← C22×Q8×ΓL(1,𝔽16) #14478850
Order 1920.
3328 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3×Q8×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×Q8×ΓL(1,𝔽16) | C2 | 1 | 24 | yes | maximal, direct product factor |
| 3 | C22×C4⋊ΓL(1,𝔽16) | C2 | 1 | 3 | yes | maximal |
| 4 | C22×C4×ΓL(1,𝔽16) | C2 | 1 | 3 | yes | maximal |
| 5 | C22×Q8×AGL(1,𝔽5) | — | 3 | 1 | no | maximal, Hall |
| 6 | C23×ΓL(1,𝔽16) | C22 | 1 | 3 | yes | — |
| 7 | C23×ΓL(1,𝔽16) | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×C3×C5×Q8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×C3×Q8×D5 | C4 | 1 | 12 | yes | complemented |
| 10 | C2×C3×Q8×D5 | C22 | 1 | 12 | yes | — |
| 11 | C22×C3×C4×D5 | C22 | 1 | 3 | yes | — |
| 12 | C22×C3×Dic10 | C4 | 1 | 3 | yes | complemented |
| 13 | Q8×ΓL(1,𝔽16) | C22 | 1 | 64 | yes | direct product factor |
| 14 | C2×C4⋊ΓL(1,𝔽16) | C22 | 1 | 36 | yes | — |
| 15 | C2×C4×ΓL(1,𝔽16) | C22 | 1 | 36 | yes | — |
| 16 | C22×Q8×Dic3 | — | 5 | 1 | no | maximal, Hall |
| 17 | C22×Q8×D5 | D3 | 1 | 1 | yes | characteristic |
| 18 | C2×Q8×AGL(1,𝔽5) | — | 3 | 24 | no | — |
| 19 | C22×C4⋊AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 20 | C22×C4×AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 21 | C23×C3×D5 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×ΓL(1,𝔽16) | — | 2 | 12 | no | — |
| 23 | C22×ΓL(1,𝔽16) | Q8 | 1 | 8 | yes | direct product factor |
| 24 | C22×ΓL(1,𝔽16) | C23 | 1 | 18 | yes | — |
| 25 | C22×ΓL(1,𝔽16) | C23 | 1 | 6 | yes | — |
| 26 | C2×C3×C5×Q8 | C2×C4 | 1 | 12 | yes | complemented |
| 27 | C22×C3×C4×C5 | C2×C4 | 1 | 3 | yes | — |
| 28 | C22×C3×Dic5 | C2×C4 | 1 | 3 | yes | — |
| 29 | C3×Q8×D5 | C2×C4 | 1 | 48 | yes | complemented |
| 30 | C3×Q8×D5 | C23 | 1 | 16 | yes | — |
| 31 | C2×C3×C4×D5 | C2×C4 | 1 | 18 | yes | — |
| 32 | C2×C3×C4×D5 | C23 | 1 | 18 | yes | — |
| 33 | C2×C3×Dic10 | C2×C4 | 1 | 36 | yes | complemented |
| 34 | C4⋊ΓL(1,𝔽16) | C23 | 1 | 48 | yes | — |
| 35 | C4×ΓL(1,𝔽16) | C23 | 1 | 48 | yes | — |
| 36 | C23×C3×Q8 | — | 5 | 1 | no | — |
| 37 | C2×Q8×Dic3 | — | 5 | 24 | no | — |
| 38 | C22×C4⋊Dic3 | — | 5 | 3 | no | — |
| 39 | C22×C4×Dic3 | — | 5 | 3 | no | — |
| 40 | C23×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 41 | C23×AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 42 | C22×C5×Q8 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 43 | C2×Q8×D5 | Dic3 | 1 | 12 | yes | complemented |
| 44 | C2×Q8×D5 | C2×D3 | 1 | 12 | yes | — |
| 45 | C22×C4×D5 | C2×D3 | 1 | 3 | yes | — |
| 46 | C22×Dic10 | Dic3 | 1 | 3 | yes | complemented |
| 47 | Q8×AGL(1,𝔽5) | — | 3 | 64 | no | — |
| 48 | C2×C4⋊AGL(1,𝔽5) | — | 3 | 36 | no | — |
| 49 | C2×C4×AGL(1,𝔽5) | — | 3 | 36 | no | — |
| 50 | C22×C4×Q8 | — | 15 | 1 | no | Sylow |
