Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
337 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #33795 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33731 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19690 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11968 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11681 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9810 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #7166 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #3231 | — | 2 | 1 | no | — |
| 9 | #3185 | — | 2 | 1 | no | — |
| 10 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2255 | — | 2 | 2 | no | — |
| 14 | #2254 | — | 2 | 1 | no | — |
| 15 | #2254 | — | 2 | 1 | no | — |
| 16 | #2252 | — | 2 | 1 | no | — |
| 17 | #2241 | — | 2 | 1 | no | — |
| 18 | #2235 | — | 2 | 2 | no | — |
| 19 | #2225 | — | 2 | 2 | no | — |
| 20 | #2162 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2159 | — | 2 | 2 | no | — |
| 22 | C4×M32 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1997 | — | 2 | 1 | no | — |
| 24 | #1723 | — | 2 | 1 | no | — |
| 25 | #1723 | — | 2 | 1 | no | — |
| 26 | #1718 | — | 2 | 1 | no | — |
| 27 | #1716 | — | 2 | 1 | no | — |
| 28 | #1566 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1549 | — | 2 | 1 | no | — |
| 30 | #1451 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1363 | — | 2 | 2 | no | — |
| 32 | #1334 | — | 2 | 2 | no | — |
| 33 | #533 | — | 2 | 1 | no | — |
| 34 | #533 | — | 2 | 1 | no | — |
| 35 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 36 | C2×M32 | — | 2 | 1 | no | — |
| 37 | #500 | — | 2 | 1 | no | — |
| 38 | #500 | — | 2 | 1 | no | — |
| 39 | #500 | — | 2 | 1 | no | — |
| 40 | #500 | — | 2 | 1 | no | — |
| 41 | #500 | — | 2 | 1 | no | — |
| 42 | #500 | D4 | 1 | 1 | yes | characteristic |
| 43 | #500 | D4 | 1 | 1 | yes | characteristic |
| 44 | #500 | D4 | 1 | 1 | yes | characteristic |
| 45 | #499 | — | 2 | 1 | no | — |
| 46 | #499 | — | 2 | 1 | no | — |
| 47 | #498 | — | 2 | 1 | no | — |
| 48 | #498 | — | 2 | 1 | no | — |
| 49 | #498 | — | 2 | 1 | no | — |
| 50 | #494 | — | 2 | 1 | no | — |
| 51 | #494 | D4 | 1 | 1 | yes | characteristic |
| 52 | #487 | — | 2 | 1 | no | — |
| 53 | #487 | — | 2 | 1 | no | — |
| 54 | #487 | — | 2 | 1 | no | — |
| 55 | #487 | — | 2 | 1 | no | — |
| 56 | #487 | — | 2 | 1 | no | — |
| 57 | #485 | — | 4 | 2 | no | — |
| 58 | #479 | — | 4 | 2 | no | — |
| 59 | #470 | — | 4 | 2 | no | — |
| 60 | #467 | — | 2 | 1 | no | — |
| 61 | #467 | — | 2 | 1 | no | — |
| 62 | #441 | — | 4 | 2 | no | — |
| 63 | C4×Q16 | — | 2 | 1 | no | — |
| 64 | C4×Q16 | — | 2 | 1 | no | — |
| 65 | #431 | D4 | 1 | 1 | yes | characteristic |
| 66 | #427 | — | 2 | 1 | no | — |
| 67 | #427 | — | 2 | 1 | no | — |
| 68 | C2×#107 | — | 2 | 1 | no | — |
| 69 | C2×#107 | D4 | 1 | 2 | yes | — |
| 70 | #423 | — | 2 | 1 | no | — |
| 71 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 73 | #367 | — | 2 | 4 | no | — |
| 74 | #367 | — | 2 | 4 | no | — |
| 75 | #366 | — | 2 | 4 | no | — |
