Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
745 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 144 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 | #62374 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #21734 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19350 | C2 | 1 | 2 | yes | maximal |
| 4 | #18703 | C2 | 1 | 2 | yes | maximal |
| 5 | #10534 | C2 | 1 | 2 | yes | maximal |
| 6 | #10303 | C2 | 1 | 2 | yes | maximal |
| 7 | #9850 | C2 | 1 | 2 | yes | maximal |
| 8 | #8637 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #8533 | C2 | 1 | 2 | yes | maximal |
| 10 | #3556 | C22 | 1 | 2 | yes | — |
| 11 | #3395 | — | 2 | 2 | no | — |
| 12 | #3383 | — | 2 | 2 | no | — |
| 13 | #3374 | — | 2 | 4 | no | — |
| 14 | #3261 | — | 2 | 2 | no | — |
| 15 | #3255 | C22 | 1 | 2 | yes | — |
| 16 | #3255 | C22 | 1 | 4 | yes | — |
| 17 | #3177 | — | 2 | 2 | no | — |
| 18 | #3170 | C22 | 1 | 2 | yes | — |
| 19 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1994 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1994 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1994 | C22 | 1 | 4 | yes | — |
| 25 | #1946 | — | 2 | 2 | no | — |
| 26 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1773 | — | 2 | 2 | no | — |
| 28 | C4×#97 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1694 | — | 2 | 2 | no | — |
| 30 | #1690 | — | 2 | 2 | no | — |
| 31 | #1680 | — | 2 | 4 | no | — |
| 32 | #1653 | — | 2 | 4 | no | — |
| 33 | #1648 | — | 2 | 4 | no | — |
| 34 | #1566 | C22 | 1 | 2 | yes | — |
| 35 | #1561 | C22 | 1 | 2 | yes | — |
| 36 | #1561 | C22 | 1 | 4 | yes | — |
| 37 | #1493 | C22 | 1 | 4 | yes | — |
| 38 | #1309 | C22 | 1 | 2 | yes | — |
| 39 | #564 | — | 2 | 4 | no | — |
| 40 | #557 | — | 2 | 4 | no | — |
| 41 | #557 | — | 2 | 4 | no | — |
| 42 | #557 | — | 2 | 4 | no | — |
| 43 | #541 | — | 2 | 4 | no | — |
| 44 | #541 | — | 4 | 4 | no | — |
| 45 | #533 | C23 | 1 | 4 | yes | — |
| 46 | #533 | C23 | 1 | 1 | yes | characteristic |
| 47 | #533 | C23 | 1 | 1 | yes | characteristic |
| 48 | #500 | D4 | 1 | 2 | yes | — |
| 49 | #500 | D4 | 1 | 2 | yes | — |
| 50 | #499 | — | 2 | 2 | no | — |
| 51 | #499 | — | 2 | 2 | no | — |
| 52 | #498 | D4 | 1 | 2 | yes | — |
| 53 | #494 | — | 2 | 2 | no | — |
| 54 | #494 | D4 | 1 | 2 | yes | — |
| 55 | #491 | — | 2 | 4 | no | — |
| 56 | #487 | D4 | 1 | 4 | yes | — |
| 57 | #479 | — | 2 | 8 | no | — |
| 58 | #475 | — | 2 | 4 | no | — |
| 59 | #475 | — | 2 | 4 | no | — |
| 60 | #475 | — | 2 | 4 | no | — |
| 61 | #470 | — | 2 | 4 | no | — |
| 62 | #470 | — | 4 | 4 | no | — |
| 63 | #467 | — | 2 | 2 | no | — |
| 64 | #467 | — | 2 | 2 | no | — |