| 51 | C23×C3×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C3×D5 | C22×C4 | 1 | 3 | yes | — |
| 53 | C22×C3×D5 | C2×Q8 | 1 | 4 | yes | — |
| 54 | C22×C3×D5 | #41 | 1 | 4 | yes | — |
| 55 | C22×C3×D5 | C24 | 1 | 3 | yes | — |
| 56 | C2×ΓL(1,𝔽16) | — | 2 | 36 | no | — |
| 57 | C2×ΓL(1,𝔽16) | C2×Q8 | 1 | 24 | yes | direct product factor |
| 58 | C2×ΓL(1,𝔽16) | C24 | 1 | 12 | yes | — |
| 59 | C2×ΓL(1,𝔽16) | C24 | 1 | 4 | yes | — |
| 60 | C3×C5×Q8 | C22×C4 | 1 | 16 | yes | complemented |
| 61 | C2×C3×C4×C5 | C22×C4 | 1 | 18 | yes | — |
| 62 | C2×C3×Dic5 | C22×C4 | 1 | 18 | yes | — |
| 63 | C3×C4×D5 | C22×C4 | 1 | 36 | yes | — |
| 64 | C3×C4×D5 | C24 | 1 | 12 | yes | — |
| 65 | C3×Dic10 | C22×C4 | 1 | 48 | yes | complemented |
| 66 | C22×C3×Q8 | — | 5 | 12 | no | — |
| 67 | C22×C3×Q8 | — | 5 | 12 | no | — |
| 68 | C22×C3×Q8 | — | 5 | 3 | no | — |
| 69 | C22×C3×Q8 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 70 | C23×C3×C4 | — | 5 | 3 | no | — |
| 71 | C23×Dic3 | — | 5 | 1 | no | — |
| 72 | C23×Dic3 | — | 5 | 3 | no | — |
| 73 | Q8×Dic3 | — | 5 | 64 | no | — |
| 74 | C2×C4⋊Dic3 | — | 5 | 36 | no | — |
| 75 | C2×C4×Dic3 | — | 5 | 36 | no | — |
| 76 | C23×D5 | C22×D3 | 1 | 1 | yes | characteristic |
| 77 | C22×AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 78 | C22×AGL(1,𝔽5) | — | 3 | 6 | no | — |
| 79 | C22×AGL(1,𝔽5) | — | 3 | 18 | no | — |
| 80 | C22×AGL(1,𝔽5) | — | 6 | 12 | no | — |
| 81 | C2×C5×Q8 | C2×Dic3 | 1 | 12 | yes | complemented |
| 82 | C22×C4×C5 | C2×Dic3 | 1 | 3 | yes | — |
| 83 | C22×Dic5 | C2×Dic3 | 1 | 3 | yes | — |
| 84 | Q8×D5 | C2×Dic3 | 1 | 48 | yes | complemented |
| 85 | Q8×D5 | C22×D3 | 1 | 16 | yes | — |
| 86 | C2×C4×D5 | C2×Dic3 | 1 | 18 | yes | — |
| 87 | C2×C4×D5 | C22×D3 | 1 | 18 | yes | — |
| 88 | C2×Dic10 | C2×Dic3 | 1 | 36 | yes | complemented |
| 89 | C4⋊AGL(1,𝔽5) | — | 3 | 48 | no | — |
| 90 | C4×AGL(1,𝔽5) | — | 3 | 48 | no | — |
| 91 | C23×Q8 | — | 5 | 1 | no | — |
| 92 | C2×C4×Q8 | — | 15 | 24 | no | — |
| 93 | C22×C4⋊C4 | — | 15 | 3 | no | — |
| 94 | C22×C42 | — | 15 | 3 | no | — |
| 95 | C22×C3×C5 | C4×Q8 | 1 | 4 | yes | complemented |
| 96 | C22×C3×C5 | C23×C4 | 1 | 3 | yes | — |
| 97 | C2×C3×D5 | C4×Q8 | 1 | 12 | yes | complemented |
| 98 | C2×C3×D5 | C23×C4 | 1 | 3 | yes | — |
| 99 | C2×C3×D5 | C22×Q8 | 1 | 6 | yes | — |
| 100 | C2×C3×D5 | C2×#41 | 1 | 6 | yes | — |
| 101 | C2×C3×D5 | C25 | 1 | 1 | yes | characteristic |
| 102 | ΓL(1,𝔽16) | — | 2 | 12 | no | — |
| 103 | ΓL(1,𝔽16) | C22×Q8 | 1 | 8 | yes | direct product factor |
| 104 | C3×C4×C5 | C23×C4 | 1 | 12 | yes | — |
| 105 | C3×Dic5 | C23×C4 | 1 | 12 | yes | — |
| 106 | C24×C3 | — | 5 | 1 | no | — |
| 107 | C2×C3×Q8 | — | 5 | 48 | no | — |
| 108 | C2×C3×Q8 | — | 5 | 16 | no | — |
