Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1044 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 176 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 | #62635 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33721 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #33608 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33532 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #25800 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13335 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #13243 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12613 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #3584 | C22 | 1 | 2 | yes | — |
| 10 | #3417 | — | 2 | 2 | no | — |
| 11 | #3416 | — | 2 | 4 | no | — |
| 12 | #3415 | — | 2 | 2 | no | — |
| 13 | #3414 | — | 2 | 2 | no | — |
| 14 | #3414 | C22 | 1 | 2 | yes | — |
| 15 | C2×#510 | C22 | 1 | 4 | yes | complemented |
| 16 | C2×#508 | — | 2 | 2 | no | — |
| 17 | C2×#508 | C22 | 1 | 2 | yes | — |
| 18 | C22×Q32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×#504 | C22 | 1 | 1 | yes | characteristic |
| 20 | #3156 | C22 | 1 | 4 | yes | complemented |
| 21 | #3059 | C22 | 1 | 2 | yes | — |
| 22 | #3028 | — | 2 | 2 | no | — |
| 23 | #3022 | — | 2 | 2 | no | — |
| 24 | #2251 | — | 2 | 4 | no | — |
| 25 | #2249 | — | 2 | 2 | no | — |
| 26 | #2247 | — | 2 | 2 | no | — |
| 27 | #2219 | — | 2 | 2 | no | — |
| 28 | #2216 | — | 2 | 2 | no | — |
| 29 | #2191 | — | 2 | 8 | no | — |
| 30 | #2158 | C22 | 1 | 1 | yes | characteristic, complemented |
| 31 | #2145 | C22 | 1 | 4 | yes | complemented |
| 32 | #2141 | C22 | 1 | 2 | yes | complemented |
| 33 | #2115 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2108 | C22 | 1 | 2 | yes | complemented |
| 35 | #2009 | — | 2 | 4 | no | — |
| 36 | #1976 | C22 | 1 | 4 | yes | complemented |
| 37 | #1757 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1378 | C22 | 1 | 2 | yes | complemented |
| 39 | #585 | — | 2 | 2 | no | — |
| 40 | #578 | — | 2 | 4 | no | — |
| 41 | #577 | — | 4 | 4 | no | — |
| 42 | #576 | — | 2 | 4 | no | — |
| 43 | #575 | — | 2 | 4 | no | — |
| 44 | #575 | — | 2 | 2 | no | — |
| 45 | C2×#137 | — | 2 | 4 | no | — |
| 46 | C2×#135 | — | 2 | 4 | no | — |
| 47 | C2×#135 | C23 | 1 | 4 | yes | — |
| 48 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 49 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 50 | #536 | — | 2 | 2 | no | — |
| 51 | #536 | — | 2 | 4 | no | — |
| 52 | #510 | — | 2 | 8 | no | — |
| 53 | #510 | — | 2 | 4 | no | — |
| 54 | #510 | — | 2 | 8 | no | — |
| 55 | #510 | — | 4 | 4 | no | — |
| 56 | #509 | — | 2 | 8 | no | — |
| 57 | #509 | — | 2 | 4 | no | — |
| 58 | #508 | — | 2 | 4 | no | — |
