirect |
| 77. C4⋊C4 | C2×C33 | semidirect |
| 78. C22⋊C4 | C2×C33 | semidirect |
| 84. C22×C3 | C32×D4 | semidirect |
| 88. C3×C4 | C32×D4 | semidirect |
| 89. C32 | C3×#121 | direct (N × H) |
| 108. C22 | C33×D4 | semidirect |
| 112. C4 | C33×D4 | semidirect |
| 113. C3 | C32×#121 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×D4 | 2 |
| C22×C33×C4 | 2 |
| C33×C4⋊C4 | 2 |
| C33×C22⋊C4 | 2 |
| C32×#121 | 3 |