Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1148 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 224 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 | #10619194 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10617547 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10604995 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10604993 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10604960 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×C5×2−1+4 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×#4149 | C4 | 1 | 1 | yes | characteristic |
| 8 | #66166 | C22 | 1 | 3 | yes | — |
| 9 | #66164 | C4 | 1 | 3 | yes | — |
| 10 | #66163 | — | 2 | 6 | no | — |
| 11 | #41×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 12 | #66151 | — | 2 | 2 | no | — |
| 13 | #66150 | C4 | 1 | 3 | yes | — |
| 14 | C2×#4093 | C22 | 1 | 3 | yes | — |
| 15 | Q8×#170 | — | 2 | 2 | no | — |
| 16 | #65774 | — | 2 | 6 | no | — |
| 17 | #65759 | C22 | 1 | 6 | yes | complemented |
| 18 | C4×#704 | C22 | 1 | 6 | yes | complemented |
| 19 | #64927 | C22 | 1 | 6 | yes | complemented |
| 20 | #64927 | C22 | 1 | 6 | yes | complemented |
| 21 | #64921 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #64921 | C22 | 1 | 3 | yes | complemented |
| 23 | #17688 | — | 5 | 1 | no | maximal, Sylow |
| 24 | C5×2−1+4 | — | 4 | 4 | no | — |
| 25 | C2×C5×#41 | — | 2 | 3 | no | — |
| 26 | C2×C5×#41 | C2×C4 | 1 | 3 | yes | — |
| 27 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×C5×Q8 | — | 2 | 1 | no | — |
| 29 | C22×C5×Q8 | C2×C4 | 1 | 3 | yes | — |
| 30 | #4149 | — | 4 | 4 | no | — |
| 31 | C2×#713 | — | 2 | 1 | no | — |
| 32 | C2×D4×D5 | — | 2 | 3 | no | — |
| 33 | C2×#709 | C2×C4 | 1 | 3 | yes | — |
| 34 | C22×D20 | C2×C4 | 1 | 3 | yes | — |
| 35 | C2×#704 | — | 2 | 6 | no | — |
| 36 | #4093 | — | 2 | 6 | no | — |
| 37 | #4093 | C2×C4 | 1 | 6 | yes | — |
| 38 | #4093 | C23 | 1 | 3 | yes | — |
| 39 | #4092 | — | 2 | 6 | no | — |
| 40 | Q8×Dic5 | D4 | 1 | 2 | yes | — |
| 41 | #4090 | C2×C4 | 1 | 3 | yes | — |
| 42 | #4086 | C2×C4 | 1 | 3 | yes | — |
| 43 | #4085 | — | 2 | 6 | no | — |
| 44 | #4083 | — | 2 | 6 | no | — |
| 45 | #4082 | C23 | 1 | 1 | yes | characteristic |
| 46 | D4×Dic5 | D4 | 1 | 6 | yes | — |
| 47 | #4076 | — | 2 | 6 | no | — |
| 48 | #4075 | — | 2 | 6 | no | — |
| 49 | C4×C22⋊D5 | — | 2 | 6 | no | — |
| 50 | C2×#685 | — | 2 | 6 | no | — |
| 51 | #4072 | C2×C4 | 1 | 3 | yes | — |
| 52 | C2×C4×Dic5 | C23 | 1 | 3 | yes | — |
| 53 | C22⋊#170 | — | 2 | 6 | no | — |
| 54 | #4002 | C23 | 1 | 2 | yes | — |
| 55 | #4002 | C23 | 1 | 6 | yes | — |
| 56 | C4⋊#170 | — | 2 | 6 | no | — |
| 57 | C4×#170 | — | 2 | 6 | no | — |
| 58 | #3500 | — | 5 | 1 | no | — |
| 59 | #2971 | — | 5 | 2 | no | — |
| 60 | #1536 | — | 5 | 3 | no | — |
| 61 | #1527 | — | 5 | 6 | no | — |
| 62 | #1525 | — | 5 | 3 | no | — |
| 63 | C23×D5 | — | 2 | 3 | no | — |
| 64 | C5×#41 | — | 2 | 4 | no | — |
| 65 | C5×#41 | — | 4 | 6 | no | — |
| 66 | C5×#41 | — | 4 | 12 | no | — |
| 67 | C2×C5×Q8 | — | 2 | 3 | no | — |
| 68 | C2×C5×Q8 | — | 4 | 6 | no | — |
