Finite Group DatabasebetaFinite Group DatabasebetaOrder 3840.
286 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AΓL(1,𝔽16) | C2 | 1 | 2 | yes | maximal |
| 2 | C4×((C24⋊C5)⋊C2)⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4×#66297 | — | 3 | 1 | no | maximal, Hall |
| 4 | C2×((C24⋊C5)⋊C2)⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 5 | AΓL(1,𝔽16) | C4 | 1 | 4 | yes | direct product factor |
| 6 | C4×AGL(1,𝔽16) | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #66245⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C4×#4625 | — | 5 | 1 | no | maximal, Hall |
| 9 | C2×#66297 | — | 3 | 2 | no | — |
| 10 | C4×(C24⋊C5)⋊C2 | D3 | 1 | 1 | yes | characteristic |
| 11 | C2×AGL(1,𝔽16) | C2×C4 | 1 | 1 | yes | characteristic |
| 12 | ((C24⋊C5)⋊C2)⋊C3 | C2×C4 | 1 | 2 | yes | — |
| 13 | C4×#120⋊C3 | — | 5 | 1 | no | — |
| 14 | C2×#4625 | — | 5 | 2 | no | — |
| 15 | C2×(C24⋊C5)⋊C2 | C2×D3 | 1 | 1 | yes | characteristic |
| 16 | #66297 | — | 3 | 4 | no | — |
| 17 | C4×C24⋊C5 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 18 | #66245 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C4×AΣL(1,𝔽16) | — | 15 | 1 | no | Sylow |
| 20 | AGL(1,𝔽16) | C42 | 1 | 1 | yes | characteristic, complemented |
| 21 | C4×ΓL(1,𝔽16) | — | 16 | 1 | no | maximal |
| 22 | C4×C22⋊A4 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×#120⋊C3 | — | 5 | 1 | no | — |
| 24 | C4×SL(2,ℤ/4) | — | 20 | 1 | no | — |
| 25 | #379⋊C3 | — | 5 | 1 | no | — |
| 26 | #4625 | — | 5 | 4 | no | — |
| 27 | C2×C24⋊C5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 28 | (C24⋊C5)⋊C2 | C4×D3 | 1 | 2 | yes | — |
| 29 | C4×#120 | — | 5 | 1 | no | — |
| 30 | C2×AΣL(1,𝔽16) | — | 15 | 2 | no | — |
| 31 | C4×#100 | — | 15 | 1 | no | — |
| 32 | C4×#99 | — | 15 | 1 | no | — |
| 33 | #1390 | — | 15 | 2 | no | — |
| 34 | C2×ΓL(1,𝔽16) | — | 16 | 2 | no | — |
| 35 | C3×C4×D5 | — | 16 | 1 | no | — |
| 36 | C2×C22⋊A4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 37 | C2×C4×A4 | — | 20 | 1 | no | — |
| 38 | C2×SL(2,ℤ/4) | — | 20 | 2 | no | — |
| 39 | #120⋊C3 | — | 5 | 2 | no | — |
| 40 | C24⋊C5 | C4×Dic3 | 1 | 1 | yes | characteristic, complemented |
| 41 | C4×AGL(1,𝔽5) | — | 48 | 1 | no | — |
| 42 | C24×C4 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 43 | C2×#120 | — | 5 | 1 | no | — |
| 44 | C2×C4×D4 | — | 15 | 1 | no | — |
| 45 | C2×#100 | — | 15 | 2 | no | — |
| 46 | C2×#99 | — | 15 | 2 | no | — |
| 47 | C4×M16 | — | 30 | 1 | no | — |
| 48 | #386 | — | 15 | 1 | no | — |
| 49 | #385 | — | 15 | 1 | no | — |
| 50 | #379 | — | 5 | 1 | no | — |
| 51 | C4×C22⋊C4 | — | 15 | 1 | no | — |
| 52 | C4×C22⋊C4 | — | 30 | 1 | no | — |
| 53 | AΣL(1,𝔽16) | — | 15 | 4 | no | — |
| 54 | AΣL(1,𝔽16) | — | 15 | 4 | no | — |
| 55 | #343 | — | 30 | 1 | no | — |
| 56 | #342 | — | 30 | 1 | no | — |
| 57 | C2×C3×D5 | — | 16 | 1 | no | — |
| 58 | ΓL(1,𝔽16) | — | 16 | 4 | no | — |
| 59 | C3×C4×C5 | — | 16 | 1 | no | — |
