Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
884 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 884 conjugacy classes of subgroups — too many to draw. Only its 190 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 | #10617431 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #10613491 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604821 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10604676 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10604618 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10603851 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #66260 | C22 | 1 | 2 | yes | — |
| 8 | #66256 | C22 | 1 | 2 | yes | — |
| 9 | C2×#4062 | C22 | 1 | 4 | yes | — |
| 10 | #66059 | C22 | 1 | 1 | yes | characteristic |
| 11 | #66050 | C22 | 1 | 1 | yes | characteristic |
| 12 | #65785 | C22 | 1 | 2 | yes | — |
| 13 | #65784 | C22 | 1 | 4 | yes | — |
| 14 | #65771 | C22 | 1 | 2 | yes | — |
| 15 | #65769 | C22 | 1 | 4 | yes | — |
| 16 | #65737 | — | 2 | 8 | no | — |
| 17 | #65736 | — | 2 | 8 | no | — |
| 18 | D5⋊#108 | C22 | 1 | 2 | yes | — |
| 19 | #65721 | C22 | 1 | 2 | yes | — |
| 20 | #65718 | C22 | 1 | 4 | yes | — |
| 21 | C5×#492 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #65457 | C4 | 1 | 4 | yes | complemented |
| 23 | #65381 | C4 | 1 | 1 | yes | characteristic, complemented |
| 24 | #65362 | C4 | 1 | 1 | yes | characteristic, complemented |
| 25 | #64982 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 27 | #64862 | C22 | 1 | 4 | yes | — |
| 28 | #23196 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C2×#713 | C23 | 1 | 2 | yes | — |
| 30 | C2×#711 | C23 | 1 | 2 | yes | — |
| 31 | C2×#704 | — | 4 | 2 | no | — |
| 32 | #4133 | — | 2 | 8 | no | — |
| 33 | #4131 | — | 2 | 8 | no | — |
| 34 | #4130 | C23 | 1 | 4 | yes | — |
| 35 | D5⋊M16 | — | 4 | 2 | no | — |
| 36 | D5⋊M16 | C23 | 1 | 4 | yes | — |
| 37 | C2×C5×SD16 | C2×C4 | 1 | 4 | yes | — |
| 38 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C5×#127 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | #4093 | C2×C4 | 1 | 2 | yes | — |
| 41 | C2×#689 | C2×C4 | 1 | 4 | yes | — |
| 42 | #4084 | — | 4 | 1 | no | — |
| 43 | #4082 | C2×C4 | 1 | 2 | yes | — |
| 44 | D4×Dic5 | — | 4 | 1 | no | — |
| 45 | C2×#686 | C2×C4 | 1 | 4 | yes | — |
| 46 | #4062 | — | 2 | 4 | no | — |
| 47 | #4062 | — | 2 | 4 | no | — |
| 48 | #4062 | — | 2 | 4 | no | — |
| 49 | #4062 | — | 2 | 4 | no | — |
| 50 | C2×#677 | C2×C4 | 1 | 4 | yes | — |
| 51 | C2×C8×D5 | C23 | 1 | 2 | yes | — |
| 52 | #4041 | — | 4 | 1 | no | — |
| 53 | #4039 | — | 4 | 1 | no | — |
| 54 | #4020 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 56 | #4015 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | #4011 | — | 4 | 2 | no | — |
| 58 | #4010 | — | 2 | 8 | no | — |
| 59 | #4008 | — | 2 | 8 | no | — |
| 60 | #3999 | — | 4 | 2 | no | — |