| 109 | C2×C3×Q8 | — | 5 | 36 | no | — |
| 110 | C2×C3×Q8 | C2×AGL(1,𝔽5) | 1 | 12 | yes | complemented |
| 111 | C22×C3×C4 | — | 5 | 3 | no | — |
| 112 | C22×C3×C4 | — | 5 | 18 | no | — |
| 113 | C22×C3×C4 | — | 5 | 18 | no | — |
| 114 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 115 | C22×Dic3 | — | 5 | 6 | no | — |
| 116 | C22×Dic3 | — | 5 | 8 | no | — |
| 117 | C22×Dic3 | — | 5 | 18 | no | — |
| 118 | C22×Dic3 | — | 10 | 12 | no | — |
| 119 | C4⋊Dic3 | — | 5 | 48 | no | — |
| 120 | C4×Dic3 | — | 5 | 48 | no | — |
| 121 | C23×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 122 | C22×D5 | D3×Q8 | 1 | 4 | yes | — |
| 123 | C22×D5 | #239 | 1 | 4 | yes | — |
| 124 | C22×D5 | C22×Dic3 | 1 | 3 | yes | — |
| 125 | C22×D5 | C23×D3 | 1 | 3 | yes | — |
| 126 | C2×AGL(1,𝔽5) | — | 3 | 24 | no | — |
| 127 | C2×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 128 | C2×AGL(1,𝔽5) | — | 3 | 12 | no | — |
| 129 | C2×AGL(1,𝔽5) | — | 6 | 36 | no | — |
| 130 | C5×Q8 | C22×Dic3 | 1 | 16 | yes | complemented |
| 131 | C2×C4×C5 | C22×Dic3 | 1 | 18 | yes | — |
| 132 | C2×Dic5 | C22×Dic3 | 1 | 18 | yes | — |
| 133 | C4×D5 | C22×Dic3 | 1 | 36 | yes | — |
| 134 | C4×D5 | C23×D3 | 1 | 12 | yes | — |
| 135 | Dic10 | C22×Dic3 | 1 | 48 | yes | complemented |
| 136 | C22×Q8 | — | 5 | 12 | no | — |
| 137 | C22×Q8 | — | 5 | 12 | no | — |
| 138 | C22×Q8 | — | 5 | 3 | no | — |
| 139 | C22×Q8 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, direct product factor |
| 140 | C23×C4 | — | 5 | 3 | no | — |
| 141 | C23×C4 | — | 15 | 1 | no | — |
| 142 | C23×C4 | — | 15 | 3 | no | — |
| 143 | C4×Q8 | — | 15 | 64 | no | — |
| 144 | C2×C4⋊C4 | — | 15 | 36 | no | — |
| 145 | C2×C42 | — | 15 | 36 | no | — |
| 146 | C2×C3×C5 | C2×C4×Q8 | 1 | 6 | yes | complemented |
| 147 | C2×C3×C5 | C24×C4 | 1 | 1 | yes | characteristic |
| 148 | C3×D5 | C2×C4×Q8 | 1 | 6 | yes | complemented |
| 149 | C3×D5 | C23×Q8 | 1 | 1 | yes | characteristic |
| 150 | C3×D5 | C22×#41 | 1 | 1 | yes | characteristic |
| 151 | C23×C3 | — | 5 | 4 | no | — |
| 152 | C23×C3 | — | 5 | 4 | no | — |
| 153 | C23×C3 | — | 5 | 3 | no | — |
| 154 | C23×C3 | — | 5 | 3 | no | — |
| 155 | C23×C3 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 156 | C3×Q8 | — | 5 | 48 | no | — |
| 157 | C3×Q8 | C22×AGL(1,𝔽5) | 1 | 16 | yes | complemented |
| 158 | C2×C3×C4 | — | 5 | 18 | no | — |
| 159 | C2×C3×C4 | — | 5 | 36 | no | — |
| 160 | C2×C3×C4 | — | 5 | 12 | no | — |
| 161 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 162 | C2×Dic3 | — | 5 | 24 | no | — |
| 163 | C2×Dic3 | — | 5 | 4 | no | — |
| 164 | C2×Dic3 | — | 5 | 12 | no | — |
| 165 | C2×Dic3 | — | 10 | 36 | no | — |
| 166 | C22×C5 | Q8×Dic3 | 1 | 4 | yes | complemented |
| 167 | C22×C5 | C23×Dic3 | 1 | 3 | yes | — |
