Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
658 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 63 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×#91956 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C5×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic5×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11751611 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10615663 | — | 3 | 1 | no | maximal, Hall |
| 6 | C2×C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #91956 | — | 2 | 2 | no | — |
| 8 | C2×C5⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 9 | C5×#986 | — | 2 | 2 | no | — |
| 10 | C2×C5×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 11 | #91451 | — | 4 | 1 | no | maximal |
| 12 | Dic5×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 13 | #85346 | — | 5 | 1 | no | maximal, Hall |
| 14 | C2×C5×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 15 | #41×Dic5 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#4095 | — | 3 | 1 | no | — |
| 17 | C2×#4086 | — | 3 | 1 | no | — |
| 18 | #65504 | — | 3 | 1 | no | — |
| 19 | #65452 | — | 3 | 4 | no | — |
| 20 | #65439 | — | 3 | 4 | no | — |
| 21 | #65420 | — | 3 | 1 | no | — |
| 22 | #65407 | — | 3 | 1 | no | — |
| 23 | C2×D60 | — | 4 | 1 | no | — |
| 24 | C2×C5×D12 | — | 4 | 1 | no | — |
| 25 | C5×#41⋊C3 | D4 | 1 | 2 | yes | complemented |
| 26 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#1203 | — | 4 | 4 | no | — |
| 28 | C5⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 29 | C5×GL(2,𝔽3) | — | 2 | 2 | no | — |
| 30 | C3×C4×Dic5 | — | 4 | 1 | no | — |
| 31 | C2×#986 | — | 5 | 1 | no | — |
| 32 | C2×#986 | D5 | 1 | 1 | yes | characteristic, complemented |
| 33 | C4×#41⋊C3 | — | 5 | 1 | no | — |
| 34 | C4⋊GL(2,𝔽3) | — | 5 | 4 | no | — |
| 35 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | C22×C5×D4 | — | 3 | 1 | no | — |
| 37 | C22×C22⋊D5 | — | 6 | 2 | no | — |
| 38 | C22×D20 | — | 3 | 1 | no | — |
| 39 | C5×AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 40 | C5×AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 41 | C2×C5×SD16 | — | 3 | 2 | no | — |
| 42 | C2×C5×D8 | — | 3 | 2 | no | — |
| 43 | C2×C5×M16 | — | 3 | 1 | no | — |
| 44 | #4095 | — | 6 | 2 | no | — |
| 45 | #4095 | — | 6 | 2 | no | — |
| 46 | Q8×Dic5 | C2×D3 | 1 | 2 | yes | — |
| 47 | C2×#688 | — | 3 | 2 | no | — |
| 48 | #4086 | — | 3 | 4 | no | — |
| 49 | #4086 | — | 6 | 1 | no | — |
| 50 | #4086 | — | 6 | 1 | no | — |
| 51 | D4×Dic5 | — | 3 | 2 | no | — |
| 52 | C2×#686 | — | 3 | 2 | no | — |
| 53 | #4072 | — | 3 | 1 | no | — |
| 54 | C2×C4×Dic5 | — | 3 | 1 | no | — |
| 55 | C2×#681 | — | 3 | 1 | no | — |
| 56 | #3964 | — | 3 | 4 | no | — |
| 57 | #3953 | — | 3 | 4 | no | — |