| 61 | #3996 | D4 | 1 | 4 | yes | complemented |
| 62 | #3995 | D4 | 1 | 4 | yes | complemented |
| 63 | #3994 | — | 2 | 2 | no | — |
| 64 | #3993 | — | 2 | 2 | no | — |
| 65 | #3992 | — | 2 | 4 | no | — |
| 66 | C8×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 67 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C8×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 69 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 70 | #3147 | — | 5 | 1 | no | — |
| 71 | #2957 | — | 5 | 4 | no | — |
| 72 | #1976 | — | 5 | 4 | no | — |
| 73 | #1884 | — | 5 | 2 | no | — |
| 74 | #1884 | — | 5 | 2 | no | — |
| 75 | #1772 | — | 5 | 2 | no | — |
| 76 | C2×C5×Q8 | C22×C4 | 1 | 2 | yes | — |
| 77 | C2×C5×D4 | — | 4 | 1 | no | — |
| 78 | C2×C5×D4 | C22×C4 | 1 | 2 | yes | — |
| 79 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 80 | C22×Dic5 | — | 2 | 2 | no | — |
| 81 | #713 | — | 2 | 4 | no | — |
| 82 | #713 | — | 2 | 4 | no | — |
| 83 | #711 | — | 2 | 4 | no | — |
| 84 | #711 | — | 2 | 4 | no | — |
| 85 | C2×D20 | — | 2 | 2 | no | — |
| 86 | C2×D20 | — | 4 | 1 | no | — |
| 87 | C2×D20 | C22×C4 | 1 | 2 | yes | — |
| 88 | C2×C4×D5 | — | 2 | 1 | no | — |
| 89 | C2×C4×D5 | — | 2 | 2 | no | — |
| 90 | C2×C4×D5 | C24 | 1 | 1 | yes | characteristic |
| 91 | C2×Dic10 | C22×C4 | 1 | 2 | yes | — |
| 92 | C22⋊AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 93 | #704 | — | 2 | 4 | no | — |
| 94 | #704 | — | 4 | 4 | no | — |
| 95 | C2×#170 | — | 4 | 4 | no | — |
| 96 | C4⋊AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 97 | C4×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 98 | #700 | — | 4 | 4 | no | — |
| 99 | #700 | C2×D4 | 1 | 8 | yes | — |
| 100 | D5⋊C8 | — | 2 | 4 | no | — |
| 101 | D5⋊C8 | — | 4 | 4 | no | — |
| 102 | C5×SD16 | — | 2 | 4 | no | — |
| 103 | C5×SD16 | — | 2 | 4 | no | — |
| 104 | C2×C5×C8 | C22×C4 | 1 | 2 | yes | — |
| 105 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 107 | C22⋊Dic5 | — | 4 | 2 | no | — |
| 108 | #689 | — | 2 | 4 | no | — |
| 109 | #689 | — | 2 | 4 | no | — |
| 110 | #686 | — | 2 | 4 | no | — |
| 111 | #686 | — | 2 | 4 | no | — |
| 112 | #685 | — | 4 | 2 | no | — |
| 113 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 114 | C4×Dic5 | — | 2 | 1 | no | — |
| 115 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 116 | C2×#168 | C22×C4 | 1 | 2 | yes | — |
| 117 | #677 | — | 2 | 4 | no | — |
| 118 | #677 | — | 2 | 4 | no | — |
| 119 | C8×D5 | C2×D4 | 1 | 2 | yes | — |
| 120 | C8×D5 | C2×D4 | 1 | 2 | yes | — |
| 121 | C8×D5 | #41 | 1 | 2 | yes | — |
| 122 | C8×D5 | #41 | 1 | 2 | yes | — |
| 123 | C2×#135 | — | 5 | 4 | no | — |
| 124 | #568 | — | 5 | 2 | no | — |
| 125 | #568 | — | 5 | 2 | no | — |
| 126 | #532 | — | 5 | 1 | no | — |
| 127 | #532 | — | 5 | 1 | no | — |
| 128 | #495 | — | 5 | 4 | no | — |
| 129 | #494 | — | 5 | 1 | no | — |
| 130 | #493 | — | 5 | 1 | no | — |