| 168 | C2×D5 | Q8×Dic3 | 1 | 12 | yes | complemented |
| 169 | C2×D5 | C2×D3×Q8 | 1 | 6 | yes | — |
| 170 | C2×D5 | C2×#239 | 1 | 6 | yes | — |
| 171 | C2×D5 | C23×Dic3 | 1 | 3 | yes | — |
| 172 | C2×D5 | C24×D3 | 1 | 1 | yes | characteristic |
| 173 | AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 174 | AGL(1,𝔽5) | — | 6 | 12 | no | — |
| 175 | C4×C5 | C23×Dic3 | 1 | 12 | yes | — |
| 176 | Dic5 | C23×Dic3 | 1 | 12 | yes | — |
| 177 | C24 | — | 5 | 1 | no | — |
| 178 | C2×Q8 | — | 5 | 16 | no | — |
| 179 | C2×Q8 | — | 5 | 48 | no | — |
| 180 | C2×Q8 | — | 5 | 36 | no | — |
| 181 | C2×Q8 | C2×ΓL(1,𝔽16) | 1 | 12 | yes | direct product factor |
| 182 | C22×C4 | — | 5 | 3 | no | — |
| 183 | C22×C4 | — | 5 | 18 | no | — |
| 184 | C22×C4 | — | 5 | 18 | no | — |
| 185 | C22×C4 | — | 15 | 6 | no | — |
| 186 | C22×C4 | — | 15 | 8 | no | — |
| 187 | C22×C4 | — | 15 | 18 | no | — |
| 188 | C22×C4 | — | 30 | 12 | no | — |
| 189 | C22×C4 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 190 | C4⋊C4 | — | 15 | 48 | no | — |
| 191 | C42 | — | 15 | 48 | no | — |
| 192 | C3×C5 | C22×C4×Q8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 193 | C22×C3 | — | 5 | 12 | no | — |
| 194 | C22×C3 | — | 5 | 6 | no | — |
| 195 | C22×C3 | — | 5 | 6 | no | — |
| 196 | C22×C3 | — | 5 | 3 | no | — |
| 197 | C22×C3 | — | 5 | 1 | no | — |
| 198 | C22×C3 | Q8×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 199 | C22×C3 | C23×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 200 | C3×C4 | — | 5 | 12 | no | — |
| 201 | C3×C4 | C23×AGL(1,𝔽5) | 1 | 12 | yes | — |
| 202 | Dic3 | — | 5 | 8 | no | — |
| 203 | Dic3 | — | 10 | 12 | no | — |
| 204 | C2×C5 | C2×Q8×Dic3 | 1 | 6 | yes | complemented |
| 205 | C2×C5 | C24×Dic3 | 1 | 1 | yes | characteristic |
| 206 | D5 | C2×Q8×Dic3 | 1 | 6 | yes | complemented |
| 207 | D5 | C22×D3×Q8 | 1 | 1 | yes | characteristic |
| 208 | D5 | C22×#239 | 1 | 1 | yes | characteristic |
| 209 | C23 | — | 5 | 4 | no | — |
| 210 | C23 | — | 5 | 4 | no | — |
| 211 | C23 | — | 5 | 3 | no | — |
| 212 | C23 | — | 5 | 3 | no | — |
| 213 | C23 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 214 | Q8 | — | 5 | 48 | no | — |
| 215 | Q8 | C22×ΓL(1,𝔽16) | 1 | 16 | yes | direct product factor |
| 216 | C2×C4 | — | 5 | 18 | no | — |
| 217 | C2×C4 | — | 5 | 12 | no | — |
| 218 | C2×C4 | — | 5 | 36 | no | — |
| 219 | C2×C4 | — | 15 | 4 | no | — |
| 220 | C2×C4 | — | 15 | 24 | no | — |
| 221 | C2×C4 | — | 15 | 12 | no | — |
| 222 | C2×C4 | — | 30 | 36 | no | — |
| 223 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 18 | yes | — |
| 224 | C2×C3 | — | 5 | 1 | no | — |
| 225 | C2×C3 | — | 5 | 1 | no | — |
| 226 | C2×C3 | — | 5 | 6 | no | — |
