Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
378 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 74 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 | #33981 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2263 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33720 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13622 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13617 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13243 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7469 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#509 | — | 2 | 1 | no | — |
| 9 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 12 | #3155 | — | 2 | 1 | no | — |
| 13 | #2264 | — | 2 | 1 | no | — |
| 14 | #2264 | — | 2 | 1 | no | — |
| 15 | #2263 | — | 2 | 2 | no | — |
| 16 | #2263 | — | 2 | 1 | no | — |
| 17 | #2263 | — | 2 | 1 | no | — |
| 18 | #2263 | — | 2 | 2 | no | — |
| 19 | #2262 | — | 2 | 2 | no | — |
| 20 | C2×SD64 | — | 2 | 1 | no | — |
| 21 | C2×D32 | — | 2 | 1 | no | — |
| 22 | C2×M64 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×M64 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2250 | — | 2 | 2 | no | — |
| 25 | #2249 | — | 2 | 1 | no | — |
| 26 | #2246 | — | 2 | 1 | no | — |
| 27 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2145 | — | 2 | 1 | no | — |
| 29 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1976 | — | 2 | 1 | no | — |
| 31 | #1418 | — | 2 | 2 | no | — |
| 32 | #1415 | — | 2 | 2 | no | — |
| 33 | C2×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 34 | C2×#135 | — | 2 | 1 | no | — |
| 35 | C2×#135 | — | 2 | 1 | no | — |
| 36 | C22×D8 | — | 2 | 1 | no | — |
| 37 | #535 | — | 4 | 1 | no | — |
| 38 | #509 | — | 4 | 2 | no | — |
| 39 | #509 | — | 4 | 2 | no | — |
| 40 | #508 | — | 2 | 1 | no | — |
| 41 | #508 | — | 2 | 1 | no | — |
| 42 | #508 | — | 2 | 1 | no | — |
| 43 | #508 | — | 2 | 1 | no | — |
| 44 | #508 | — | 2 | 1 | no | — |
| 45 | #508 | — | 2 | 1 | no | — |
| 46 | #508 | — | 2 | 1 | no | — |
| 47 | #508 | — | 2 | 1 | no | — |
| 48 | C2×Q32 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×Q32 | D4 | 1 | 1 | yes | characteristic, complemented |
| 50 | C2×SD32 | — | 2 | 2 | no | — |
| 51 | C2×SD32 | — | 2 | 1 | no | — |
| 52 | C2×SD32 | — | 2 | 1 | no | — |
| 53 | C2×D16 | — | 2 | 2 | no | — |
| 54 | C2×D16 | — | 2 | 1 | no | — |
| 55 | C2×D16 | — | 2 | 1 | no | — |
| 56 | C2×D16 | — | 2 | 1 | no | — |
| 57 | C2×D16 | — | 2 | 1 | no | — |
| 58 | C2×D16 | — | 2 | 1 | no | — |
| 59 | C2×D16 | D4 | 1 | 2 | yes | — |
| 60 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×D16 | D4 | 1 | 1 | yes | characteristic, complemented |
| 62 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 63 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 64 | #496 | — | 4 | 2 | no | — |
| 65 | #495 | — | 2 | 2 | no | — |
| 66 | #493 | — | 2 | 2 | no | — |
| 67 | #427 | — | 2 | 1 | no | — |
| 68 | #421 | — | 4 | 1 | no | — |
| 69 | #410 | — | 4 | 1 | no | — |
| 70 | #405 | D4 | 1 | 1 | yes | characteristic |
| 71 | Q64 | — | 2 | 2 | no | — |
| 72 | SD64 | — | 2 | 4 | no | — |
| 73 | SD64 | — | 2 | 2 | no | — |
| 74 | SD64 | — | 2 | 2 | no | — |
| 75 | D32 | — | 2 | 4 | no | — |
| 76 | D32 | — | 2 | 2 | no | — |
| 77 | M64 | — | 2 | 1 | no | — |
| 78 | M64 | — | 2 | 1 | no | — |
| 79 | M64 | — | 2 | 1 | no | — |
| 80 | M64 | D4 | 1 | 2 | yes | complemented |
| 81 | C2×C32 | — | 2 | 1 | no | — |
| 82 | C2×C32 | D4 | 1 | 2 | yes | complemented |
| 83 | #367 | — | 2 | 1 | no | — |
| 84 | #366 | — | 2 | 1 | no | — |
| 85 | #360 | — | 4 | 2 | no | — |
| 86 | #346 | — | 2 | 1 | no | — |
| 87 | C4×C16 | — | 2 | 1 | no | — |
| 88 | C2×#41 | — | 4 | 1 | no | — |
| 89 | C2×#41 | — | 4 | 1 | no | — |
| 90 | C22×D4 | — | 4 | 1 | no | — |
| 91 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 92 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 93 | #135 | — | 4 | 1 | no | — |
| 94 | #135 | — | 4 | 1 | no | — |
| 95 | #135 | — | 4 | 1 | no | — |
| 96 | #135 | — | 4 | 1 | no | — |
| 97 | #135 | — | 4 | 1 | no | — |
| 98 | #135 | — | 4 | 1 | no | — |
| 99 | #135 | — | 4 | 1 | no | — |
| 100 | #135 | — | 4 | 1 | no | — |
| 101 | C2×Q16 | — | 2 | 1 | no | — |
| 102 | C2×Q16 | — | 2 | 1 | no | — |
| 103 | C2×SD16 | — | 4 | 2 | no | — |
| 104 | C2×SD16 | — | 4 | 1 | no | — |
| 105 | C2×SD16 | — | 4 | 1 | no | — |
| 106 | C2×D8 | — | 2 | 2 | no | — |
| 107 | C2×D8 | — | 2 | 1 | no | — |
| 108 | C2×D8 | — | 2 | 1 | no | — |
| 109 | C2×D8 | — | 4 | 2 | no | — |
| 110 | C2×D8 | — | 4 | 1 | no | — |
| 111 | C2×D8 | — | 4 | 1 | no | — |
| 112 | C2×D8 | — | 4 | 1 | no | — |
| 113 | C2×D8 | — | 4 | 1 | no | — |
| 114 | C2×D8 | — | 4 | 1 | no | — |
| 115 | C2×M16 | D8 | 1 | 1 | yes | characteristic |
| 116 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | #127 | — | 4 | 2 | no | — |
| 118 | #124 | — | 4 | 2 | no | — |
| 119 | #121 | — | 8 | 2 | no | — |
| 120 | #120 | — | 8 | 2 | no | — |
| 121 | #117 | — | 4 | 1 | no | — |
| 122 | #117 | D8 | 1 | 1 | yes | characteristic |
| 123 | Q32 | — | 2 | 1 | no | — |
| 124 | Q32 | — | 2 | 1 | no | — |
| 125 | Q32 | — | 2 | 1 | no | — |
| 126 | Q32 | — | 2 | 1 | no | — |
| 127 | SD32 | — | 4 | 4 | no | — |
| 128 | SD32 | — | 4 | 1 | no | — |
| 129 | SD32 | — | 4 | 1 | no | — |
| 130 | SD32 | — | 4 | 1 | no | — |
| 131 | SD32 | — | 4 | 1 | no | — |
| 132 | D16 | — | 2 | 1 | no | — |
| 133 | D16 | — | 2 | 2 | no | — |
| 134 | D16 | — | 2 | 1 | no | — |
| 135 | D16 | — | 2 | 1 | no | — |
| 136 | D16 | — | 2 | 2 | no | — |
| 137 | D16 | — | 2 | 1 | no | — |
| 138 | D16 | — | 4 | 4 | no | — |
| 139 | D16 | — | 4 | 1 | no | — |
| 140 | D16 | — | 4 | 1 | no | — |
| 141 | M32 | — | 2 | 1 | no | — |
| 142 | M32 | D8 | 1 | 2 | yes | — |
| 143 | C2×C16 | — | 2 | 1 | no | — |
| 144 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 145 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 147 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 148 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 150 | #107 | — | 4 | 1 | no | — |