| 131 | #492 | — | 5 | 1 | no | — |
| 132 | #492 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 133 | #444 | — | 10 | 8 | no | — |
| 134 | #443 | — | 10 | 8 | no | — |
| 135 | C2×#108 | — | 5 | 2 | no | — |
| 136 | #427 | — | 5 | 2 | no | — |
| 137 | #421 | — | 5 | 4 | no | — |
| 138 | #421 | — | 5 | 4 | no | — |
| 139 | #410 | — | 5 | 2 | no | — |
| 140 | #410 | — | 5 | 2 | no | — |
| 141 | #405 | — | 5 | 4 | no | — |
| 142 | C2×C4×C8 | — | 5 | 1 | no | — |
| 143 | #339 | — | 5 | 4 | no | — |
| 144 | C23×C5 | — | 2 | 2 | no | — |
| 145 | C22×D5 | — | 2 | 2 | no | — |
| 146 | C22×D5 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 147 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 148 | C5×Q8 | — | 2 | 4 | no | — |
| 149 | C5×D4 | — | 2 | 4 | no | — |
| 150 | C5×D4 | — | 8 | 2 | no | — |
| 151 | C2×C4×C5 | — | 2 | 1 | no | — |
| 152 | C2×C4×C5 | — | 2 | 2 | no | — |
| 153 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 154 | C22⋊D5 | — | 4 | 4 | no | — |
| 155 | C2×Dic5 | — | 2 | 1 | no | — |
| 156 | C2×Dic5 | — | 2 | 2 | no | — |
| 157 | C2×Dic5 | — | 4 | 2 | no | — |
| 158 | C2×Dic5 | — | 4 | 2 | no | — |
| 159 | C2×Dic5 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 160 | D20 | — | 2 | 4 | no | — |
| 161 | D20 | — | 4 | 4 | no | — |
| 162 | D20 | — | 8 | 2 | no | — |
| 163 | C4×D5 | — | 4 | 2 | no | — |
| 164 | C4×D5 | — | 4 | 2 | no | — |
| 165 | C4×D5 | — | 4 | 2 | no | — |
| 166 | C4×D5 | C22×D4 | 1 | 2 | yes | — |
| 167 | C4×D5 | C2×#41 | 1 | 2 | yes | — |
| 168 | Dic10 | — | 2 | 4 | no | — |
| 169 | #170 | — | 2 | 8 | no | — |
| 170 | C5×C8 | C4×D4 | 1 | 2 | yes | — |
| 171 | C5×C8 | C4×D4 | 1 | 2 | yes | — |
| 172 | #168 | C4×D4 | 1 | 2 | yes | — |
| 173 | #168 | C4×D4 | 1 | 2 | yes | — |
| 174 | C2×#41 | — | 5 | 2 | no | — |
| 175 | C2×#41 | — | 5 | 2 | no | — |
| 176 | #135 | — | 10 | 4 | no | — |
| 177 | #135 | — | 10 | 4 | no | — |
| 178 | #135 | — | 10 | 4 | no | — |
| 179 | #135 | — | 10 | 4 | no | — |
| 180 | C2×Q16 | — | 5 | 4 | no | — |
| 181 | C2×SD16 | — | 5 | 4 | no | — |
| 182 | C2×SD16 | C2×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 183 | C2×D8 | — | 5 | 4 | no | — |
| 184 | #131 | — | 10 | 8 | no | — |
| 185 | #131 | — | 10 | 8 | no | — |
| 186 | C2×M16 | — | 5 | 4 | no | — |
| 187 | C2×M16 | — | 20 | 2 | no | — |
| 188 | C2×M16 | — | 20 | 2 | no | — |
| 189 | C22×C8 | — | 5 | 2 | no | — |
| 190 | C4⋊Q8 | — | 5 | 1 | no | — |
| 191 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 192 | #127 | — | 5 | 1 | no | — |
| 193 | #127 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 194 | #124 | — | 5 | 2 | no | — |
| 195 | #124 | — | 5 | 2 | no | — |
| 196 | #121 | — | 20 | 1 | no | — |
| 197 | #121 | — | 20 | 1 | no | — |
| 198 | C4×D4 | — | 20 | 1 | no | — |
| 199 | C4×D4 | — | 20 | 1 | no | — |
| 200 | #117 | — | 5 | 4 | no | — |
| 201 | C2×C42 | — | 5 | 1 | no | — |