| 227 | C2×C3 | C2×Q8×AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 228 | C2×C3 | C24×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 229 | C5 | C22×Q8×Dic3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 230 | C22 | — | 5 | 1 | no | — |
| 231 | C22 | — | 5 | 3 | no | — |
| 232 | C22 | — | 5 | 6 | no | — |
| 233 | C22 | — | 5 | 6 | no | — |
| 234 | C22 | — | 5 | 12 | no | — |
| 235 | C22 | Q8×ΓL(1,𝔽16) | 1 | 4 | yes | direct product factor |
| 236 | C22 | C23×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 237 | C4 | — | 5 | 12 | no | — |
| 238 | C4 | — | 15 | 8 | no | — |
| 239 | C4 | — | 30 | 12 | no | — |
| 240 | C4 | C23×ΓL(1,𝔽16) | 1 | 12 | yes | — |
| 241 | C3 | C22×Q8×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 242 | C2 | — | 5 | 1 | no | — |
| 243 | C2 | — | 5 | 1 | no | — |
| 244 | C2 | — | 5 | 6 | no | — |
| 245 | C2 | C2×Q8×ΓL(1,𝔽16) | 1 | 6 | yes | direct product factor |
| 246 | C2 | C24×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 2. C2×Q8×ΓL(1,𝔽16) | C2 | direct (N × H) |
| 8. C22×C3×C5×Q8 | C4 | semidirect |
| 9. C2×C3×Q8×D5 | C4 | semidirect |
| 12. C22×C3×Dic10 | C4 | semidirect |
| 13. Q8×ΓL(1,𝔽16) | C22 | direct (N × H) |
| 23. C22×ΓL(1,𝔽16) | Q8 | direct (N × H) |
| 26. C2×C3×C5×Q8 | C2×C4 | semidirect |
| 29. C3×Q8×D5 | C2×C4 | semidirect |
| 33. C2×C3×Dic10 | C2×C4 | semidirect |
| 42. C22×C5×Q8 | Dic3 | semidirect |
| 43. C2×Q8×D5 | Dic3 | semidirect |
| 46. C22×Dic10 | Dic3 | semidirect |
| 57. C2×ΓL(1,𝔽16) | C2×Q8 | direct (N × H) |
| 60. C3×C5×Q8 | C22×C4 | semidirect |
| 65. C3×Dic10 | C22×C4 | semidirect |
| 69. C22×C3×Q8 | AGL(1,𝔽5) | semidirect |
| 81. C2×C5×Q8 | C2×Dic3 | semidirect |
| 84. Q8×D5 | C2×Dic3 | semidirect |
| 88. C2×Dic10 | C2×Dic3 | semidirect |
| 95. C22×C3×C5 | C4×Q8 | semidirect |
| 97. C2×C3×D5 | C4×Q8 | semidirect |
| 103. ΓL(1,𝔽16) | C22×Q8 | direct (N × H) |
| 110. C2×C3×Q8 | C2×AGL(1,𝔽5) | semidirect |
| 130. C5×Q8 | C22×Dic3 | semidirect |
| 135. Dic10 | C22×Dic3 | semidirect |
| 139. C22×Q8 | ΓL(1,𝔽16) | direct (N × H) |
| 146. C2×C3×C5 | C2×C4×Q8 | semidirect |
| 148. C3×D5 | C2×C4×Q8 | semidirect |
| 157. C3×Q8 | C22×AGL(1,𝔽5) | semidirect |
| 166. C22×C5 | Q8×Dic3 | semidirect |
| 168. C2×D5 | Q8×Dic3 | semidirect |
| 181. C2×Q8 | C2×ΓL(1,𝔽16) | direct (N × H) |
| 192. C3×C5 | C22×C4×Q8 | semidirect |
| 198. C22×C3 | Q8×AGL(1,𝔽5) | semidirect |
| 204. C2×C5 | C2×Q8×Dic3 | semidirect |
| 206. D5 | C2×Q8×Dic3 | semidirect |
| 215. Q8 | C22×ΓL(1,𝔽16) | direct (N × H) |
| 227. C2×C3 | C2×Q8×AGL(1,𝔽5) | semidirect |
| 229. C5 | C22×Q8×Dic3 | semidirect |
| 235. C22 | Q8×ΓL(1,𝔽16) | direct (N × H) |
| 241. C3 | C22×Q8×AGL(1,𝔽5) | semidirect |
| 245. C2 | C2×Q8×ΓL(1,𝔽16) | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta