Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
570 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 75 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×C5×#986 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #66161⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11752143 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Dic5×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11751600 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10615661 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | C5×#986 | C4 | 1 | 4 | yes | complemented |
| 9 | C2×C5×GL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 10 | #91435 | — | 4 | 1 | no | maximal |
| 11 | Dic5×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | #91184 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #91184 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #85366 | — | 5 | 1 | no | maximal, Hall |
| 15 | C2×C5×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 16 | #66161 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | #66132 | — | 3 | 1 | no | — |
| 18 | #65522 | — | 3 | 1 | no | — |
| 19 | C2×#3964 | — | 3 | 1 | no | — |
| 20 | #65453 | — | 3 | 2 | no | — |
| 21 | SD16×Dic5 | — | 3 | 2 | no | — |
| 22 | #65441 | — | 3 | 2 | no | — |
| 23 | D8×Dic5 | — | 3 | 2 | no | — |
| 24 | #65411 | — | 3 | 1 | no | — |
| 25 | #65410 | — | 3 | 1 | no | — |
| 26 | C2×C5×D12 | — | 4 | 1 | no | — |
| 27 | C5×#41⋊C3 | C2×C4 | 1 | 2 | yes | complemented |
| 28 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×D3×Dic5 | — | 4 | 2 | no | — |
| 30 | C5×GL(2,𝔽3) | C2×C4 | 1 | 4 | yes | complemented |
| 31 | C4×Dic15 | — | 4 | 1 | no | — |
| 32 | C3×C4⋊Dic5 | — | 4 | 1 | no | — |
| 33 | (C2×D3)⋊Dic5 | — | 4 | 2 | no | — |
| 34 | C2×#986 | D5 | 1 | 1 | yes | characteristic |
| 35 | #520⋊C3 | — | 5 | 1 | no | — |
| 36 | #5424 | — | 5 | 1 | no | — |
| 37 | #5394 | — | 5 | 2 | no | — |
| 38 | C4×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 39 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | C22×C5×D4 | — | 3 | 1 | no | — |
| 41 | C5×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 42 | C5×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 43 | C2×C5×SD16 | — | 3 | 2 | no | — |
| 44 | C2×C5×D8 | — | 3 | 2 | no | — |
| 45 | C2×C5×M16 | — | 3 | 1 | no | — |
| 46 | C2×C22⋊Dic5 | — | 6 | 2 | no | — |
| 47 | Q8×Dic5 | C2×D3 | 1 | 2 | yes | — |
| 48 | D4×Dic5 | — | 3 | 2 | no | — |
| 49 | D4×Dic5 | — | 3 | 2 | no | — |
| 50 | D4×Dic5 | — | 3 | 2 | no | — |
| 51 | D4×Dic5 | — | 6 | 1 | no | — |
| 52 | D4×Dic5 | — | 6 | 1 | no | — |
| 53 | #4072 | — | 3 | 1 | no | — |
| 54 | #4072 | — | 3 | 1 | no | — |
| 55 | C2×C4⋊Dic5 | — | 3 | 1 | no | — |
| 56 | C22×#168 | — | 3 | 1 | no | — |
| 57 | #3967 | — | 3 | 1 | no | — |
| 58 | #3967 | — | 3 | 1 | no | — |