| 151 | #106 | — | 4 | 1 | no | — |
| 152 | #104 | — | 8 | 2 | no | — |
| 153 | #99 | — | 8 | 2 | no | — |
| 154 | #97 | — | 2 | 1 | no | — |
| 155 | C4×C8 | — | 2 | 1 | no | — |
| 156 | C32 | — | 2 | 2 | no | — |
| 157 | C32 | — | 2 | 2 | no | — |
| 158 | C24 | — | 8 | 1 | no | — |
| 159 | #41 | — | 8 | 1 | no | — |
| 160 | #41 | — | 8 | 1 | no | — |
| 161 | #41 | — | 8 | 1 | no | — |
| 162 | #41 | — | 8 | 1 | no | — |
| 163 | #41 | — | 8 | 1 | no | — |
| 164 | #41 | — | 8 | 1 | no | — |
| 165 | #41 | — | 8 | 1 | no | — |
| 166 | #41 | — | 8 | 1 | no | — |
| 167 | C2×Q8 | — | 4 | 1 | no | — |
| 168 | C2×Q8 | — | 4 | 1 | no | — |
| 169 | C2×D4 | — | 4 | 2 | no | — |
| 170 | C2×D4 | — | 4 | 1 | no | — |
| 171 | C2×D4 | — | 4 | 1 | no | — |
| 172 | C2×D4 | — | 8 | 4 | no | — |
| 173 | C2×D4 | — | 8 | 1 | no | — |
| 174 | C2×D4 | — | 8 | 1 | no | — |
| 175 | C2×D4 | — | 8 | 1 | no | — |
| 176 | C2×D4 | — | 8 | 1 | no | — |
| 177 | C2×D4 | — | 8 | 1 | no | — |
| 178 | C2×D4 | — | 8 | 1 | no | — |
| 179 | C2×D4 | — | 8 | 1 | no | — |
| 180 | C22×C4 | — | 8 | 1 | no | — |
| 181 | C22×C4 | — | 8 | 1 | no | — |
| 182 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 183 | Q16 | — | 4 | 1 | no | — |
| 184 | Q16 | — | 4 | 1 | no | — |
| 185 | Q16 | — | 4 | 1 | no | — |
| 186 | Q16 | — | 4 | 1 | no | — |
| 187 | SD16 | — | 8 | 4 | no | — |
| 188 | SD16 | — | 8 | 1 | no | — |
| 189 | SD16 | — | 8 | 1 | no | — |
| 190 | SD16 | — | 8 | 1 | no | — |
| 191 | SD16 | — | 8 | 1 | no | — |
| 192 | D8 | — | 4 | 2 | no | — |
| 193 | D8 | — | 4 | 1 | no | — |
| 194 | D8 | — | 4 | 1 | no | — |
| 195 | D8 | — | 4 | 1 | no | — |
| 196 | D8 | — | 4 | 2 | no | — |
| 197 | D8 | — | 4 | 1 | no | — |
| 198 | D8 | — | 8 | 4 | no | — |
| 199 | D8 | — | 8 | 1 | no | — |
| 200 | D8 | — | 8 | 1 | no | — |
| 201 | M16 | — | 2 | 1 | no | — |
| 202 | M16 | D16 | 1 | 2 | yes | — |
| 203 | C2×C8 | — | 2 | 1 | no | — |
| 204 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 207 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 209 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 210 | C4⋊C4 | — | 2 | 1 | no | — |
| 211 | C4⋊C4 | — | 8 | 1 | no | — |
| 212 | C22⋊C4 | — | 8 | 4 | no | — |
| 213 | C22⋊C4 | — | 8 | 1 | no | — |
| 214 | C22⋊C4 | D16 | 1 | 2 | yes | — |
| 215 | C42 | — | 2 | 1 | no | — |
| 216 | C42 | — | 8 | 1 | no | — |
| 217 | C16 | — | 2 | 2 | no | — |
| 218 | C16 | #120 | 1 | 1 | yes | characteristic |
| 219 | C16 | #120 | 1 | 1 | yes | characteristic |
| 220 | C16 | #121 | 1 | 1 | yes | characteristic |
| 221 | C16 | #121 | 1 | 1 | yes | characteristic |
| 222 | C23 | — | 8 | 2 | no | — |
| 223 | C23 | — | 8 | 1 | no | — |
| 224 | C23 | — | 8 | 1 | no | — |
| 225 | C23 | — | 16 | 1 | no | — |
| 226 | C23 | — | 16 | 1 | no | — |
| 227 | C23 | — | 32 | 2 | no | — |