| 202 | #108 | — | 5 | 4 | no | — |
| 203 | #108 | — | 5 | 4 | no | — |
| 204 | #107 | — | 10 | 2 | no | — |
| 205 | #106 | — | 10 | 2 | no | — |
| 206 | #104 | — | 10 | 8 | no | — |
| 207 | #104 | — | 10 | 8 | no | — |
| 208 | #99 | — | 20 | 2 | no | — |
| 209 | #99 | — | 20 | 2 | no | — |
| 210 | #97 | — | 10 | 4 | no | — |
| 211 | C4×C8 | — | 5 | 2 | no | — |
| 212 | C4×C8 | — | 5 | 1 | no | — |
| 213 | C4×C8 | — | 10 | 4 | no | — |
| 214 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 215 | C22×C5 | — | 4 | 2 | no | — |
| 216 | C22×C5 | — | 8 | 2 | no | — |
| 217 | C22×C5 | #518 | 1 | 1 | yes | characteristic |
| 218 | C2×D5 | — | 4 | 1 | no | — |
| 219 | C2×D5 | — | 4 | 2 | no | — |
| 220 | C2×D5 | — | 8 | 2 | no | — |
| 221 | C2×D5 | #550 | 1 | 2 | yes | — |
| 222 | AGL(1,𝔽5) | — | 8 | 4 | no | — |
| 223 | C4×C5 | — | 2 | 2 | no | — |
| 224 | C4×C5 | — | 4 | 2 | no | — |
| 225 | C4×C5 | C2×C4×D4 | 1 | 2 | yes | — |
| 226 | Dic5 | — | 4 | 1 | no | — |
| 227 | Dic5 | — | 4 | 2 | no | — |
| 228 | Dic5 | D42 | 1 | 1 | yes | characteristic |
| 229 | Dic5 | #559 | 1 | 1 | yes | characteristic |
| 230 | #41 | — | 10 | 4 | no | — |
| 231 | #41 | — | 10 | 4 | no | — |
| 232 | #41 | — | 10 | 4 | no | — |
| 233 | #41 | — | 10 | 4 | no | — |
| 234 | C2×Q8 | — | 5 | 2 | no | — |
| 235 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 236 | C2×D4 | — | 4 | 1 | no | — |
| 237 | C2×D4 | — | 5 | 2 | no | — |
| 238 | C2×D4 | — | 10 | 2 | no | — |
| 239 | C2×D4 | — | 10 | 2 | no | — |
| 240 | C2×D4 | — | 20 | 1 | no | — |
| 241 | C2×D4 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 242 | C22×C4 | — | 5 | 1 | no | — |
| 243 | C22×C4 | — | 10 | 2 | no | — |
| 244 | C22×C4 | — | 10 | 1 | no | — |
| 245 | C22×C4 | — | 10 | 2 | no | — |
| 246 | Q16 | — | 10 | 4 | no | — |
| 247 | Q16 | — | 10 | 4 | no | — |
| 248 | SD16 | — | 2 | 4 | no | — |
| 249 | SD16 | — | 2 | 4 | no | — |
| 250 | SD16 | — | 10 | 4 | no | — |
| 251 | SD16 | — | 10 | 4 | no | — |
| 252 | D8 | — | 10 | 4 | no | — |
| 253 | D8 | — | 10 | 4 | no | — |
| 254 | M16 | — | 5 | 8 | no | — |
| 255 | M16 | — | 10 | 4 | no | — |
| 256 | M16 | — | 20 | 4 | no | — |
| 257 | M16 | — | 20 | 4 | no | — |
| 258 | C2×C8 | — | 5 | 2 | no | — |
| 259 | C2×C8 | — | 5 | 2 | no | — |
| 260 | C2×C8 | — | 5 | 2 | no | — |
| 261 | C2×C8 | — | 5 | 2 | no | — |
| 262 | C2×C8 | — | 5 | 2 | no | — |
| 263 | C2×C8 | — | 10 | 4 | no | — |
| 264 | C2×C8 | — | 20 | 4 | no | — |
| 265 | C2×C8 | — | 20 | 4 | no | — |
| 266 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 267 | C4⋊C4 | — | 4 | 1 | no | — |
| 268 | C4⋊C4 | — | 10 | 4 | no | — |
| 269 | C4⋊C4 | — | 20 | 1 | no | — |
| 270 | C22⋊C4 | — | 10 | 4 | no | — |
| 271 | C22⋊C4 | — | 20 | 2 | no | — |
| 272 | C22⋊C4 | — | 20 | 2 | no | — |
| 273 | C42 | — | 5 | 1 | no | — |
| 274 | C42 | — | 10 | 1 | no | — |
| 275 | C42 | — | 10 | 4 | no | — |