| 69 | C2×C5×Q8 | — | 4 | 4 | no | — |
| 70 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C5×Q8 | C22×C4 | 1 | 3 | yes | — |
| 72 | C2×C5×D4 | — | 2 | 3 | no | — |
| 73 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 74 | C2×C5×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C5 | — | 2 | 3 | no | — |
| 76 | C22×C4×C5 | — | 2 | 3 | no | — |
| 77 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 78 | C2×C22⋊D5 | — | 2 | 3 | no | — |
| 79 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 80 | C22×Dic5 | C2×D4 | 1 | 6 | yes | — |
| 81 | #713 | — | 4 | 4 | no | — |
| 82 | D4×D5 | — | 4 | 12 | no | — |
| 83 | #709 | — | 4 | 6 | no | — |
| 84 | C2×D20 | — | 2 | 3 | no | — |
| 85 | C2×D20 | — | 4 | 6 | no | — |
| 86 | C2×D20 | C22×C4 | 1 | 3 | yes | — |
| 87 | C2×C4×D5 | — | 2 | 3 | no | — |
| 88 | C2×Dic10 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | #704 | — | 4 | 12 | no | — |
| 90 | C2×#170 | — | 2 | 2 | no | — |
| 91 | C2×#170 | — | 2 | 6 | no | — |
| 92 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 93 | C22⋊Dic5 | C22⋊C4 | 1 | 6 | yes | — |
| 94 | #685 | — | 2 | 12 | no | — |
| 95 | #685 | — | 2 | 6 | no | — |
| 96 | C4⋊Dic5 | C22⋊C4 | 1 | 6 | yes | — |
| 97 | #683 | — | 2 | 6 | no | — |
| 98 | C4×Dic5 | C22×C4 | 1 | 3 | yes | — |
| 99 | C4×Dic5 | C2×D4 | 1 | 6 | yes | — |
| 100 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 101 | C2×2−1+4 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 102 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 103 | #548 | — | 10 | 2 | no | — |
| 104 | #546 | — | 10 | 6 | no | — |
| 105 | #536 | — | 5 | 3 | no | — |
| 106 | #535 | — | 5 | 3 | no | — |
| 107 | #532 | — | 5 | 3 | no | — |
| 108 | C2×#124 | — | 5 | 3 | no | — |
| 109 | C4×#41 | — | 5 | 1 | no | — |
| 110 | C8×Q8 | — | 10 | 2 | no | — |
| 111 | #436 | — | 10 | 6 | no | — |
| 112 | #432 | — | 5 | 6 | no | — |
| 113 | C4×M16 | — | 5 | 6 | no | — |
| 114 | #329 | — | 5 | 3 | no | — |
| 115 | #329 | — | 5 | 6 | no | — |
| 116 | #329 | — | 5 | 1 | no | — |
| 117 | #329 | — | 5 | 6 | no | — |
| 118 | C23×C5 | — | 2 | 1 | no | — |
| 119 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 120 | C22×D5 | — | 2 | 3 | no | — |
| 121 | C22×D5 | — | 4 | 6 | no | — |
| 122 | C22×D5 | — | 8 | 6 | no | — |
| 123 | C5×Q8 | — | 4 | 3 | no | — |
| 124 | C5×Q8 | — | 4 | 6 | no | — |
| 125 | C5×Q8 | #100 | 1 | 4 | yes | — |
| 126 | C5×D4 | — | 2 | 6 | no | — |
| 127 | C5×D4 | — | 4 | 1 | no | — |
| 128 | C5×D4 | — | 4 | 6 | no | — |
| 129 | C2×C4×C5 | — | 2 | 6 | no | — |
| 130 | C2×C4×C5 | — | 2 | 3 | no | — |
| 131 | C2×C4×C5 | — | 4 | 6 | no | — |
| 132 | C2×C4×C5 | — | 4 | 6 | no | — |
| 133 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 134 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 135 | C22⋊D5 | — | 4 | 6 | no | — |
| 136 | C22⋊D5 | — | 8 | 6 | no | — |
| 137 | C2×Dic5 | — | 2 | 1 | no | — |
| 138 | C2×Dic5 | — | 4 | 6 | no | — |
| 139 | C2×Dic5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 140 | C2×Dic5 | C22×D4 | 1 | 2 | yes | — |
| 141 | D20 | — | 4 | 3 | no | — |
| 142 | D20 | — | 4 | 6 | no | — |
