Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1217 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1217 conjugacy classes of subgroups — too many to draw. Only its 218 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×#2811 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2024 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×#2016 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7759 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 5 | C2×#1272 | C2 | 1 | 2 | yes | maximal |
| 6 | C2×#1272 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C23×Q16 | — | 2 | 1 | no | — |
| 8 | C22×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#103 | — | 2 | 2 | no | — |
| 10 | C22×#103 | — | 2 | 1 | no | — |
| 11 | #2811 | — | 2 | 2 | no | — |
| 12 | #2811 | — | 2 | 4 | no | — |
| 13 | #2811 | C22 | 1 | 4 | yes | — |
| 14 | C2×#395 | — | 2 | 1 | no | — |
| 15 | C2×#395 | — | 2 | 2 | no | — |
| 16 | C2×#391 | — | 2 | 1 | no | — |
| 17 | C2×#391 | — | 2 | 2 | no | — |
| 18 | C2×#391 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#391 | C22 | 1 | 2 | yes | — |
| 20 | #2024 | — | 2 | 4 | no | — |
| 21 | #2024 | C22 | 1 | 8 | yes | — |
| 22 | #2016 | — | 2 | 2 | no | — |
| 23 | #2016 | C22 | 1 | 4 | yes | — |
| 24 | C2×#340 | — | 2 | 2 | no | — |
| 25 | C2×#336 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#336 | C22 | 1 | 2 | yes | — |
| 27 | #1272 | C22 | 1 | 8 | yes | — |
| 28 | #1272 | C22 | 1 | 4 | yes | — |
| 29 | C23×Q8 | — | 2 | 1 | no | — |
| 30 | C23×Q8 | — | 2 | 1 | no | — |
| 31 | C23×Q8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C23×Q8 | D4 | 1 | 2 | yes | — |
| 33 | C22×Q16 | — | 2 | 4 | no | — |
| 34 | C22×Q16 | — | 4 | 4 | no | — |
| 35 | C22×Q16 | — | 4 | 4 | no | — |
| 36 | C22×Q16 | — | 4 | 4 | no | — |
| 37 | C23×C8 | — | 2 | 1 | no | — |
| 38 | C23×C8 | — | 2 | 2 | no | — |
| 39 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 40 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 41 | C2×C4⋊Q8 | — | 4 | 4 | no | — |
| 42 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 43 | C22×C4⋊C4 | — | 2 | 2 | no | — |
| 44 | C22×C4⋊C4 | — | 2 | 1 | no | — |
| 45 | C22×C4⋊C4 | — | 2 | 2 | no | — |
| 46 | C22×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 47 | C22×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 48 | C22×C42 | — | 2 | 1 | no | — |
| 49 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#103 | — | 2 | 8 | no | — |
| 51 | C2×#103 | — | 2 | 4 | no | — |
| 52 | C2×#103 | — | 4 | 8 | no | — |
| 53 | C2×#103 | — | 4 | 4 | no | — |
| 54 | #395 | — | 2 | 4 | no | — |
| 55 | #395 | — | 2 | 8 | no | — |
| 56 | #395 | — | 2 | 8 | no | — |
| 57 | #395 | — | 2 | 4 | no | — |
| 58 | #391 | — | 2 | 4 | no | — |
| 59 | #391 | — | 2 | 8 | no | — |
| 60 | #391 | — | 2 | 4 | no | — |
| 61 | #391 | — | 2 | 8 | no | — |
| 62 | #391 | — | 2 | 2 | no | — |
| 63 | #391 | — | 2 | 2 | no | — |
| 64 | #391 | — | 2 | 2 | no | — |
| 65 | #391 | D4 | 1 | 2 | yes | — |
| 66 | #391 | D4 | 1 | 4 | yes | — |
| 67 | #391 | C23 | 1 | 2 | yes | — |
| 68 | #391 | C23 | 1 | 4 | yes | — |
| 69 | C2×#95 | — | 2 | 2 | no | — |
| 70 | C2×#95 | — | 2 | 1 | no | — |
| 71 | C2×#95 | — | 2 | 2 | no | — |
| 72 | C2×#95 | — | 2 | 2 | no | — |
| 73 | C2×#95 | — | 2 | 2 | no | — |
| 74 | #340 | — | 2 | 8 | no | — |
| 75 | #336 | — | 2 | 1 | no | — |
| 76 | #336 | — | 2 | 2 | no | — |
| 77 | #336 | C23 | 1 | 2 | yes | — |
| 78 | #336 | C23 | 1 | 4 | yes | — |
| 79 | C22×Q8 | — | 2 | 4 | no | — |
| 80 | C22×Q8 | — | 2 | 1 | no | — |
| 81 | C22×Q8 | — | 2 | 2 | no | — |
| 82 | C22×Q8 | — | 2 | 2 | no | — |
| 83 | C22×Q8 | — | 2 | 4 | no | — |
| 84 | C22×Q8 | — | 2 | 4 | no | — |
| 85 | C22×Q8 | — | 2 | 2 | no | — |
| 86 | C22×Q8 | — | 4 | 2 | no | — |
| 87 | C22×Q8 | — | 4 | 2 | no | — |
| 88 | C22×Q8 | — | 4 | 2 | no | — |
| 89 | C22×Q8 | — | 4 | 4 | no | — |
| 90 | C22×Q8 | — | 4 | 4 | no | — |
| 91 | C22×Q8 | — | 4 | 2 | no | — |
| 92 | C22×Q8 | — | 4 | 2 | no | — |
| 93 | C22×Q8 | — | 4 | 1 | no | — |
| 94 | C22×Q8 | — | 4 | 2 | no | — |
| 95 | C22×Q8 | — | 4 | 2 | no | — |
| 96 | C22×Q8 | — | 8 | 2 | no | — |
| 97 | C22×Q8 | — | 8 | 4 | no | — |
| 98 | C22×Q8 | SD16 | 1 | 4 | yes | — |
| 99 | C22×Q8 | Q16 | 1 | 8 | yes | complemented |
| 100 | C22×Q8 | C2×D4 | 1 | 2 | yes | — |
| 101 | C22×Q8 | C2×D4 | 1 | 4 | yes | — |
| 102 | C23×C4 | — | 2 | 1 | no | — |
| 103 | C23×C4 | — | 2 | 2 | no | — |
| 104 | C23×C4 | — | 2 | 1 | no | — |
| 105 | C23×C4 | — | 2 | 2 | no | — |
| 106 | C23×C4 | — | 4 | 1 | no | — |
| 107 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 109 | C2×Q16 | — | 4 | 8 | no | — |
| 110 | C2×Q16 | — | 4 | 8 | no | — |
| 111 | C2×Q16 | — | 4 | 32 | no | — |
| 112 | C2×Q16 | — | 4 | 8 | no | — |
| 113 | C22×C8 | — | 2 | 4 | no | — |
| 114 | C22×C8 | — | 2 | 2 | no | — |
| 115 | C22×C8 | — | 2 | 4 | no | — |
| 116 | C22×C8 | — | 4 | 2 | no | — |
| 117 | C22×C8 | — | 4 | 2 | no | — |
| 118 | C22×C8 | — | 4 | 4 | no | — |
| 119 | C22×C8 | — | 4 | 4 | no | — |
| 120 | C22×C8 | — | 4 | 4 | no | — |
| 121 | C4⋊Q8 | — | 4 | 4 | no | — |
| 122 | C4⋊Q8 | — | 4 | 8 | no | — |
| 123 | C4⋊Q8 | — | 4 | 4 | no | — |
| 124 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 125 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 126 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 127 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 128 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 129 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 130 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 131 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 132 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 133 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 134 | C2×C4⋊C4 | — | 8 | 2 | no | — |
| 135 | C2×C4⋊C4 | — | 8 | 4 | no | — |
| 136 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 137 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 138 | C2×C42 | — | 2 | 4 | no | — |
| 139 | C2×C42 | — | 4 | 2 | no | — |
| 140 | C2×C42 | — | 8 | 2 | no | — |
| 141 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 142 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 143 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 144 | #103 | — | 4 | 16 | no | — |
| 145 | #103 | — | 4 | 8 | no | — |
| 146 | #95 | — | 2 | 8 | no | — |
| 147 | #95 | — | 2 | 4 | no | — |
| 148 | #95 | — | 2 | 4 | no | — |
| 149 | #95 | — | 2 | 4 | no | — |
| 150 | #95 | — | 2 | 4 | no | — |
| 151 | #95 | — | 2 | 4 | no | — |
| 152 | #95 | — | 2 | 4 | no | — |
| 153 | #95 | — | 2 | 4 | no | — |
| 154 | C24 | #120 | 1 | 1 | yes | characteristic |
| 155 | C2×Q8 | — | 2 | 2 | no | — |
| 156 | C2×Q8 | — | 2 | 4 | no | — |
| 157 | C2×Q8 | — | 2 | 4 | no | — |
| 158 | C2×Q8 | — | 2 | 8 | no | — |
| 159 | C2×Q8 | — | 2 | 4 | no | — |
| 160 | C2×Q8 | — | 4 | 2 | no | — |
| 161 | C2×Q8 | — | 4 | 4 | no | — |
| 162 | C2×Q8 | — | 4 | 4 | no | — |
| 163 | C2×Q8 | — | 4 | 8 | no | — |
| 164 | C2×Q8 | — | 4 | 2 | no | — |
| 165 | C2×Q8 | — | 4 | 2 | no | — |
| 166 | C2×Q8 | — | 4 | 16 | no | — |
| 167 | C2×Q8 | — | 4 | 4 | no | — |
| 168 | C2×Q8 | — | 4 | 2 | no | — |
| 169 | C2×Q8 | — | 4 | 8 | no | — |
| 170 | C2×Q8 | — | 4 | 2 | no | — |
| 171 | C2×Q8 | — | 4 | 8 | no | — |
| 172 | C2×Q8 | — | 4 | 2 | no | — |
| 173 | C2×Q8 | — | 4 | 2 | no | — |
| 174 | C2×Q8 | — | 8 | 4 | no | — |
| 175 | C2×Q8 | — | 8 | 8 | no | — |
| 176 | C2×Q8 | — | 8 | 1 | no | — |
| 177 | C2×Q8 | — | 8 | 2 | no | — |
| 178 | C2×Q8 | — | 8 | 2 | no | — |
| 179 | C2×Q8 | — | 8 | 1 | no | — |
| 180 | C2×Q8 | — | 8 | 4 | no | — |
| 181 | C2×Q8 | — | 8 | 4 | no | — |
| 182 | C2×Q8 | — | 8 | 2 | no | — |
| 183 | C2×Q8 | — | 8 | 4 | no | — |
| 184 | C2×Q8 | — | 8 | 8 | no | — |
| 185 | C2×Q8 | — | 8 | 16 | no | — |
| 186 | C2×Q8 | — | 8 | 2 | no | — |
| 187 | C2×Q8 | C2×SD16 | 1 | 4 | yes | — |
| 188 | C2×Q8 | C2×Q16 | 1 | 8 | yes | complemented |
| 189 | C22×C4 | — | 2 | 2 | no | — |
| 190 | C22×C4 | — | 2 | 4 | no | — |
| 191 | C22×C4 | — | 2 | 4 | no | — |
| 192 | C22×C4 | — | 2 | 2 | no | — |
| 193 | C22×C4 | — | 2 | 2 | no | — |
| 194 | C22×C4 | — | 2 | 4 | no | — |
| 195 | C22×C4 | — | 2 | 4 | no | — |
| 196 | C22×C4 | — | 2 | 8 | no | — |
| 197 | C22×C4 | — | 2 | 2 | no | — |
| 198 | C22×C4 | — | 2 | 4 | no | — |
| 199 | C22×C4 | — | 4 | 2 | no | — |
| 200 | C22×C4 | — | 4 | 4 | no | — |
| 201 | C22×C4 | — | 4 | 2 | no | — |
| 202 | C22×C4 | — | 4 | 4 | no | — |
| 203 | C22×C4 | — | 4 | 2 | no | — |
| 204 | C22×C4 | — | 4 | 2 | no | — |
| 205 | C22×C4 | — | 4 | 4 | no | — |
| 206 | C22×C4 | — | 4 | 4 | no | — |
| 207 | C22×C4 | — | 4 | 4 | no | — |
| 208 | C22×C4 | — | 4 | 4 | no | — |
| 209 | C22×C4 | — | 4 | 4 | no | — |
| 210 | C22×C4 | — | 8 | 2 | no | — |
| 211 | C22×C4 | — | 8 | 2 | no | — |
| 212 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 213 | C22×C4 | #120 | 1 | 2 | yes | — |
| 214 | C22×C4 | C2×SD16 | 1 | 2 | yes | — |
| 215 | C22×C4 | C2×Q16 | 1 | 4 | yes | — |
| 216 | C22×C4 | #137 | 1 | 2 | yes | — |
| 217 | C22×C4 | #137 | 1 | 4 | yes | — |
| 218 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 219 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 220 | Q16 | — | 4 | 32 | no | — |
| 221 | C2×C8 | — | 2 | 4 | no | — |
| 222 | C2×C8 | — | 4 | 8 | no | — |
| 223 | C2×C8 | — | 4 | 2 | no | — |
| 224 | C2×C8 | — | 4 | 2 | no | — |
| 225 | C2×C8 | — | 4 | 4 | no | — |
| 226 | C2×C8 | — | 4 | 4 | no | — |
| 227 | C2×C8 | — | 4 | 4 | no | — |
| 228 | C2×C8 | — | 8 | 8 | no | — |
| 229 | C4⋊C4 | — | 2 | 8 | no | — |
| 230 | C4⋊C4 | — | 4 | 2 | no | — |
| 231 | C4⋊C4 | — | 4 | 4 | no | — |
| 232 | C4⋊C4 | — | 4 | 4 | no | — |
| 233 | C4⋊C4 | — | 4 | 8 | no | — |
| 234 | C4⋊C4 | — | 8 | 1 | no | — |
| 235 | C4⋊C4 | — | 8 | 2 | no | — |
| 236 | C4⋊C4 | — | 8 | 4 | no | — |
| 237 | C4⋊C4 | — | 8 | 8 | no | — |
| 238 | C42 | — | 4 | 2 | no | — |
| 239 | C42 | — | 4 | 1 | no | — |
| 240 | C42 | — | 4 | 1 | no | — |
| 241 | C42 | — | 8 | 4 | no | — |
| 242 | C23 | #448 | 1 | 2 | yes | — |
| 243 | C23 | #451 | 1 | 4 | yes | — |
| 244 | C23 | #453 | 1 | 2 | yes | — |
| 245 | C23 | #455 | 1 | 4 | yes | — |
| 246 | C23 | B(4,𝔽2) | 1 | 2 | yes | — |
| 247 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 248 | Q8 | — | 4 | 4 | no | — |
| 249 | Q8 | — | 4 | 8 | no | — |
| 250 | Q8 | — | 4 | 4 | no | — |
| 251 | Q8 | — | 4 | 4 | no | — |
| 252 | Q8 | — | 8 | 2 | no | — |
| 253 | Q8 | — | 8 | 4 | no | — |
| 254 | Q8 | — | 8 | 4 | no | — |
| 255 | Q8 | — | 8 | 8 | no | — |
| 256 | Q8 | — | 8 | 2 | no | — |
| 257 | Q8 | — | 8 | 2 | no | — |
| 258 | Q8 | — | 8 | 8 | no | — |