| 59 | #508 | — | 2 | 4 | no | — |
| 60 | #508 | — | 2 | 2 | no | — |
| 61 | #508 | — | 2 | 4 | no | — |
| 62 | #508 | — | 2 | 2 | no | — |
| 63 | #508 | — | 2 | 2 | no | — |
| 64 | #508 | — | 2 | 2 | no | — |
| 65 | C2×Q32 | — | 2 | 2 | no | — |
| 66 | C2×Q32 | — | 2 | 4 | no | — |
| 67 | C2×Q32 | — | 2 | 1 | no | — |
| 68 | C2×Q32 | — | 2 | 1 | no | — |
| 69 | C2×Q32 | — | 2 | 2 | no | — |
| 70 | C2×Q32 | — | 2 | 2 | no | — |
| 71 | C2×Q32 | D4 | 1 | 2 | yes | — |
| 72 | C2×SD32 | — | 2 | 4 | no | — |
| 73 | C2×SD32 | — | 2 | 4 | no | — |
| 74 | C2×SD32 | D4 | 1 | 4 | yes | — |
| 75 | C2×D16 | — | 2 | 2 | no | — |
| 76 | C2×D16 | D4 | 1 | 2 | yes | — |
| 77 | #504 | — | 2 | 2 | no | — |
| 78 | #504 | — | 2 | 2 | no | — |
| 79 | C2×M32 | D4 | 1 | 2 | yes | — |
| 80 | C2×M32 | C23 | 1 | 2 | yes | — |
| 81 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 82 | C22×C16 | D4 | 1 | 2 | yes | — |
| 83 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 84 | #497 | — | 4 | 4 | no | — |
| 85 | #495 | — | 2 | 4 | no | — |
| 86 | #494 | — | 2 | 4 | no | — |
| 87 | #470 | — | 2 | 4 | no | — |
| 88 | #467 | — | 2 | 4 | no | — |
| 89 | #458 | — | 4 | 8 | no | — |
| 90 | #456 | — | 4 | 4 | no | — |
| 91 | #455 | — | 4 | 4 | no | — |
| 92 | #427 | C23 | 1 | 2 | yes | — |
| 93 | #421 | — | 2 | 4 | no | — |
| 94 | #419 | — | 2 | 2 | no | — |
| 95 | #413 | D4 | 1 | 1 | yes | characteristic |
| 96 | #410 | — | 2 | 4 | no | — |
| 97 | #410 | D4 | 1 | 1 | yes | characteristic |
| 98 | #405 | C23 | 1 | 2 | yes | — |
| 99 | #367 | — | 2 | 1 | no | — |
| 100 | #366 | — | 2 | 1 | no | — |
| 101 | #360 | — | 2 | 8 | no | — |
| 102 | #358 | — | 2 | 4 | no | — |
| 103 | #349 | D4 | 1 | 4 | yes | — |
| 104 | #346 | — | 2 | 2 | no | — |
| 105 | C4×C16 | — | 2 | 2 | no | — |
| 106 | #344 | D4 | 1 | 2 | yes | — |
| 107 | 2−1+4 | — | 4 | 4 | no | — |
| 108 | 2−1+4 | — | 4 | 2 | no | — |
| 109 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 110 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 111 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 112 | C2×#41 | — | 2 | 4 | no | — |
| 113 | C2×#41 | — | 4 | 4 | no | — |
| 114 | C2×#41 | — | 8 | 4 | no | — |
| 115 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C22×Q8 | — | 2 | 2 | no | — |
| 117 | #137 | — | 4 | 8 | no | — |
| 118 | #137 | — | 4 | 4 | no | — |
| 119 | #137 | — | 4 | 8 | no | — |
| 120 | #137 | — | 8 | 4 | no | — |
| 121 | AGL(1,ℤ/8) | — | 4 | 8 | no | — |
| 122 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 123 | #135 | — | 2 | 8 | no | — |
| 124 | #135 | — | 2 | 4 | no | — |
| 125 | #135 | — | 2 | 4 | no | — |
| 126 | #135 | — | 2 | 4 | no | — |
| 127 | #135 | — | 2 | 4 | no | — |
| 128 | #135 | — | 4 | 4 | no | — |