| 60 | C3×Dic5 | — | 16 | 1 | no | — |
| 61 | C22⋊A4 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 62 | C22×A4 | — | 20 | 1 | no | — |
| 63 | C4×A4 | — | 20 | 1 | no | — |
| 64 | C4×A4 | — | 20 | 1 | no | — |
| 65 | SL(2,ℤ/4) | — | 20 | 4 | no | — |
| 66 | C4×Dic3 | — | 80 | 1 | no | — |
| 67 | C2×AGL(1,𝔽5) | — | 48 | 2 | no | — |
| 68 | C4×D5 | — | 16 | 1 | no | — |
| 69 | C25 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 70 | C22×D4 | — | 15 | 1 | no | — |
| 71 | C23×C4 | — | 5 | 1 | no | — |
| 72 | C23×C4 | — | 15 | 1 | no | — |
| 73 | C23×C4 | — | 15 | 1 | no | — |
| 74 | C2×M16 | — | 30 | 2 | no | — |
| 75 | #120 | — | 5 | 2 | no | — |
| 76 | #120 | — | 15 | 2 | no | — |
| 77 | C4×D4 | — | 60 | 1 | no | — |
| 78 | C4×D4 | — | 60 | 1 | no | — |
| 79 | C2×C4⋊C4 | — | 15 | 1 | no | — |
| 80 | C2×C22⋊C4 | — | 15 | 1 | no | — |
| 81 | C2×C22⋊C4 | — | 15 | 1 | no | — |
| 82 | C2×C22⋊C4 | — | 15 | 1 | no | — |
| 83 | C2×C22⋊C4 | — | 15 | 1 | no | — |
| 84 | C2×C22⋊C4 | — | 30 | 2 | no | — |
| 85 | C2×C42 | — | 15 | 1 | no | — |
| 86 | C2×C42 | — | 60 | 1 | no | — |
| 87 | #100 | — | 15 | 4 | no | — |
| 88 | #100 | — | 30 | 2 | no | — |
| 89 | #99 | — | 15 | 4 | no | — |
| 90 | #99 | — | 30 | 2 | no | — |
| 91 | #97 | — | 60 | 1 | no | — |
| 92 | C4×C8 | — | 60 | 1 | no | — |
| 93 | #95 | — | 30 | 1 | no | — |
| 94 | #95 | — | 60 | 1 | no | — |
| 95 | C2×C3×C5 | — | 16 | 1 | no | — |
| 96 | C3×D5 | — | 16 | 2 | no | — |
| 97 | C2×A4 | — | 20 | 1 | no | — |
| 98 | C2×A4 | — | 20 | 2 | no | — |
| 99 | C2×C3×C4 | — | 80 | 1 | no | — |
| 100 | C2×Dic3 | — | 80 | 2 | no | — |
| 101 | C2×D5 | — | 16 | 1 | no | — |
| 102 | AGL(1,𝔽5) | — | 48 | 4 | no | — |
| 103 | C4×C5 | — | 16 | 1 | no | — |
| 104 | Dic5 | — | 16 | 1 | no | — |
| 105 | C24 | — | 5 | 1 | no | — |
| 106 | C24 | — | 15 | 1 | no | — |
| 107 | C24 | — | 15 | 1 | no | — |
| 108 | C24 | C4×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 109 | C2×D4 | — | 15 | 2 | no | — |
| 110 | C2×D4 | — | 15 | 2 | no | — |
| 111 | C2×D4 | — | 60 | 1 | no | — |
| 112 | C2×D4 | — | 60 | 1 | no | — |
| 113 | C22×C4 | — | 5 | 1 | no | — |
| 114 | C22×C4 | — | 5 | 1 | no | — |
| 115 | C22×C4 | — | 15 | 1 | no | — |
| 116 | C22×C4 | — | 15 | 1 | no | — |
| 117 | C22×C4 | — | 15 | 1 | no | — |
| 118 | C22×C4 | — | 30 | 1 | no | — |
| 119 | C22×C4 | — | 30 | 1 | no | — |
| 120 | C22×C4 | — | 30 | 1 | no | — |
| 121 | C22×C4 | — | 30 | 1 | no | — |
| 122 | C22×C4 | — | 60 | 1 | no | — |
| 123 | C22×C4 | — | 60 | 2 | no | — |
| 124 | M16 | — | 30 | 4 | no | — |
| 125 | M16 | — | 60 | 2 | no | — |
| 126 | C2×C8 | — | 60 | 2 | no | — |
| 127 | C4⋊C4 | — | 60 | 1 | no | — |
| 128 | C22⋊C4 | — | 15 | 2 | no | — |
| 129 | C22⋊C4 | — | 30 | 1 | no | — |
| 130 | C22⋊C4 | — | 30 | 4 | no | — |
| 131 | C22⋊C4 | — | 60 | 1 | no | — |
| 132 | C22⋊C4 | — | 60 | 1 | no | — |
| 133 | C22⋊C4 | — | 60 | 2 | no | — |
| 134 | C22⋊C4 | — | 60 | 1 | no | — |