| 259 | Q8 | — | 8 | 16 | no | — |
| 260 | C2×C4 | — | 2 | 4 | no | — |
| 261 | C2×C4 | — | 2 | 4 | no | — |
| 262 | C2×C4 | — | 2 | 8 | no | — |
| 263 | C2×C4 | — | 2 | 8 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 2 | no | — |
| 268 | C2×C4 | — | 2 | 2 | no | — |
| 269 | C2×C4 | — | 2 | 2 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 4 | no | — |
| 273 | C2×C4 | — | 2 | 8 | no | — |
| 274 | C2×C4 | — | 4 | 8 | no | — |
| 275 | C2×C4 | — | 4 | 16 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 4 | 4 | no | — |
| 278 | C2×C4 | — | 4 | 2 | no | — |
| 279 | C2×C4 | — | 4 | 4 | no | — |
| 280 | C2×C4 | — | 4 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 4 | 4 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 8 | 8 | no | — |
| 287 | C2×C4 | — | 8 | 4 | no | — |
| 288 | C2×C4 | — | 8 | 8 | no | — |
| 289 | C2×C4 | — | 8 | 2 | no | — |
| 290 | C2×C4 | — | 8 | 2 | no | — |
| 291 | C2×C4 | #448 | 1 | 2 | yes | — |
| 292 | C2×C4 | #451 | 1 | 4 | yes | — |
| 293 | C2×C4 | C22×SD16 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C22×Q16 | 1 | 2 | yes | — |
| 295 | C2×C4 | C2×#137 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 297 | C8 | — | 4 | 4 | no | — |
| 298 | C8 | — | 8 | 4 | no | — |
| 299 | C22 | #1596 | 1 | 4 | yes | — |
| 300 | C22 | #1602 | 1 | 8 | yes | — |
| 301 | C22 | #1633 | 1 | 4 | yes | — |
| 302 | C22 | #1645 | 1 | 8 | yes | — |
| 303 | C22 | #1659 | 1 | 4 | yes | — |
| 304 | C22 | C2×#448 | 1 | 1 | yes | characteristic |
| 305 | C22 | C2×#451 | 1 | 2 | yes | — |
| 306 | C22 | C2×#453 | 1 | 1 | yes | characteristic |
| 307 | C22 | C2×#455 | 1 | 2 | yes | — |
| 308 | C22 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 1 | no | — |
| 310 | C4 | — | 2 | 1 | no | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 2 | 2 | no | — |
| 313 | C4 | — | 2 | 2 | no | — |
| 314 | C4 | — | 2 | 4 | no | — |
| 315 | C4 | — | 4 | 4 | no | — |
| 316 | C4 | — | 4 | 8 | no | — |
| 317 | C4 | — | 8 | 2 | no | — |
| 318 | C4 | — | 8 | 4 | no | — |
| 319 | C4 | — | 8 | 4 | no | — |
| 320 | C2 | #7759 | 1 | 8 | yes | direct product factor |
| 321 | C2 | C2×#1596 | 1 | 1 | yes | characteristic |
| 322 | C2 | C2×#1602 | 1 | 2 | yes | — |
| 323 | C2 | C2×#1633 | 1 | 1 | yes | characteristic |
| 324 | C2 | C2×#1645 | 1 | 2 | yes | — |
| 325 | C2 | C2×#1659 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7759 | C2 | direct (N × H) |
| 99. C22×Q8 | Q16 | semidirect |
| 188. C2×Q8 | C2×Q16 | semidirect |
| 320. C2 | #7759 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2811 | 2 |
| C2×#2024 | 2 |
| C2×#2016 | 2 |
| #7759 | 2 |
| C2×#1272 | 2 |