| 65 | #462 | — | 2 | 8 | no | — |
| 66 | #462 | — | 2 | 4 | no | — |
| 67 | #462 | — | 2 | 4 | no | — |
| 68 | #462 | — | 2 | 4 | no | — |
| 69 | #446 | — | 2 | 2 | no | — |
| 70 | #446 | — | 2 | 2 | no | — |
| 71 | C8×Q8 | — | 2 | 2 | no | — |
| 72 | C8×Q8 | — | 2 | 2 | no | — |
| 73 | #441 | — | 2 | 4 | no | — |
| 74 | C4×Q16 | — | 2 | 4 | no | — |
| 75 | C4×Q16 | — | 2 | 2 | no | — |
| 76 | C4×Q16 | — | 2 | 2 | no | — |
| 77 | C2×#108 | — | 2 | 2 | no | — |
| 78 | #427 | — | 2 | 2 | no | — |
| 79 | #424 | — | 2 | 1 | no | — |
| 80 | #424 | D4 | 1 | 2 | yes | — |
| 81 | #423 | C23 | 1 | 1 | yes | characteristic |
| 82 | #423 | C23 | 1 | 4 | yes | — |
| 83 | #423 | C23 | 1 | 1 | yes | characteristic |
| 84 | #405 | — | 2 | 1 | no | — |
| 85 | #405 | D4 | 1 | 2 | yes | — |
| 86 | C2×#97 | — | 2 | 1 | no | — |
| 87 | C2×#97 | — | 2 | 1 | no | — |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 90 | C43 | C23 | 1 | 1 | yes | characteristic |
| 91 | #339 | — | 2 | 2 | no | — |
| 92 | #339 | — | 2 | 2 | no | — |
| 93 | #339 | — | 4 | 4 | no | — |
| 94 | #334 | — | 2 | 2 | no | — |
| 95 | #334 | — | 2 | 2 | no | — |
| 96 | #332 | — | 2 | 4 | no | — |
| 97 | #332 | — | 2 | 4 | no | — |
| 98 | #332 | — | 2 | 4 | no | — |
| 99 | Q8⋊C8 | — | 2 | 4 | no | — |
| 100 | Q8⋊C8 | — | 2 | 4 | no | — |
| 101 | C2×Q16 | — | 2 | 4 | no | — |
| 102 | C2×Q16 | — | 2 | 2 | no | — |
| 103 | C2×Q16 | — | 2 | 2 | no | — |
| 104 | C2×Q16 | — | 4 | 4 | no | — |
| 105 | C2×M16 | — | 2 | 1 | no | — |
| 106 | C2×M16 | — | 2 | 4 | no | — |
| 107 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 108 | C22×C8 | — | 2 | 1 | no | — |
| 109 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C4⋊Q8 | — | 2 | 4 | no | — |
| 112 | C4⋊Q8 | — | 4 | 4 | no | — |
| 113 | C4⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 114 | C4⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 115 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 116 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 117 | #126 | — | 4 | 8 | no | — |
| 118 | #125 | — | 2 | 4 | no | — |
| 119 | #125 | — | 2 | 4 | no | — |
| 120 | #125 | — | 2 | 1 | no | — |
| 121 | #125 | — | 4 | 4 | no | — |
| 122 | #125 | — | 4 | 8 | no | — |
| 123 | #125 | — | 4 | 8 | no | — |
| 124 | #125 | — | 4 | 8 | no | — |
| 125 | #125 | #41 | 1 | 2 | yes | — |
| 126 | C22⋊Q8 | — | 2 | 2 | no | — |
| 127 | C22⋊Q8 | — | 2 | 4 | no | — |
| 128 | C22⋊Q8 | — | 2 | 4 | no | — |
| 129 | C22⋊Q8 | — | 2 | 2 | no | — |
| 130 | C4×Q8 | — | 2 | 2 | no | — |
| 131 | C4×Q8 | — | 2 | 8 | no | — |
| 132 | C4×Q8 | — | 2 | 4 | no | — |
| 133 | C4×Q8 | — | 2 | 4 | no | — |