| 59 | #3964 | — | 3 | 1 | no | — |
| 60 | #3964 | — | 3 | 2 | no | — |
| 61 | #3964 | — | 3 | 1 | no | — |
| 62 | #3964 | — | 3 | 2 | no | — |
| 63 | #3950 | — | 3 | 2 | no | — |
| 64 | #3949 | — | 3 | 2 | no | — |
| 65 | #3947 | — | 3 | 2 | no | — |
| 66 | C8×Dic5 | — | 3 | 2 | no | — |
| 67 | #2946 | — | 15 | 1 | no | Sylow |
| 68 | C22×C5×D3 | — | 4 | 2 | no | — |
| 69 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 70 | C2×Dic15 | — | 4 | 1 | no | — |
| 71 | C2×Dic15 | — | 4 | 1 | no | — |
| 72 | C5×D12 | — | 4 | 4 | no | — |
| 73 | C2×C3×Dic5 | — | 4 | 2 | no | — |
| 74 | C5×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | D3×Dic5 | — | 8 | 4 | no | — |
| 76 | C2×#41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 77 | #986 | Dic5 | 1 | 4 | yes | complemented |
| 78 | C2×GL(2,𝔽3) | C2×D5 | 1 | 2 | yes | — |
| 79 | #893 | — | 20 | 1 | no | — |
| 80 | C4×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 81 | #859 | — | 5 | 1 | no | — |
| 82 | #859 | — | 5 | 1 | no | — |
| 83 | C24×C5 | — | 6 | 1 | no | — |
| 84 | C5×#41 | — | 3 | 2 | no | — |
| 85 | C5×#41 | C4×D3 | 1 | 2 | yes | complemented |
| 86 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 87 | C2×C5×D4 | — | 3 | 1 | no | — |
| 88 | C2×C5×D4 | — | 3 | 2 | no | — |
| 89 | C2×C5×D4 | — | 3 | 4 | no | — |
| 90 | C2×C5×D4 | — | 6 | 2 | no | — |
| 91 | C2×C5×D4 | — | 6 | 1 | no | — |
| 92 | C22×C4×C5 | — | 3 | 1 | no | — |
| 93 | C22×Dic5 | — | 6 | 1 | no | — |
| 94 | C22×Dic5 | — | 6 | 2 | no | — |
| 95 | C22×Dic5 | — | 6 | 2 | no | — |
| 96 | C5×SD16 | — | 3 | 4 | no | — |
| 97 | C5×SD16 | — | 3 | 4 | no | — |
| 98 | C5×D8 | — | 3 | 8 | no | — |
| 99 | C5×M16 | — | 3 | 4 | no | — |
| 100 | C2×C5×C8 | — | 3 | 2 | no | — |
| 101 | C22⋊Dic5 | — | 3 | 2 | no | — |
| 102 | C22⋊Dic5 | — | 6 | 2 | no | — |
| 103 | C22⋊Dic5 | — | 6 | 1 | no | — |
| 104 | C22⋊Dic5 | — | 6 | 2 | no | — |
| 105 | C4⋊Dic5 | — | 3 | 1 | no | — |
| 106 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 107 | C4⋊Dic5 | — | 3 | 1 | no | — |
| 108 | C4⋊Dic5 | — | 3 | 1 | no | — |
| 109 | C4⋊Dic5 | S4 | 1 | 1 | yes | characteristic |
| 110 | C4×Dic5 | — | 3 | 2 | no | — |
| 111 | C4×Dic5 | — | 6 | 1 | no | — |
| 112 | C2×#168 | — | 3 | 1 | no | — |
| 113 | C2×#168 | — | 3 | 1 | no | — |
| 114 | C2×#168 | — | 6 | 2 | no | — |
| 115 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 116 | #520 | — | 5 | 1 | no | — |
| 117 | #518 | — | 15 | 1 | no | — |
| 118 | #442 | — | 15 | 2 | no | — |
| 119 | #440 | — | 15 | 2 | no | — |
| 120 | C4×SD16 | — | 15 | 2 | no | — |
| 121 | C4×D8 | — | 15 | 2 | no | — |
| 122 | #428 | — | 15 | 1 | no | — |
| 123 | #416 | — | 15 | 1 | no | — |
| 124 | C2×#102 | — | 15 | 1 | no | — |
| 125 | #405 | — | 15 | 1 | no | — |
| 126 | C22×C3×C5 | — | 4 | 1 | no | — |
| 127 | C2×C5×D3 | — | 4 | 4 | no | — |