| 58 | #3946 | — | 3 | 4 | no | — |
| 59 | #3038 | — | 15 | 1 | no | Sylow |
| 60 | C22×D15 | — | 4 | 2 | no | — |
| 61 | C22×C5×D3 | — | 4 | 2 | no | — |
| 62 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 63 | D60 | — | 8 | 2 | no | — |
| 64 | C5×D12 | — | 8 | 2 | no | — |
| 65 | C2×C3×Dic5 | — | 4 | 2 | no | — |
| 66 | C5×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1203 | — | 8 | 8 | no | — |
| 68 | C2×#41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 69 | #986 | — | 2 | 2 | no | — |
| 70 | #986 | — | 10 | 2 | no | — |
| 71 | C2×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 72 | C2×GL(2,𝔽3) | C2×D5 | 1 | 2 | yes | complemented |
| 73 | #874 | — | 20 | 1 | no | — |
| 74 | C4×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 75 | C24×C5 | — | 6 | 1 | no | — |
| 76 | C23×D5 | — | 6 | 1 | no | — |
| 77 | C5×#41 | — | 3 | 2 | no | — |
| 78 | C5×#41 | D12 | 1 | 2 | yes | complemented |
| 79 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 80 | C2×C5×D4 | — | 3 | 1 | no | — |
| 81 | C2×C5×D4 | — | 3 | 2 | no | — |
| 82 | C2×C5×D4 | — | 6 | 2 | no | — |
| 83 | C2×C5×D4 | — | 6 | 2 | no | — |
| 84 | C2×C5×D4 | — | 6 | 1 | no | — |
| 85 | C22×C4×C5 | — | 3 | 1 | no | — |
| 86 | C2×C22⋊D5 | — | 6 | 4 | no | — |
| 87 | C2×C22⋊D5 | — | 6 | 4 | no | — |
| 88 | C2×C22⋊D5 | — | 12 | 8 | no | — |
| 89 | C22×Dic5 | — | 3 | 2 | no | — |
| 90 | C2×D20 | — | 3 | 2 | no | — |
| 91 | C2×D20 | — | 6 | 2 | no | — |
| 92 | C2×D20 | — | 6 | 2 | no | — |
| 93 | C2×D20 | — | 6 | 1 | no | — |
| 94 | C5×SD16 | — | 6 | 2 | no | — |
| 95 | C5×SD16 | — | 6 | 2 | no | — |
| 96 | C5×D8 | — | 6 | 4 | no | — |
| 97 | C5×M16 | — | 6 | 2 | no | — |
| 98 | C2×C5×C8 | — | 3 | 2 | no | — |
| 99 | C22⋊Dic5 | — | 6 | 1 | no | — |
| 100 | #688 | — | 6 | 2 | no | — |
| 101 | #688 | — | 6 | 2 | no | — |
| 102 | #686 | — | 6 | 4 | no | — |
| 103 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 104 | C4×Dic5 | — | 3 | 1 | no | — |
| 105 | C4×Dic5 | — | 3 | 2 | no | — |
| 106 | C4×Dic5 | S4 | 1 | 1 | yes | characteristic |
| 107 | #681 | — | 6 | 2 | no | — |
| 108 | C2×#168 | — | 3 | 2 | no | — |
| 109 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 110 | C2×AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 111 | C2×#127 | — | 15 | 1 | no | — |
| 112 | C4×#41 | — | 5 | 1 | no | — |
| 113 | #460 | — | 15 | 4 | no | — |
| 114 | #459 | — | 15 | 4 | no | — |
| 115 | #423 | — | 15 | 1 | no | — |
| 116 | #418 | — | 15 | 1 | no | — |
| 117 | #418 | — | 15 | 1 | no | — |
| 118 | C22×C3×C5 | — | 4 | 1 | no | — |
| 119 | C2×D15 | — | 8 | 2 | no | — |
| 120 | C2×D15 | — | 8 | 4 | no | — |
| 121 | C2×C5×D3 | — | 8 | 2 | no | — |
| 122 | C2×C5×D3 | — | 8 | 4 | no | — |
| 123 | C3×C4×C5 | — | 4 | 2 | no | — |