| 134 | C4×Q8 | — | 2 | 2 | no | — |
| 135 | C4×Q8 | — | 4 | 4 | no | — |
| 136 | C4×Q8 | — | 4 | 4 | no | — |
| 137 | #117 | — | 2 | 1 | no | — |
| 138 | #117 | — | 2 | 4 | no | — |
| 139 | #117 | C2×D4 | 1 | 2 | yes | — |
| 140 | C2×C42 | — | 2 | 1 | no | — |
| 141 | C2×C42 | — | 2 | 2 | no | — |
| 142 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 143 | #108 | — | 4 | 4 | no | — |
| 144 | #107 | — | 2 | 2 | no | — |
| 145 | #107 | — | 2 | 2 | no | — |
| 146 | #107 | — | 2 | 4 | no | — |
| 147 | #106 | — | 2 | 4 | no | — |
| 148 | #106 | — | 2 | 4 | no | — |
| 149 | #106 | — | 4 | 4 | no | — |
| 150 | C4⋊C8 | — | 2 | 4 | no | — |
| 151 | C4⋊C8 | — | 2 | 2 | no | — |
| 152 | C4⋊C8 | — | 2 | 4 | no | — |
| 153 | C4⋊C8 | — | 2 | 2 | no | — |
| 154 | C4⋊C8 | — | 2 | 4 | no | — |
| 155 | C4⋊C8 | — | 4 | 2 | no | — |
| 156 | #103 | — | 2 | 8 | no | — |
| 157 | #103 | — | 2 | 4 | no | — |
| 158 | #103 | — | 2 | 4 | no | — |
| 159 | #103 | — | 4 | 4 | no | — |
| 160 | #103 | — | 4 | 4 | no | — |
| 161 | #97 | — | 2 | 4 | no | — |
| 162 | #97 | — | 2 | 2 | no | — |
| 163 | #97 | — | 4 | 2 | no | — |
| 164 | #97 | — | 4 | 1 | no | — |
| 165 | C4×C8 | — | 2 | 2 | no | — |
| 166 | C4×C8 | — | 2 | 1 | no | — |
| 167 | C4×C8 | — | 4 | 1 | no | — |
| 168 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 171 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 172 | C2×Q8 | — | 2 | 2 | no | — |
| 173 | C2×Q8 | — | 2 | 4 | no | — |
| 174 | C2×Q8 | — | 2 | 2 | no | — |
| 175 | C2×Q8 | — | 2 | 4 | no | — |
| 176 | C22×C4 | — | 4 | 1 | no | — |
| 177 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 178 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 179 | Q16 | — | 4 | 4 | no | — |
| 180 | Q16 | — | 8 | 2 | no | — |
| 181 | Q16 | — | 8 | 4 | no | — |
| 182 | Q16 | — | 16 | 4 | no | — |
| 183 | M16 | — | 4 | 2 | no | — |
| 184 | M16 | — | 4 | 2 | no | — |
| 185 | M16 | — | 8 | 4 | no | — |
| 186 | C2×C8 | — | 2 | 2 | no | — |
| 187 | C2×C8 | — | 2 | 2 | no | — |
| 188 | C2×C8 | — | 2 | 1 | no | — |
| 189 | C2×C8 | — | 2 | 1 | no | — |
| 190 | C2×C8 | — | 2 | 2 | no | — |
| 191 | C2×C8 | — | 2 | 2 | no | — |
| 192 | C2×C8 | — | 4 | 1 | no | — |
| 193 | C2×C8 | — | 4 | 1 | no | — |
| 194 | C2×C8 | — | 4 | 4 | no | — |
| 195 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 196 | C2×C8 | #120 | 1 | 2 | yes | — |
| 197 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 198 | C4⋊C4 | — | 2 | 2 | no | — |
| 199 | C4⋊C4 | — | 2 | 4 | no | — |
| 200 | C4⋊C4 | — | 2 | 4 | no | — |
| 201 | C4⋊C4 | — | 2 | 2 | no | — |
| 202 | C4⋊C4 | — | 2 | 2 | no | — |
| 203 | C4⋊C4 | — | 2 | 4 | no | — |