| 128 | C2×C5×D3 | — | 8 | 2 | no | — |
| 129 | C2×C5×D3 | — | 8 | 2 | no | — |
| 130 | C3×C4×C5 | — | 4 | 2 | no | — |
| 131 | Dic15 | — | 4 | 2 | no | — |
| 132 | Dic15 | — | 8 | 1 | no | — |
| 133 | C3×Dic5 | — | 8 | 2 | no | — |
| 134 | C2×D12 | — | 4 | 1 | no | — |
| 135 | C2×C4×D3 | — | 20 | 2 | no | — |
| 136 | #41⋊C3 | C2×Dic5 | 1 | 2 | yes | complemented |
| 137 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 138 | GL(2,𝔽3) | C2×Dic5 | 1 | 4 | yes | complemented |
| 139 | C3×C4⋊C4 | — | 20 | 1 | no | — |
| 140 | (C2×D3)⋊C4 | — | 20 | 2 | no | — |
| 141 | C4×Dic3 | — | 20 | 1 | no | — |
| 142 | C23×C5 | — | 3 | 1 | no | — |
| 143 | C23×C5 | — | 6 | 4 | no | — |
| 144 | C23×C5 | — | 6 | 2 | no | — |
| 145 | C23×C5 | — | 24 | 1 | no | — |
| 146 | C23×C5 | — | 24 | 1 | no | — |
| 147 | C5×Q8 | — | 3 | 1 | no | — |
| 148 | C5×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 149 | C5×D4 | — | 3 | 2 | no | — |
| 150 | C5×D4 | — | 3 | 4 | no | — |
| 151 | C5×D4 | — | 6 | 1 | no | — |
| 152 | C5×D4 | — | 6 | 2 | no | — |
| 153 | C5×D4 | — | 6 | 4 | no | — |
| 154 | C2×C4×C5 | — | 3 | 1 | no | — |
| 155 | C2×C4×C5 | — | 3 | 2 | no | — |
| 156 | C2×C4×C5 | — | 3 | 2 | no | — |
| 157 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 158 | C2×Dic5 | — | 3 | 2 | no | — |
| 159 | C2×Dic5 | — | 6 | 1 | no | — |
| 160 | C2×Dic5 | — | 6 | 1 | no | — |
| 161 | C2×Dic5 | — | 24 | 2 | no | — |
| 162 | C2×Dic5 | — | 24 | 1 | no | — |
| 163 | C2×Dic5 | — | 24 | 2 | no | — |
| 164 | C2×Dic5 | C2×S4 | 1 | 2 | yes | — |
| 165 | C5×C8 | — | 3 | 4 | no | — |
| 166 | #168 | — | 6 | 1 | no | — |
| 167 | #168 | — | 6 | 1 | no | — |
| 168 | C2×#41 | D3×D5 | 1 | 1 | yes | characteristic |
| 169 | C22×D4 | — | 3 | 1 | no | — |
| 170 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 171 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 172 | C2×SD16 | — | 3 | 2 | no | — |
| 173 | C2×D8 | — | 3 | 2 | no | — |
| 174 | C2×M16 | — | 3 | 1 | no | — |
| 175 | C22×C8 | — | 15 | 1 | no | — |
| 176 | C4×Q8 | — | 5 | 2 | no | — |
| 177 | C4×D4 | — | 15 | 2 | no | — |
| 178 | C4×D4 | — | 15 | 2 | no | — |
| 179 | C4×D4 | — | 15 | 2 | no | — |
| 180 | C4×D4 | — | 30 | 1 | no | — |
| 181 | C4×D4 | — | 30 | 1 | no | — |
| 182 | #117 | — | 15 | 1 | no | — |
| 183 | #117 | — | 15 | 1 | no | — |
| 184 | C2×C4⋊C4 | — | 15 | 1 | no | — |
| 185 | C2×C22⋊C4 | — | 30 | 2 | no | — |
| 186 | #107 | — | 15 | 2 | no | — |
| 187 | #106 | — | 15 | 2 | no | — |
| 188 | #103 | — | 15 | 1 | no | — |
| 189 | #103 | — | 15 | 1 | no | — |
| 190 | #102 | — | 15 | 2 | no | — |
| 191 | #102 | — | 15 | 2 | no | — |
| 192 | #102 | — | 15 | 1 | no | — |
| 193 | #102 | — | 15 | 1 | no | — |
| 194 | #97 | — | 15 | 2 | no | — |
| 195 | C4×C8 | — | 15 | 2 | no | — |
| 196 | C2×C3×C5 | — | 4 | 1 | no | — |
| 197 | C2×C3×C5 | — | 4 | 1 | no | — |
| 198 | C2×C3×C5 | — | 4 | 1 | no | — |
| 199 | C5×D3 | — | 8 | 4 | no | — |
| 200 | C22×D3 | — | 4 | 2 | no | — |
| 201 | C2×C3×C4 | — | 4 | 1 | no | — |
| 202 | C2×C3×C4 | — | 20 | 2 | no | — |
| 203 | C2×Dic3 | — | 20 | 1 | no | — |
| 204 | C2×Dic3 | — | 20 | 1 | no | — |
| 205 | D12 | — | 4 | 4 | no | — |
| 206 | C4×D3 | — | 40 | 4 | no | — |
| 207 | SL(2,𝔽3) | C22×Dic5 | 1 | 1 | yes | characteristic |
| 208 | C22×C5 | — | 3 | 2 | no | — |
| 209 | C22×C5 | — | 6 | 1 | no | — |
| 210 | C22×C5 | — | 6 | 1 | no | — |
| 211 | C22×C5 | — | 6 | 4 | no | — |
| 212 | C22×C5 | — | 12 | 2 | no | — |
| 213 | C22×C5 | — | 12 | 2 | no | — |
| 214 | C22×C5 | — | 24 | 4 | no | — |
| 215 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 216 | C4×C5 | — | 3 | 1 | no | — |
| 217 | C4×C5 | — | 3 | 1 | no | — |
| 218 | C4×C5 | C4×S4 | 1 | 2 | yes | — |
| 219 | Dic5 | — | 2 | 2 | no | — |
| 220 | Dic5 | — | 6 | 2 | no | — |
| 221 | Dic5 | — | 12 | 1 | no | — |
| 222 | Dic5 | — | 12 | 1 | no | — |
| 223 | C24 | — | 6 | 1 | no | — |
| 224 | #41 | — | 3 | 2 | no | — |
| 225 | #41 | D3×Dic5 | 1 | 2 | yes | complemented |
| 226 | C2×Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 227 | C2×D4 | — | 3 | 1 | no | — |
| 228 | C2×D4 | — | 3 | 2 | no | — |
| 229 | C2×D4 | — | 3 | 4 | no | — |
| 230 | C2×D4 | — | 6 | 2 | no | — |
| 231 | C2×D4 | — | 6 | 1 | no | — |
| 232 | C22×C4 | — | 3 | 1 | no | — |
| 233 | C22×C4 | — | 30 | 2 | no | — |
| 234 | C22×C4 | — | 30 | 1 | no | — |
| 235 | C22×C4 | — | 30 | 2 | no | — |
| 236 | SD16 | — | 3 | 4 | no | — |
| 237 | SD16 | — | 3 | 4 | no | — |
| 238 | D8 | — | 3 | 8 | no | — |
| 239 | M16 | — | 3 | 4 | no | — |
| 240 | C2×C8 | — | 3 | 2 | no | — |
| 241 | C2×C8 | — | 15 | 1 | no | — |
| 242 | C2×C8 | — | 15 | 1 | no | — |
| 243 | C2×C8 | — | 30 | 2 | no | — |
| 244 | C4⋊C4 | — | 5 | 1 | no | — |
| 245 | C4⋊C4 | — | 15 | 2 | no | — |
| 246 | C4⋊C4 | — | 15 | 1 | no | — |
| 247 | C4⋊C4 | — | 15 | 1 | no | — |
| 248 | C4⋊C4 | — | 15 | 1 | no | — |
| 249 | C22⋊C4 | — | 15 | 2 | no | — |
| 250 | C22⋊C4 | — | 30 | 2 | no | — |
| 251 | C22⋊C4 | — | 30 | 2 | no | — |
| 252 | C22⋊C4 | — | 30 | 1 | no | — |
| 253 | C42 | — | 15 | 2 | no | — |
| 254 | C42 | — | 30 | 1 | no | — |
| 255 | C3×C5 | — | 4 | 1 | no | Hall |
| 256 | C22×C3 | — | 4 | 1 | no | — |
| 257 | C2×D3 | — | 4 | 4 | no | — |
| 258 | C2×D3 | — | 8 | 2 | no | — |
| 259 | C2×D3 | — | 8 | 2 | no | — |
| 260 | C3×C4 | — | 4 | 2 | no | — |
| 261 | C3×C4 | — | 40 | 2 | no | — |
| 262 | Dic3 | — | 20 | 2 | no | — |
| 263 | Dic3 | — | 40 | 1 | no | — |
| 264 | C2×C5 | — | 6 | 2 | no | — |
| 265 | C2×C5 | — | 12 | 4 | no | — |
| 266 | C2×C5 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 267 | C2×C5 | #5925 | 1 | 1 | yes | characteristic |
| 268 | C2×C5 | #5926 | 1 | 1 | yes | characteristic |
| 269 | C23 | — | 3 | 1 | no | — |
| 270 | C23 | — | 6 | 4 | no | — |
| 271 | C23 | — | 6 | 2 | no | — |
| 272 | C23 | — | 24 | 1 | no | — |
| 273 | C23 | — | 24 | 1 | no | — |
| 274 | Q8 | — | 3 | 1 | no | — |
| 275 | Q8 | C2×D3×Dic5 | 1 | 1 | yes | characteristic |
| 276 | D4 | — | 3 | 2 | no | — |
| 277 | D4 | — | 3 | 4 | no | — |
| 278 | D4 | — | 6 | 1 | no | — |
| 279 | D4 | — | 6 | 2 | no | — |
| 280 | D4 | — | 6 | 4 | no | — |
| 281 | C2×C4 | — | 3 | 1 | no | — |
| 282 | C2×C4 | — | 3 | 2 | no | — |
| 283 | C2×C4 | — | 3 | 2 | no | — |
| 284 | C2×C4 | — | 5 | 2 | no | — |
| 285 | C2×C4 | — | 15 | 2 | no | — |
| 286 | C2×C4 | — | 30 | 1 | no | — |
| 287 | C2×C4 | — | 30 | 1 | no | — |
| 288 | C2×C4 | — | 120 | 2 | no | — |
| 289 | C2×C4 | — | 120 | 1 | no | — |
| 290 | C2×C4 | — | 120 | 2 | no | — |
| 291 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 292 | C8 | — | 3 | 4 | no | — |
| 293 | C8 | — | 30 | 1 | no | — |
| 294 | C8 | — | 30 | 1 | no | — |
| 295 | C2×C3 | — | 4 | 1 | no | — |
| 296 | C2×C3 | — | 4 | 1 | no | — |
| 297 | C2×C3 | — | 4 | 1 | no | — |
| 298 | D3 | — | 8 | 4 | no | — |
| 299 | C5 | #85366 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 300 | C22 | — | 3 | 2 | no | — |
| 301 | C22 | — | 6 | 1 | no | — |
| 302 | C22 | — | 6 | 1 | no | — |
| 303 | C22 | — | 6 | 4 | no | — |
| 304 | C22 | — | 12 | 2 | no | — |
| 305 | C22 | — | 12 | 2 | no | — |
| 306 | C22 | — | 24 | 4 | no | — |
| 307 | C22 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 3 | 1 | no | — |
| 309 | C4 | — | 3 | 1 | no | — |
| 310 | C4 | — | 10 | 2 | no | — |
| 311 | C4 | — | 30 | 2 | no | — |
| 312 | C4 | — | 60 | 1 | no | — |
| 313 | C4 | — | 60 | 1 | no | — |
| 314 | C4 | Dic5×S4 | 1 | 2 | yes | — |
| 315 | C3 | — | 4 | 1 | no | Sylow |
| 316 | C2 | — | 6 | 2 | no | — |
| 317 | C2 | — | 12 | 4 | no | — |
| 318 | C2 | #11756774 | 1 | 1 | yes | characteristic |
| 319 | C2 | #11756786 | 1 | 1 | yes | characteristic |
| 320 | C2 | C2×Dic5×S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #66161⋊C3 | C2 | semidirect |
| 3. #11752143 | C2 | semidirect |
| 4. Dic5×GL(2,𝔽3) | C2 | semidirect |
| 5. #11751600 | C2 | semidirect |
| 8. C5×#986 | C4 | semidirect |
| 11. Dic5×SL(2,𝔽3) | C22 | semidirect |
| 12. #91184 | C22 | semidirect |
| 13. #91184 | C22 | semidirect |
| 16. #66161 | D3 | semidirect |
| 27. C5×#41⋊C3 | C2×C4 | semidirect |
| 30. C5×GL(2,𝔽3) | C2×C4 | semidirect |
| 77. #986 | Dic5 | semidirect |
| 85. C5×#41 | C4×D3 | semidirect |
| 136. #41⋊C3 | C2×Dic5 | semidirect |
| 138. GL(2,𝔽3) | C2×Dic5 | semidirect |
| 225. #41 | D3×Dic5 | semidirect |
| 299. C5 | #85366 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#986 | 2 |
| #66161⋊C3 | 2 |
| #11752143 | 2 |
| Dic5×GL(2,𝔽3) | 2 |
| #11751600 | 2 |
| #10615661 | 3 |
| #91435 | 4 |
| #85366 | 5 |