| 124 | C3×Dic5 | — | 4 | 4 | no | — |
| 125 | C2×D12 | — | 4 | 1 | no | — |
| 126 | C2×D12 | — | 20 | 1 | no | — |
| 127 | C2×D12 | — | 20 | 4 | no | — |
| 128 | #41⋊C3 | C22⋊D5 | 1 | 2 | yes | complemented |
| 129 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 130 | GL(2,𝔽3) | — | 2 | 2 | no | — |
| 131 | GL(2,𝔽3) | — | 10 | 2 | no | — |
| 132 | C3×C42 | — | 20 | 1 | no | — |
| 133 | C23×C5 | — | 3 | 1 | no | — |
| 134 | C23×C5 | — | 6 | 2 | no | — |
| 135 | C23×C5 | — | 12 | 2 | no | — |
| 136 | C23×C5 | — | 12 | 4 | no | — |
| 137 | C22×D5 | — | 6 | 2 | no | — |
| 138 | C22×D5 | — | 12 | 2 | no | — |
| 139 | C22×D5 | — | 12 | 4 | no | — |
| 140 | C5×Q8 | — | 3 | 1 | no | — |
| 141 | C5×Q8 | C2×D12 | 1 | 1 | yes | characteristic |
| 142 | C5×D4 | — | 3 | 2 | no | — |
| 143 | C5×D4 | — | 6 | 1 | no | — |
| 144 | C5×D4 | — | 6 | 2 | no | — |
| 145 | C5×D4 | — | 6 | 2 | no | — |
| 146 | C5×D4 | — | 12 | 2 | no | — |
| 147 | C2×C4×C5 | — | 3 | 1 | no | — |
| 148 | C2×C4×C5 | — | 3 | 2 | no | — |
| 149 | C2×C4×C5 | — | 3 | 2 | no | — |
| 150 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 151 | C22⋊D5 | — | 12 | 8 | no | — |
| 152 | C22⋊D5 | — | 12 | 8 | no | — |
| 153 | C2×Dic5 | — | 3 | 2 | no | — |
| 154 | C2×Dic5 | — | 6 | 4 | no | — |
| 155 | C2×Dic5 | C2×S4 | 1 | 2 | yes | — |
| 156 | D20 | — | 6 | 2 | no | — |
| 157 | D20 | — | 6 | 2 | no | — |
| 158 | D20 | — | 12 | 2 | no | — |
| 159 | C5×C8 | — | 6 | 2 | no | — |
| 160 | #168 | — | 6 | 2 | no | — |
| 161 | C2×#41 | D3×D5 | 1 | 1 | yes | characteristic |
| 162 | C22×D4 | — | 3 | 1 | no | — |
| 163 | C22×D4 | — | 15 | 1 | no | — |
| 164 | C22×D4 | — | 30 | 2 | no | — |
| 165 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 166 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 167 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 168 | AGL(1,ℤ/8) | — | 30 | 2 | no | — |
| 169 | C2×SD16 | — | 3 | 2 | no | — |
| 170 | C2×SD16 | — | 15 | 2 | no | — |
| 171 | C2×D8 | — | 3 | 2 | no | — |
| 172 | C2×D8 | — | 15 | 2 | no | — |
| 173 | C2×M16 | — | 3 | 1 | no | — |
| 174 | C2×M16 | — | 15 | 1 | no | — |
| 175 | #127 | — | 15 | 4 | no | — |
| 176 | #127 | — | 30 | 1 | no | — |
| 177 | #127 | — | 30 | 1 | no | — |
| 178 | C4×Q8 | — | 5 | 2 | no | — |
| 179 | C4×D4 | — | 15 | 2 | no | — |
| 180 | #117 | — | 15 | 1 | no | — |
| 181 | C2×C42 | — | 15 | 1 | no | — |
| 182 | C4⋊C8 | — | 15 | 4 | no | — |
| 183 | #102 | — | 15 | 4 | no | — |
| 184 | #102 | — | 15 | 4 | no | — |
| 185 | C2×C3×C5 | — | 4 | 1 | no | — |
| 186 | C2×C3×C5 | — | 4 | 2 | no | — |
| 187 | D15 | — | 16 | 2 | no | — |
| 188 | C5×D3 | — | 16 | 2 | no | — |
| 189 | C22×D3 | — | 4 | 2 | no | — |
| 190 | C22×D3 | — | 20 | 2 | no | — |