| 129 | #135 | — | 4 | 4 | no | — |
| 130 | C2×Q16 | — | 2 | 4 | no | — |
| 131 | C2×Q16 | — | 2 | 2 | no | — |
| 132 | C2×Q16 | — | 2 | 2 | no | — |
| 133 | C2×Q16 | — | 2 | 4 | no | — |
| 134 | C2×Q16 | — | 2 | 4 | no | — |
| 135 | C2×Q16 | — | 4 | 4 | no | — |
| 136 | C2×Q16 | C2×D4 | 1 | 4 | yes | — |
| 137 | C2×SD16 | — | 2 | 4 | no | — |
| 138 | C2×SD16 | — | 4 | 4 | no | — |
| 139 | C2×SD16 | — | 4 | 4 | no | — |
| 140 | C2×D8 | — | 2 | 4 | no | — |
| 141 | C2×D8 | C2×D4 | 1 | 4 | yes | — |
| 142 | #131 | — | 2 | 2 | no | — |
| 143 | #131 | — | 2 | 2 | no | — |
| 144 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 145 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 147 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 148 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 149 | C4⋊Q8 | — | 4 | 4 | no | — |
| 150 | C4⋊Q8 | — | 4 | 2 | no | — |
| 151 | #124 | — | 4 | 4 | no | — |
| 152 | #124 | — | 4 | 2 | no | — |
| 153 | #123 | — | 4 | 4 | no | — |
| 154 | #123 | — | 4 | 8 | no | — |
| 155 | C22⋊Q8 | — | 4 | 4 | no | — |
| 156 | C22⋊Q8 | — | 8 | 4 | no | — |
| 157 | #117 | — | 2 | 2 | no | — |
| 158 | #117 | C2×D4 | 1 | 2 | yes | — |
| 159 | Q32 | — | 2 | 4 | no | — |
| 160 | Q32 | — | 2 | 2 | no | — |
| 161 | Q32 | — | 2 | 2 | no | — |
| 162 | Q32 | — | 4 | 8 | no | — |
| 163 | Q32 | — | 4 | 1 | no | — |
| 164 | Q32 | — | 4 | 1 | no | — |
| 165 | Q32 | — | 4 | 2 | no | — |
| 166 | SD32 | — | 2 | 8 | no | — |
| 167 | SD32 | — | 2 | 4 | no | — |
| 168 | SD32 | — | 2 | 4 | no | — |
| 169 | SD32 | — | 4 | 8 | no | — |
| 170 | D16 | — | 2 | 4 | no | — |
| 171 | D16 | — | 2 | 2 | no | — |
| 172 | D16 | — | 2 | 2 | no | — |
| 173 | M32 | — | 2 | 2 | no | — |
| 174 | M32 | — | 2 | 2 | no | — |
| 175 | M32 | — | 2 | 4 | no | — |
| 176 | M32 | C2×D4 | 1 | 4 | yes | — |
| 177 | C2×C16 | — | 2 | 2 | no | — |
| 178 | C2×C16 | — | 2 | 2 | no | — |
| 179 | C2×C16 | — | 2 | 2 | no | — |
| 180 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 181 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 182 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 183 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 185 | #107 | — | 2 | 2 | no | — |
| 186 | #106 | — | 2 | 2 | no | — |
| 187 | #104 | — | 4 | 8 | no | — |
| 188 | #103 | — | 4 | 4 | no | — |
| 189 | #101 | D8 | 1 | 2 | yes | — |
| 190 | #100 | D8 | 1 | 2 | yes | — |
| 191 | #99 | — | 4 | 8 | no | — |
| 192 | #99 | D8 | 1 | 4 | yes | — |
| 193 | #97 | — | 2 | 2 | no | — |
| 194 | C4×C8 | — | 2 | 2 | no | — |
| 195 | #41 | — | 2 | 2 | no | — |
| 196 | #41 | — | 2 | 2 | no | — |
| 197 | #41 | — | 4 | 8 | no | — |
| 198 | #41 | — | 4 | 4 | no | — |
| 199 | #41 | — | 4 | 4 | no | — |