| 228 | C23 | C2×D16 | 1 | 1 | yes | characteristic |
| 229 | Q8 | — | 8 | 1 | no | — |
| 230 | Q8 | — | 8 | 1 | no | — |
| 231 | Q8 | — | 8 | 1 | no | — |
| 232 | Q8 | — | 8 | 1 | no | — |
| 233 | D4 | — | 8 | 1 | no | — |
| 234 | D4 | — | 8 | 2 | no | — |
| 235 | D4 | — | 8 | 1 | no | — |
| 236 | D4 | — | 8 | 2 | no | — |
| 237 | D4 | — | 8 | 1 | no | — |
| 238 | D4 | — | 8 | 1 | no | — |
| 239 | D4 | — | 16 | 1 | no | — |
| 240 | D4 | — | 16 | 1 | no | — |
| 241 | D4 | — | 16 | 1 | no | — |
| 242 | D4 | — | 16 | 1 | no | — |
| 243 | D4 | — | 16 | 1 | no | — |
| 244 | D4 | — | 16 | 1 | no | — |
| 245 | D4 | — | 32 | 4 | no | — |
| 246 | C2×C4 | — | 2 | 2 | no | — |
| 247 | C2×C4 | — | 8 | 1 | no | — |
| 248 | C2×C4 | — | 8 | 1 | no | — |
| 249 | C2×C4 | — | 16 | 1 | no | — |
| 250 | C2×C4 | — | 16 | 1 | no | — |
| 251 | C2×C4 | — | 16 | 1 | no | — |
| 252 | C2×C4 | — | 16 | 1 | no | — |
| 253 | C2×C4 | — | 16 | 1 | no | — |
| 254 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #509 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | #509 | 1 | 1 | yes | characteristic |
| 260 | C8 | — | 2 | 2 | no | — |
| 261 | C8 | #392 | 1 | 1 | yes | characteristic |
| 262 | C8 | #447 | 1 | 1 | yes | characteristic |
| 263 | C8 | #459 | 1 | 1 | yes | characteristic |
| 264 | C8 | #496 | 1 | 1 | yes | characteristic |
| 265 | C22 | — | 4 | 1 | no | — |
| 266 | C22 | — | 16 | 2 | no | — |
| 267 | C22 | — | 16 | 4 | no | — |
| 268 | C22 | — | 16 | 1 | no | — |
| 269 | C22 | — | 16 | 1 | no | — |
| 270 | C22 | — | 16 | 2 | no | — |
| 271 | C22 | — | 32 | 1 | no | — |
| 272 | C22 | — | 32 | 2 | no | — |
| 273 | C22 | — | 32 | 2 | no | — |
| 274 | C22 | #2184 | 1 | 1 | yes | characteristic |
| 275 | C22 | #2206 | 1 | 1 | yes | characteristic |
| 276 | C22 | #2246 | 1 | 1 | yes | characteristic |
| 277 | C4 | — | 4 | 2 | no | — |
| 278 | C4 | — | 16 | 1 | no | — |
| 279 | C4 | — | 16 | 1 | no | — |
| 280 | C4 | — | 32 | 1 | no | — |
| 281 | C4 | #2011 | 1 | 1 | yes | characteristic |
| 282 | C4 | #2184 | 1 | 1 | yes | characteristic |
| 283 | C4 | #2206 | 1 | 1 | yes | characteristic |
| 284 | C4 | #2250 | 1 | 1 | yes | characteristic |
| 285 | C2 | — | 2 | 1 | no | — |
| 286 | C2 | — | 2 | 1 | no | — |
| 287 | C2 | — | 2 | 1 | no | — |
| 288 | C2 | — | 32 | 1 | no | — |
| 289 | C2 | — | 32 | 2 | no | — |
| 290 | C2 | — | 32 | 1 | no | — |
| 291 | C2 | #13325 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13325 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33981 | C2 | semidirect |
| 2. C2×#2263 | C2 | semidirect |
| 3. #33720 | C2 | semidirect |
| 4. #13622 | C2 | semidirect |
| 5. #13617 | C2 | semidirect |
| 6. #13243 | C2 | semidirect |
| 7. #7469 | C2 | semidirect |
| 49. C2×Q32 | D4 | semidirect |
| 61. C2×D16 | D4 | semidirect |
| 80. M64 | D4 | semidirect |
| 82. C2×C32 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33981 | 2 |
| C2×#2263 | 2 |
| #33720 | 2 |
| #13622 | 2 |
| #13617 | 2 |
| #13243 | 2 |
| #7469 | 2 |