| 191 | C2×C3×C4 | — | 4 | 1 | no | — |
| 192 | C2×C3×C4 | — | 20 | 2 | no | — |
| 193 | D12 | — | 8 | 2 | no | — |
| 194 | D12 | — | 40 | 2 | no | — |
| 195 | D12 | — | 40 | 8 | no | — |
| 196 | SL(2,𝔽3) | C2×C22⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 197 | C22×C5 | — | 3 | 2 | no | — |
| 198 | C22×C5 | — | 6 | 2 | no | — |
| 199 | C22×C5 | — | 12 | 2 | no | — |
| 200 | C22×C5 | — | 12 | 4 | no | — |
| 201 | C22×C5 | — | 24 | 4 | no | — |
| 202 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 203 | C2×D5 | — | 12 | 2 | no | — |
| 204 | C2×D5 | — | 12 | 4 | no | — |
| 205 | C2×D5 | — | 24 | 4 | no | — |
| 206 | C4×C5 | — | 3 | 1 | no | — |
| 207 | C4×C5 | — | 3 | 1 | no | — |
| 208 | C4×C5 | C4⋊S4 | 1 | 2 | yes | — |
| 209 | Dic5 | — | 6 | 2 | no | — |
| 210 | Dic5 | #986 | 1 | 4 | yes | complemented |
| 211 | C24 | — | 6 | 1 | no | — |
| 212 | C24 | — | 30 | 1 | no | — |
| 213 | #41 | — | 3 | 2 | no | — |
| 214 | #41 | #1203 | 1 | 2 | yes | complemented |
| 215 | C2×Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 216 | C2×D4 | — | 3 | 1 | no | — |
| 217 | C2×D4 | — | 3 | 2 | no | — |
| 218 | C2×D4 | — | 6 | 2 | no | — |
| 219 | C2×D4 | — | 6 | 2 | no | — |
| 220 | C2×D4 | — | 6 | 1 | no | — |
| 221 | C2×D4 | — | 15 | 2 | no | — |
| 222 | C2×D4 | — | 30 | 2 | no | — |
| 223 | C2×D4 | — | 30 | 2 | no | — |
| 224 | C2×D4 | — | 30 | 4 | no | — |
| 225 | C2×D4 | — | 30 | 4 | no | — |
| 226 | C2×D4 | — | 30 | 1 | no | — |
| 227 | C2×D4 | — | 60 | 8 | no | — |
| 228 | C22×C4 | — | 3 | 1 | no | — |
| 229 | C22×C4 | — | 15 | 2 | no | — |
| 230 | SD16 | — | 6 | 2 | no | — |
| 231 | SD16 | — | 6 | 2 | no | — |
| 232 | SD16 | — | 30 | 2 | no | — |
| 233 | SD16 | — | 30 | 2 | no | — |
| 234 | D8 | — | 6 | 4 | no | — |
| 235 | D8 | — | 30 | 4 | no | — |
| 236 | M16 | — | 6 | 2 | no | — |
| 237 | M16 | — | 30 | 2 | no | — |
| 238 | C2×C8 | — | 3 | 2 | no | — |
| 239 | C2×C8 | — | 15 | 2 | no | — |
| 240 | C4⋊C4 | — | 15 | 2 | no | — |
| 241 | C22⋊C4 | — | 30 | 1 | no | — |
| 242 | C42 | — | 5 | 1 | no | — |
| 243 | C42 | — | 15 | 2 | no | — |
| 244 | C42 | — | 15 | 1 | no | — |
| 245 | C3×C5 | — | 4 | 1 | no | Hall |
| 246 | C22×C3 | — | 4 | 1 | no | — |
| 247 | C2×D3 | — | 8 | 2 | no | — |
| 248 | C2×D3 | — | 8 | 4 | no | — |
| 249 | C2×D3 | — | 40 | 4 | no | — |
| 250 | C2×D3 | — | 40 | 2 | no | — |
| 251 | C3×C4 | — | 4 | 2 | no | — |
| 252 | C3×C4 | — | 20 | 4 | no | — |
| 253 | C2×C5 | — | 6 | 2 | no | — |
| 254 | C2×C5 | — | 24 | 2 | no | — |
| 255 | C2×C5 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 256 | C2×C5 | C2×#986 | 1 | 2 | yes | — |
| 257 | D5 | — | 24 | 2 | no | — |
| 258 | C23 | — | 3 | 1 | no | — |
| 259 | C23 | — | 6 | 2 | no | — |
| 260 | C23 | — | 12 | 2 | no | — |