| 276 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 277 | C2×C5 | — | 2 | 1 | no | — |
| 278 | C2×C5 | — | 8 | 2 | no | — |
| 279 | C2×C5 | #2366 | 1 | 1 | yes | characteristic |
| 280 | D5 | — | 2 | 2 | no | — |
| 281 | D5 | — | 8 | 2 | no | — |
| 282 | C23 | — | 2 | 2 | no | — |
| 283 | C23 | — | 5 | 1 | no | — |
| 284 | C23 | — | 10 | 2 | no | — |
| 285 | Q8 | — | 2 | 4 | no | — |
| 286 | Q8 | — | 10 | 4 | no | — |
| 287 | D4 | — | 2 | 4 | no | — |
| 288 | D4 | — | 8 | 2 | no | — |
| 289 | D4 | — | 10 | 4 | no | — |
| 290 | D4 | — | 20 | 4 | no | — |
| 291 | D4 | — | 20 | 4 | no | — |
| 292 | D4 | — | 40 | 2 | no | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 5 | 1 | no | — |
| 296 | C2×C4 | — | 5 | 2 | no | — |
| 297 | C2×C4 | — | 5 | 2 | no | — |
| 298 | C2×C4 | — | 10 | 1 | no | — |
| 299 | C2×C4 | — | 10 | 2 | no | — |
| 300 | C2×C4 | — | 20 | 2 | no | — |
| 301 | C2×C4 | — | 20 | 2 | no | — |
| 302 | C2×C4 | — | 20 | 2 | no | — |
| 303 | C2×C4 | — | 20 | 2 | no | — |
| 304 | C2×C4 | — | 20 | 2 | no | — |
| 305 | C2×C4 | — | 20 | 4 | no | — |
| 306 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 307 | C8 | — | 5 | 2 | no | — |
| 308 | C8 | — | 5 | 2 | no | — |
| 309 | C8 | — | 10 | 8 | no | — |
| 310 | C8 | D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 311 | C8 | D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 312 | C5 | #23196 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 313 | C22 | — | 4 | 2 | no | — |
| 314 | C22 | — | 5 | 2 | no | — |
| 315 | C22 | — | 8 | 2 | no | — |
| 316 | C22 | — | 20 | 1 | no | — |
| 317 | C22 | — | 20 | 2 | no | — |
| 318 | C22 | — | 40 | 2 | no | — |
| 319 | C22 | #66258 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 2 | no | — |
| 321 | C4 | — | 4 | 2 | no | — |
| 322 | C4 | — | 5 | 1 | no | — |
| 323 | C4 | — | 5 | 1 | no | — |
| 324 | C4 | — | 20 | 1 | no | — |
| 325 | C4 | — | 20 | 2 | no | — |
| 326 | C4 | — | 40 | 4 | no | — |
| 327 | C4 | C2×D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 328 | C2 | — | 2 | 1 | no | — |
| 329 | C2 | — | 8 | 2 | no | — |
| 330 | C2 | — | 10 | 2 | no | — |
| 331 | C2 | — | 40 | 2 | no | — |
| 332 | C2 | #10617369 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10617431 | C2 | semidirect |
| 3. #10604821 | C2 | semidirect |
| 4. #10604676 | C2 | semidirect |
| 5. #10604618 | C2 | semidirect |
| 6. #10603851 | C2 | semidirect |
| 21. C5×#492 | C4 | semidirect |
| 22. #65457 | C4 | semidirect |
| 23. #65381 | C4 | semidirect |
| 24. #65362 | C4 | semidirect |
| 25. #64982 | C4 | semidirect |
| 61. #3996 | D4 | semidirect |
| 62. #3995 | D4 | semidirect |
| 132. #492 | AGL(1,𝔽5) | semidirect |
| 312. C5 | #23196 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617431 | 2 |
| #10613491 | 2 |
| #10604821 | 2 |
| #10604676 | 2 |
| #10604618 | 2 |
| #10603851 | 2 |
| #23196 | 5 |