| 200 | #41 | — | 4 | 4 | no | — |
| 201 | #41 | — | 4 | 4 | no | — |
| 202 | #41 | — | 8 | 4 | no | — |
| 203 | #41 | — | 8 | 8 | no | — |
| 204 | #41 | — | 8 | 4 | no | — |
| 205 | #41 | — | 8 | 4 | no | — |
| 206 | #41 | — | 8 | 4 | no | — |
| 207 | C2×Q8 | — | 2 | 4 | no | — |
| 208 | C2×Q8 | — | 4 | 4 | no | — |
| 209 | C2×Q8 | — | 4 | 2 | no | — |
| 210 | C2×Q8 | — | 4 | 4 | no | — |
| 211 | C2×Q8 | — | 4 | 2 | no | — |
| 212 | C2×Q8 | — | 4 | 4 | no | — |
| 213 | C2×Q8 | — | 8 | 4 | no | — |
| 214 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 215 | C2×D4 | — | 2 | 4 | no | — |
| 216 | C2×D4 | — | 4 | 4 | no | — |
| 217 | C2×D4 | — | 4 | 4 | no | — |
| 218 | C2×D4 | — | 8 | 4 | no | — |
| 219 | C2×D4 | — | 8 | 8 | no | — |
| 220 | C2×D4 | — | 8 | 4 | no | — |
| 221 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 222 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 223 | C22×C4 | — | 4 | 4 | no | — |
| 224 | C22×C4 | — | 4 | 2 | no | — |
| 225 | C22×C4 | — | 8 | 4 | no | — |
| 226 | C22×C4 | #127 | 1 | 2 | yes | — |
| 227 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 228 | Q16 | — | 2 | 4 | no | — |
| 229 | Q16 | — | 2 | 4 | no | — |
| 230 | Q16 | — | 4 | 4 | no | — |
| 231 | Q16 | — | 4 | 2 | no | — |
| 232 | Q16 | — | 4 | 2 | no | — |
| 233 | Q16 | — | 4 | 4 | no | — |
| 234 | Q16 | — | 8 | 8 | no | — |
| 235 | SD16 | — | 4 | 8 | no | — |
| 236 | SD16 | — | 4 | 4 | no | — |
| 237 | SD16 | — | 4 | 4 | no | — |
| 238 | SD16 | — | 8 | 8 | no | — |
| 239 | D8 | — | 2 | 4 | no | — |
| 240 | D8 | — | 2 | 4 | no | — |
| 241 | D8 | — | 4 | 4 | no | — |
| 242 | M16 | — | 2 | 2 | no | — |
| 243 | M16 | — | 2 | 2 | no | — |
| 244 | M16 | — | 2 | 4 | no | — |
| 245 | M16 | C2×D8 | 1 | 4 | yes | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | #127 | 1 | 2 | yes | — |
| 250 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 252 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 8 | 8 | no | — |
| 257 | C4⋊C4 | — | 8 | 4 | no | — |
| 258 | C22⋊C4 | — | 4 | 2 | no | — |
| 259 | C22⋊C4 | — | 8 | 8 | no | — |
| 260 | C22⋊C4 | — | 8 | 4 | no | — |
| 261 | C22⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 262 | C42 | — | 2 | 2 | no | — |
| 263 | C42 | — | 4 | 2 | no | — |
| 264 | C16 | — | 2 | 2 | no | — |
| 265 | C16 | — | 2 | 4 | no | — |
| 266 | C16 | #120 | 1 | 1 | yes | characteristic |
| 267 | C16 | #120 | 1 | 1 | yes | characteristic |
| 268 | C16 | #120 | 1 | 2 | yes | — |
| 269 | C23 | — | 4 | 4 | no | — |
| 270 | C23 | — | 8 | 4 | no | — |
| 271 | C23 | #493 | 1 | 2 | yes | — |
| 272 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 273 | Q8 | — | 2 | 2 | no | — |
| 274 | Q8 | — | 4 | 4 | no | — |
| 275 | Q8 | — | 4 | 4 | no | — |
| 276 | Q8 | — | 8 | 4 | no | — |
| 277 | Q8 | — | 8 | 2 | no | — |
| 278 | Q8 | — | 8 | 2 | no | — |
| 279 | Q8 | — | 8 | 4 | no | — |
| 280 | D4 | — | 2 | 4 | no | — |
| 281 | D4 | — | 2 | 2 | no | — |
| 282 | D4 | — | 4 | 4 | no | — |
| 283 | D4 | — | 4 | 4 | no | — |
| 284 | D4 | — | 8 | 4 | no | — |
| 285 | D4 | — | 8 | 8 | no | — |
| 286 | D4 | — | 8 | 4 | no | — |
| 287 | D4 | — | 8 | 4 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 4 | 4 | no | — |
| 292 | C2×C4 | — | 8 | 8 | no | — |
| 293 | C2×C4 | — | 8 | 4 | no | — |
| 294 | C2×C4 | — | 8 | 4 | no | — |
| 295 | C2×C4 | — | 8 | 4 | no | — |
| 296 | C2×C4 | — | 8 | 2 | no | — |
| 297 | C2×C4 | — | 8 | 2 | no | — |
| 298 | C2×C4 | — | 8 | 4 | no | — |
| 299 | C2×C4 | #493 | 1 | 2 | yes | — |
| 300 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | D42 | 1 | 2 | yes | — |
| 302 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | #576 | 1 | 2 | yes | — |
| 304 | C8 | — | 2 | 2 | no | — |
| 305 | C8 | — | 2 | 4 | no | — |
| 306 | C8 | C2×#120 | 1 | 1 | yes | characteristic |
| 307 | C8 | #535 | 1 | 1 | yes | characteristic |
| 308 | C8 | D42 | 1 | 2 | yes | — |
| 309 | C22 | — | 2 | 2 | no | — |
| 310 | C22 | — | 4 | 1 | no | — |
| 311 | C22 | — | 4 | 2 | no | — |
| 312 | C22 | — | 8 | 4 | no | — |
| 313 | C22 | — | 16 | 4 | no | — |
| 314 | C22 | — | 32 | 4 | no | — |
| 315 | C22 | C2×#493 | 1 | 1 | yes | characteristic |
| 316 | C22 | D4×D8 | 1 | 2 | yes | — |
| 317 | C4 | — | 2 | 2 | no | — |
| 318 | C4 | — | 4 | 4 | no | — |
| 319 | C4 | — | 8 | 4 | no | — |
| 320 | C4 | — | 16 | 2 | no | — |
| 321 | C4 | — | 16 | 2 | no | — |
| 322 | C4 | #2433 | 1 | 1 | yes | characteristic |
| 323 | C4 | #3155 | 1 | 1 | yes | characteristic |
| 324 | C4 | D4×D8 | 1 | 2 | yes | — |
| 325 | C2 | — | 2 | 2 | no | — |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 4 | 2 | no | — |
| 328 | C2 | — | 16 | 4 | no | — |
| 329 | C2 | #25779 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25779 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33721 | C2 | semidirect |
| 3. #33608 | C2 | semidirect |
| 4. #33532 | C2 | semidirect |
| 5. #25800 | C2 | semidirect |
| 6. #13335 | C2 | semidirect |
| 7. #13243 | C2 | semidirect |
| 8. #12613 | C2 | semidirect |
| 15. C2×#510 | C22 | semidirect |
| 18. C22×Q32 | C22 | semidirect |
| 20. #3156 | C22 | semidirect |
| 30. #2158 | C22 | semidirect |
| 31. #2145 | C22 | semidirect |
| 32. #2141 | C22 | semidirect |
| 34. #2108 | C22 | semidirect |
| 36. #1976 | C22 | semidirect |
| 38. #1378 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62635 | 2 |
| #33721 | 2 |
| #33608 | 2 |
| #33532 | 2 |
| #25800 | 2 |
| #13335 | 2 |
| #13243 | 2 |
| #12613 | 2 |