| 261 | C23 | — | 12 | 4 | no | — |
| 262 | C23 | — | 30 | 2 | no | — |
| 263 | C23 | — | 60 | 2 | no | — |
| 264 | C23 | — | 60 | 4 | no | — |
| 265 | Q8 | — | 3 | 1 | no | — |
| 266 | Q8 | C2×#1203 | 1 | 1 | yes | characteristic |
| 267 | D4 | — | 3 | 2 | no | — |
| 268 | D4 | — | 6 | 1 | no | — |
| 269 | D4 | — | 6 | 2 | no | — |
| 270 | D4 | — | 6 | 2 | no | — |
| 271 | D4 | — | 12 | 2 | no | — |
| 272 | D4 | — | 30 | 2 | no | — |
| 273 | D4 | — | 30 | 2 | no | — |
| 274 | D4 | — | 60 | 8 | no | — |
| 275 | D4 | — | 60 | 8 | no | — |
| 276 | D4 | — | 60 | 2 | no | — |
| 277 | C2×C4 | — | 3 | 1 | no | — |
| 278 | C2×C4 | — | 3 | 2 | no | — |
| 279 | C2×C4 | — | 3 | 2 | no | — |
| 280 | C2×C4 | — | 5 | 2 | no | — |
| 281 | C2×C4 | — | 15 | 2 | no | — |
| 282 | C2×C4 | — | 30 | 4 | no | — |
| 283 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 284 | C8 | — | 6 | 2 | no | — |
| 285 | C8 | — | 30 | 2 | no | — |
| 286 | C2×C3 | — | 4 | 2 | no | — |
| 287 | C2×C3 | — | 4 | 1 | no | — |
| 288 | D3 | — | 16 | 2 | no | — |
| 289 | D3 | — | 80 | 2 | no | — |
| 290 | C5 | #85346 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 291 | C22 | — | 3 | 2 | no | — |
| 292 | C22 | — | 6 | 2 | no | — |
| 293 | C22 | — | 12 | 4 | no | — |
| 294 | C22 | — | 12 | 2 | no | — |
| 295 | C22 | — | 24 | 4 | no | — |
| 296 | C22 | — | 60 | 4 | no | — |
| 297 | C22 | — | 60 | 2 | no | — |
| 298 | C22 | — | 120 | 4 | no | — |
| 299 | C22 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 3 | 1 | no | — |
| 301 | C4 | — | 3 | 1 | no | — |
| 302 | C4 | — | 5 | 4 | no | — |
| 303 | C4 | — | 30 | 2 | no | — |
| 304 | C4 | #91902 | 1 | 2 | yes | — |
| 305 | C3 | — | 4 | 1 | no | Sylow |
| 306 | C2 | — | 6 | 2 | no | — |
| 307 | C2 | — | 24 | 2 | no | — |
| 308 | C2 | — | 120 | 2 | no | — |
| 309 | C2 | D5×#986 | 1 | 2 | yes | — |
| 310 | C2 | C2×#91902 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#91956 | C2 | semidirect |
| 2. C2×C5×#986 | C2 | semidirect |
| 3. Dic5×#41⋊C3 | C2 | semidirect |
| 4. #11751611 | C2 | semidirect |
| 8. C2×C5⋊GL(2,𝔽3) | C22 | semidirect |
| 10. C2×C5×GL(2,𝔽3) | C22 | semidirect |
| 12. Dic5×SL(2,𝔽3) | C22 | semidirect |
| 15. #41×Dic5 | D3 | semidirect |
| 25. C5×#41⋊C3 | D4 | semidirect |
| 32. C2×#986 | D5 | semidirect |
| 66. C5×SL(2,𝔽3) | C2×D4 | semidirect |
| 72. C2×GL(2,𝔽3) | C2×D5 | semidirect |
| 78. C5×#41 | D12 | semidirect |
| 128. #41⋊C3 | C22⋊D5 | semidirect |
| 196. SL(2,𝔽3) | C2×C22⋊D5 | semidirect |
| 210. Dic5 | #986 | semidirect |
| 214. #41 | #1203 | semidirect |
| 290. C5 | #85346 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#91956 | 2 |
| C2×C5×#986 | 2 |
| Dic5×#41⋊C3 | 2 |
| #11751611 | 2 |
| #10615663 | 3 |
| #91451 | 4 |
| #85346 | 5 |