| 204 | C4⋊C4 | — | 2 | 4 | no | — |
| 205 | C4⋊C4 | — | 2 | 4 | no | — |
| 206 | C4⋊C4 | — | 4 | 8 | no | — |
| 207 | C4⋊C4 | — | 4 | 8 | no | — |
| 208 | C4⋊C4 | — | 4 | 8 | no | — |
| 209 | C4⋊C4 | — | 4 | 4 | no | — |
| 210 | C4⋊C4 | — | 4 | 2 | no | — |
| 211 | C4⋊C4 | — | 4 | 4 | no | — |
| 212 | C4⋊C4 | #121 | 1 | 4 | yes | — |
| 213 | C22⋊C4 | — | 2 | 2 | no | — |
| 214 | C22⋊C4 | — | 4 | 1 | no | — |
| 215 | C22⋊C4 | — | 4 | 4 | no | — |
| 216 | C42 | — | 2 | 2 | no | — |
| 217 | C42 | — | 2 | 2 | no | — |
| 218 | C42 | — | 4 | 1 | no | — |
| 219 | C42 | — | 4 | 8 | no | — |
| 220 | C42 | — | 4 | 4 | no | — |
| 221 | C42 | — | 4 | 1 | no | — |
| 222 | C42 | — | 4 | 4 | no | — |
| 223 | C42 | C22×D4 | 1 | 2 | yes | — |
| 224 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 225 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 226 | C23 | #535 | 1 | 1 | yes | characteristic |
| 227 | Q8 | — | 4 | 4 | no | — |
| 228 | Q8 | — | 4 | 4 | no | — |
| 229 | Q8 | — | 4 | 4 | no | — |
| 230 | Q8 | — | 4 | 4 | no | — |
| 231 | Q8 | — | 4 | 4 | no | — |
| 232 | Q8 | — | 4 | 4 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 2 | 4 | no | — |
| 236 | C2×C4 | — | 2 | 4 | no | — |
| 237 | C2×C4 | — | 4 | 2 | no | — |
| 238 | C2×C4 | — | 4 | 4 | no | — |
| 239 | C2×C4 | — | 4 | 4 | no | — |
| 240 | C2×C4 | — | 4 | 2 | no | — |
| 241 | C2×C4 | — | 8 | 2 | no | — |
| 242 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | D42 | 1 | 2 | yes | — |
| 245 | C2×C4 | #546 | 1 | 2 | yes | — |
| 246 | C8 | — | 2 | 2 | no | — |
| 247 | C8 | — | 2 | 2 | no | — |
| 248 | C8 | — | 2 | 1 | no | — |
| 249 | C8 | — | 2 | 2 | no | — |
| 250 | C8 | — | 4 | 2 | no | — |
| 251 | C8 | — | 8 | 1 | no | — |
| 252 | C8 | — | 8 | 4 | no | — |
| 253 | C8 | #456 | 1 | 2 | yes | — |
| 254 | C22 | — | 2 | 1 | no | — |
| 255 | C22 | — | 2 | 2 | no | — |
| 256 | C22 | #2408 | 1 | 1 | yes | characteristic |
| 257 | C4 | — | 2 | 1 | no | — |
| 258 | C4 | — | 2 | 2 | no | — |
| 259 | C4 | — | 4 | 4 | no | — |
| 260 | C4 | — | 8 | 2 | no | — |
| 261 | C4 | — | 8 | 4 | no | — |
| 262 | C4 | — | 8 | 4 | no | — |
| 263 | C4 | — | 8 | 2 | no | — |
| 264 | C4 | C2×#456 | 1 | 1 | yes | characteristic |
| 265 | C4 | #3018 | 1 | 1 | yes | characteristic |
| 266 | C4 | #3289 | 1 | 4 | yes | — |
| 267 | C2 | — | 4 | 1 | no | — |
| 268 | C2 | #24225 | 1 | 1 | yes | characteristic |
| 269 | C2 | #24293 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #535 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62374 | 2 |
| #21734 | 2 |
| #19350 | 2 |
| #18703 | 2 |
| #10534 | 2 |
| #10303 | 2 |
| #9850 | 2 |
| #8637 | 2 |
| #8533 | 2 |