| 135 | C42 | — | 30 | 1 | no | — |
| 136 | C42 | — | 30 | 1 | no | — |
| 137 | C42 | — | 120 | 1 | no | — |
| 138 | C42 | — | 120 | 1 | no | — |
| 139 | C3×C5 | — | 16 | 1 | no | Hall |
| 140 | C22×C3 | — | 80 | 1 | no | — |
| 141 | A4 | — | 20 | 1 | no | — |
| 142 | C3×C4 | — | 16 | 1 | no | — |
| 143 | C3×C4 | — | 80 | 1 | no | — |
| 144 | Dic3 | — | 80 | 4 | no | — |
| 145 | C2×C5 | — | 16 | 1 | no | — |
| 146 | D5 | — | 16 | 2 | no | — |
| 147 | C23 | — | 5 | 1 | no | — |
| 148 | C23 | — | 5 | 2 | no | — |
| 149 | C23 | — | 15 | 1 | no | — |
| 150 | C23 | — | 15 | 1 | no | — |
| 151 | C23 | — | 30 | 1 | no | — |
| 152 | C23 | — | 30 | 1 | no | — |
| 153 | C23 | — | 30 | 1 | no | — |
| 154 | C23 | — | 30 | 1 | no | — |
| 155 | C23 | — | 60 | 1 | no | — |
| 156 | D4 | — | 60 | 2 | no | — |
| 157 | D4 | — | 60 | 2 | no | — |
| 158 | C2×C4 | — | 15 | 1 | no | — |
| 159 | C2×C4 | — | 15 | 1 | no | — |
| 160 | C2×C4 | — | 15 | 2 | no | — |
| 161 | C2×C4 | — | 20 | 1 | no | — |
| 162 | C2×C4 | — | 30 | 1 | no | — |
| 163 | C2×C4 | — | 30 | 1 | no | — |
| 164 | C2×C4 | — | 30 | 1 | no | — |
| 165 | C2×C4 | — | 30 | 1 | no | — |
| 166 | C2×C4 | — | 30 | 1 | no | — |
| 167 | C2×C4 | — | 60 | 1 | no | — |
| 168 | C2×C4 | — | 60 | 1 | no | — |
| 169 | C2×C4 | — | 60 | 1 | no | — |
| 170 | C2×C4 | — | 60 | 4 | no | — |
| 171 | C2×C4 | — | 120 | 2 | no | — |
| 172 | C2×C4 | — | 120 | 2 | no | — |
| 173 | C8 | — | 60 | 4 | no | — |
| 174 | C2×C3 | — | 16 | 1 | no | — |
| 175 | C2×C3 | — | 80 | 2 | no | — |
| 176 | C5 | — | 16 | 1 | no | Sylow |
| 177 | C22 | — | 5 | 1 | no | — |
| 178 | C22 | — | 15 | 1 | no | — |
| 179 | C22 | — | 15 | 1 | no | — |
| 180 | C22 | — | 20 | 1 | no | — |
| 181 | C22 | — | 30 | 1 | no | — |
| 182 | C22 | — | 30 | 1 | no | — |
| 183 | C22 | — | 30 | 2 | no | — |
| 184 | C22 | — | 60 | 1 | no | — |
| 185 | C22 | — | 60 | 1 | no | — |
| 186 | C4 | — | 15 | 1 | no | — |
| 187 | C4 | — | 20 | 1 | no | — |
| 188 | C4 | — | 30 | 2 | no | — |
| 189 | C4 | — | 60 | 1 | no | — |
| 190 | C4 | — | 120 | 4 | no | — |
| 191 | C4 | AΓL(1,𝔽16) | 1 | 1 | yes | characteristic, direct product factor |
| 192 | C3 | — | 16 | 1 | no | Sylow |
| 193 | C2 | — | 15 | 1 | no | — |
| 194 | C2 | — | 15 | 1 | no | — |
| 195 | C2 | — | 20 | 2 | no | — |
| 196 | C2 | C2×AΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. AΓL(1,𝔽16) | C4 | direct (N × H) |
| 6. C4×AGL(1,𝔽16) | C4 | semidirect |
| 7. #66245⋊C3 | C4 | semidirect |
| 17. C4×C24⋊C5 | Dic3 | semidirect |
| 18. #66245 | Dic3 | semidirect |
| 20. AGL(1,𝔽16) | C42 | semidirect |
| 22. C4×C22⋊A4 | AGL(1,𝔽5) | semidirect |
| 40. C24⋊C5 | C4×Dic3 | semidirect |
| 42. C24×C4 | ΓL(1,𝔽16) | semidirect |
| 61. C22⋊A4 | C4×AGL(1,𝔽5) | semidirect |
| 108. C24 | C4×ΓL(1,𝔽16) | semidirect |
| 191. C4 | AΓL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AΓL(1,𝔽16) | 2 |
| C4×((C24⋊C5)⋊C2)⋊C3 | 2 |
| C4×#66297 | 3 |
| C4×#4625 | 5 |
| C4×ΓL(1,𝔽16) | 16 |