| 143 | C4×D5 | — | 4 | 6 | no | — |
| 144 | Dic10 | — | 4 | 1 | no | — |
| 145 | #170 | — | 2 | 4 | no | — |
| 146 | #170 | — | 4 | 6 | no | — |
| 147 | 2−1+4 | — | 4 | 4 | no | — |
| 148 | H(5,𝔽2) | — | 20 | 4 | no | — |
| 149 | C2×#41 | — | 2 | 3 | no | — |
| 150 | C2×#41 | — | 5 | 3 | no | — |
| 151 | C2×#41 | — | 10 | 1 | no | — |
| 152 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 153 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 154 | C22×Q8 | — | 2 | 1 | no | — |
| 155 | C22×Q8 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 156 | C22×D4 | — | 5 | 3 | no | — |
| 157 | C22×D4 | — | 10 | 3 | no | — |
| 158 | C2×M16 | — | 10 | 6 | no | — |
| 159 | C4⋊Q8 | — | 5 | 3 | no | — |
| 160 | #127 | — | 5 | 3 | no | — |
| 161 | #124 | — | 5 | 3 | no | — |
| 162 | #124 | — | 5 | 6 | no | — |
| 163 | #124 | — | 5 | 1 | no | — |
| 164 | #124 | — | 10 | 6 | no | — |
| 165 | #123 | — | 10 | 6 | no | — |
| 166 | C22⋊Q8 | — | 10 | 6 | no | — |
| 167 | #121 | — | 10 | 6 | no | — |
| 168 | #121 | — | 10 | 6 | no | — |
| 169 | #120 | — | 10 | 6 | no | — |
| 170 | C4×Q8 | — | 5 | 2 | no | — |
| 171 | C4×D4 | — | 5 | 6 | no | — |
| 172 | C4×D4 | — | 10 | 6 | no | — |
| 173 | #117 | — | 5 | 3 | no | — |
| 174 | C2×C22⋊C4 | — | 10 | 6 | no | — |
| 175 | C2×C42 | — | 5 | 3 | no | — |
| 176 | C4⋊C8 | — | 10 | 6 | no | — |
| 177 | C22⋊C8 | — | 10 | 6 | no | — |
| 178 | #97 | — | 5 | 2 | no | — |
| 179 | #97 | — | 5 | 6 | no | — |
| 180 | C4×C8 | — | 10 | 6 | no | — |
| 181 | C22×C5 | — | 2 | 3 | no | — |
| 182 | C22×C5 | — | 2 | 6 | no | — |
| 183 | C22×C5 | — | 4 | 1 | no | — |
| 184 | C22×C5 | — | 4 | 2 | no | — |
| 185 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 186 | C2×D5 | — | 4 | 3 | no | — |
| 187 | C2×D5 | — | 4 | 6 | no | — |
| 188 | C2×D5 | — | 16 | 6 | no | — |
| 189 | C4×C5 | — | 2 | 1 | no | — |
| 190 | C4×C5 | — | 4 | 3 | no | — |
| 191 | C4×C5 | C2×#100 | 1 | 6 | yes | — |
| 192 | Dic5 | — | 2 | 2 | no | — |
| 193 | Dic5 | — | 2 | 6 | no | — |
| 194 | Dic5 | — | 4 | 1 | no | — |
| 195 | C24 | — | 10 | 3 | no | — |
| 196 | #41 | — | 2 | 4 | no | — |
| 197 | #41 | — | 4 | 6 | no | — |
| 198 | #41 | — | 4 | 12 | no | — |
| 199 | #41 | — | 20 | 6 | no | — |
| 200 | #41 | — | 20 | 4 | no | — |
| 201 | C2×Q8 | — | 2 | 3 | no | — |
| 202 | C2×Q8 | — | 4 | 6 | no | — |
| 203 | C2×Q8 | — | 4 | 4 | no | — |
| 204 | C2×Q8 | — | 5 | 1 | no | — |
| 205 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 206 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 207 | C2×D4 | — | 2 | 3 | no | — |
| 208 | C2×D4 | — | 5 | 3 | no | — |
| 209 | C2×D4 | — | 10 | 6 | no | — |
| 210 | C2×D4 | — | 10 | 3 | no | — |
| 211 | C2×D4 | — | 10 | 3 | no | — |
| 212 | C2×D4 | — | 20 | 6 | no | — |
| 213 | C2×D4 | — | 20 | 12 | no | — |
| 214 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 215 | C2×D4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | — | 2 | 3 | no | — |
| 217 | C22×C4 | — | 2 | 3 | no | — |
| 218 | C22×C4 | — | 5 | 6 | no | — |
| 219 | C22×C4 | — | 10 | 3 | no | — |
| 220 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 221 | M16 | — | 20 | 12 | no | — |
