Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
10246 conjugacy classes of subgroups; only the 5774 normal subgroups are shown.
This group has 10246 conjugacy classes of subgroups — too many to draw. Only its 5774 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C23 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C23×H(3,𝔽4) | C2 | direct (N × H) |
| 2. C22×C23×#124 | C2 | semidirect |
| 3. C22×C23×#120 | C2 | semidirect |
| 4. C26×C23 | C22 | semidirect |
| 5. C23×Q8×C23 | C22 | semidirect |
| 7. C23×H(3,𝔽4) | C22 | direct (N × H) |
| 8. C2×C23×#124 | C22 | semidirect |
| 9. C2×C23×#120 | C22 | semidirect |
| 10. C22×C22⋊C4×C23 | C22 | semidirect |
| 11. C22×C42×C23 | C22 | semidirect |
| 14. C25×C23 | C23 | semidirect |
| 16. C22×Q8×C23 | C23 | semidirect |
| 20. C23×#124 | C23 | semidirect |
| 21. C23×#120 | C23 | semidirect |
| 22. C2×C22⋊C4×C23 | C23 | semidirect |
| 23. C2×C42×C23 | C23 | semidirect |
| 29. C24×C23 | C24 | semidirect |
| 31. C2×Q8×C23 | C24 | semidirect |
| 36. C22⋊C4×C23 | C24 | semidirect |
| 37. C42×C23 | C24 | semidirect |
| 38. C22×H(3,𝔽4) | C23 | direct (N × H) |
| 50. C2×H(3,𝔽4) | C2×C23 | direct (N × H) |
| 51. C22×#124 | C2×C23 | semidirect |
| 52. C22×#120 | C2×C23 | semidirect |
| 56. C22×C23 | H(3,𝔽4) | direct (N × H) |
| 60. C26 | C22×C23 | semidirect |
| 61. C23×Q8 | C22×C23 | semidirect |
| 63. H(3,𝔽4) | C22×C23 | direct (N × H) |
| 64. C2×#124 | C22×C23 | semidirect |
| 65. C2×#120 | C22×C23 | semidirect |
| 66. C22×C22⋊C4 | C22×C23 | semidirect |
| 67. C22×C42 | C22×C23 | semidirect |
| 69. C2×C23 | C2×H(3,𝔽4) | direct (N × H) |
| 73. C25 | C23×C23 | semidirect |
| 75. C22×Q8 | C23×C23 | semidirect |
| 79. #124 | C23×C23 | semidirect |
| 80. #120 | C23×C23 | semidirect |
| 81. C2×C22⋊C4 | C23×C23 | semidirect |
| 82. C2×C42 | C23×C23 | semidirect |
| 83. C23 | C22×H(3,𝔽4) | direct (N × H) |
| 89. C24 | C24×C23 | semidirect |
| 91. C2×Q8 | C24×C23 | semidirect |
| 96. C22⋊C4 | C24×C23 | semidirect |
| 97. C42 | C24×C23 | semidirect |
| 112. C22 | C23×H(3,𝔽4) | direct (N × H) |
| 117. C2 | C2×C23×H(3,𝔽4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C23×H(3,𝔽4) | 2 |
| C22×C23×#124 | 2 |
| C22×C23×#120 | 2 |
| C22×H(3,𝔽4) | 23 |