| 222 | C2×C8 | — | 10 | 2 | no | — |
| 223 | C2×C8 | — | 10 | 6 | no | — |
| 224 | C4⋊C4 | — | 5 | 6 | no | — |
| 225 | C4⋊C4 | — | 10 | 6 | no | — |
| 226 | C22⋊C4 | — | 5 | 6 | no | — |
| 227 | C22⋊C4 | — | 10 | 6 | no | — |
| 228 | C22⋊C4 | — | 10 | 12 | no | — |
| 229 | C22⋊C4 | — | 10 | 2 | no | — |
| 230 | C42 | — | 5 | 6 | no | — |
| 231 | C42 | — | 5 | 1 | no | — |
| 232 | C42 | — | 5 | 3 | no | — |
| 233 | C2×C5 | — | 4 | 3 | no | — |
| 234 | C2×C5 | — | 8 | 1 | no | — |
| 235 | C2×C5 | C22×#100 | 1 | 1 | yes | characteristic |
| 236 | C2×C5 | #2901 | 1 | 2 | yes | — |
| 237 | D5 | — | 8 | 3 | no | — |
| 238 | C23 | — | 2 | 1 | no | — |
| 239 | C23 | — | 10 | 3 | no | — |
| 240 | C23 | — | 20 | 6 | no | — |
| 241 | C23 | — | 40 | 6 | no | — |
| 242 | C23 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 243 | Q8 | — | 4 | 3 | no | — |
| 244 | Q8 | — | 4 | 6 | no | — |
| 245 | Q8 | — | 20 | 1 | no | — |
| 246 | Q8 | #4013 | 1 | 4 | yes | — |
| 247 | D4 | — | 2 | 6 | no | — |
| 248 | D4 | — | 4 | 1 | no | — |
| 249 | D4 | — | 4 | 6 | no | — |
| 250 | D4 | — | 20 | 3 | no | — |
| 251 | D4 | — | 20 | 6 | no | — |
| 252 | D4 | — | 20 | 6 | no | — |
| 253 | D4 | — | 40 | 6 | no | — |
| 254 | C2×C4 | — | 2 | 6 | no | — |
| 255 | C2×C4 | — | 2 | 3 | no | — |
| 256 | C2×C4 | — | 4 | 6 | no | — |
| 257 | C2×C4 | — | 4 | 6 | no | — |
| 258 | C2×C4 | — | 5 | 2 | no | — |
| 259 | C2×C4 | — | 5 | 6 | no | — |
| 260 | C2×C4 | — | 10 | 1 | no | — |
| 261 | C2×C4 | — | 20 | 6 | no | — |
| 262 | C2×C4 | — | 20 | 6 | no | — |
| 263 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 264 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 265 | C8 | — | 10 | 4 | no | — |
| 266 | C8 | — | 20 | 6 | no | — |
| 267 | C5 | #17688 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 268 | C22 | — | 2 | 6 | no | — |
| 269 | C22 | — | 2 | 3 | no | — |
| 270 | C22 | — | 4 | 2 | no | — |
| 271 | C22 | — | 4 | 1 | no | — |
| 272 | C22 | — | 20 | 6 | no | — |
| 273 | C22 | — | 20 | 3 | no | — |
| 274 | C22 | — | 80 | 6 | no | — |
| 275 | C22 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 276 | C4 | — | 2 | 1 | no | — |
| 277 | C4 | — | 4 | 3 | no | — |
| 278 | C4 | — | 10 | 2 | no | — |
| 279 | C4 | — | 10 | 6 | no | — |
| 280 | C4 | — | 20 | 1 | no | — |
| 281 | C4 | C2×#4013 | 1 | 6 | yes | — |
| 282 | C2 | — | 4 | 3 | no | — |
| 283 | C2 | — | 8 | 1 | no | — |
| 284 | C2 | — | 40 | 3 | no | — |
| 285 | C2 | #10617562 | 1 | 2 | yes | — |
| 286 | C2 | C22×#4013 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10617547 | C2 | semidirect |
| 3. #10604995 | C2 | semidirect |
| 4. #10604993 | C2 | semidirect |
| 5. #10604960 | C2 | semidirect |
| 17. #65759 | C22 | semidirect |
| 18. C4×#704 | C22 | semidirect |
| 19. #64927 | C22 | semidirect |
| 20. #64927 | C22 | semidirect |
| 21. #64921 | C22 | semidirect |
| 22. #64921 | C22 | semidirect |
| 267. C5 | #17688 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619194 | 2 |
| #10617547 | 2 |
| #10604995 | 2 |
| #10604993 | 2 |
| #10604960 | 2 |
| #17688 | 5 |