| 76 | C4⋊C16 | D4 | 1 | 2 | yes | — |
| 77 | #358 | — | 2 | 4 | no | — |
| 78 | C22⋊C16 | — | 2 | 1 | no | — |
| 79 | #346 | D4 | 1 | 2 | yes | — |
| 80 | C4×C16 | — | 2 | 1 | no | — |
| 81 | C4×C16 | — | 2 | 1 | no | — |
| 82 | #343 | — | 4 | 2 | no | — |
| 83 | #334 | — | 2 | 1 | no | — |
| 84 | #334 | — | 2 | 1 | no | — |
| 85 | #332 | — | 4 | 2 | no | — |
| 86 | #322 | — | 2 | 1 | no | — |
| 87 | #322 | — | 2 | 1 | no | — |
| 88 | C22×Q8 | — | 4 | 1 | no | — |
| 89 | C2×Q16 | — | 2 | 1 | no | — |
| 90 | C2×Q16 | — | 2 | 1 | no | — |
| 91 | C2×Q16 | — | 4 | 2 | no | — |
| 92 | C2×M16 | — | 4 | 1 | no | — |
| 93 | C2×M16 | D8 | 1 | 2 | yes | — |
| 94 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 96 | C4⋊Q8 | — | 2 | 1 | no | — |
| 97 | C4⋊Q8 | — | 2 | 1 | no | — |
| 98 | C4⋊Q8 | — | 2 | 1 | no | — |
| 99 | C4⋊Q8 | — | 2 | 1 | no | — |
| 100 | C4⋊Q8 | — | 4 | 1 | no | — |
| 101 | C4⋊Q8 | — | 4 | 1 | no | — |
| 102 | C4⋊Q8 | — | 4 | 1 | no | — |
| 103 | C4⋊Q8 | — | 4 | 1 | no | — |
| 104 | #125 | — | 4 | 1 | no | — |
| 105 | #125 | — | 4 | 1 | no | — |
| 106 | C22⋊Q8 | — | 4 | 1 | no | — |
| 107 | C22⋊Q8 | — | 4 | 1 | no | — |
| 108 | C22⋊Q8 | — | 4 | 1 | no | — |
| 109 | C22⋊Q8 | — | 4 | 1 | no | — |
| 110 | C4×Q8 | — | 4 | 1 | no | — |
| 111 | C4×Q8 | — | 4 | 1 | no | — |
| 112 | C4×Q8 | — | 4 | 1 | no | — |
| 113 | C4×Q8 | — | 4 | 1 | no | — |
| 114 | #117 | — | 4 | 1 | no | — |
| 115 | #117 | — | 4 | 1 | no | — |
| 116 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 117 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 118 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | M32 | — | 2 | 4 | no | — |
| 120 | C2×C16 | — | 2 | 2 | no | — |
| 121 | C2×C16 | — | 2 | 2 | no | — |
| 122 | #107 | — | 2 | 2 | no | — |
| 123 | #107 | — | 2 | 1 | no | — |
| 124 | #107 | — | 2 | 2 | no | — |
| 125 | #107 | — | 2 | 1 | no | — |
| 126 | #107 | — | 2 | 1 | no | — |
| 127 | #107 | — | 2 | 1 | no | — |
| 128 | #107 | — | 2 | 1 | no | — |
| 129 | #107 | — | 2 | 1 | no | — |
| 130 | #107 | — | 4 | 2 | no | — |
| 131 | #107 | — | 4 | 1 | no | — |
| 132 | #107 | — | 4 | 1 | no | — |
| 133 | #106 | — | 4 | 2 | no | — |
| 134 | #106 | — | 4 | 2 | no | — |
| 135 | #106 | — | 4 | 1 | no | — |
| 136 | #106 | — | 4 | 1 | no | — |
| 137 | C4⋊C8 | — | 4 | 1 | no | — |
| 138 | C4⋊C8 | — | 4 | 1 | no | — |
| 139 | C4⋊C8 | D8 | 1 | 2 | yes | — |
| 140 | #103 | — | 4 | 2 | no | — |
| 141 | #103 | — | 4 | 2 | no | — |
| 142 | #103 | — | 4 | 2 | no | — |
| 143 | #103 | — | 4 | 2 | no | — |
| 144 | #103 | — | 4 | 1 | no | — |
| 145 | #103 | — | 4 | 1 | no | — |
| 146 | C22⋊C8 | — | 2 | 1 | no | — |
| 147 | #97 | — | 2 | 1 | no | — |
| 148 | C4×C8 | — | 2 | 1 | no | — |
| 149 | C4×C8 | — | 2 | 1 | no | — |
| 150 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 151 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 152 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 153 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×Q8 | — | 4 | 1 | no | — |
| 155 | C2×Q8 | — | 4 | 1 | no | — |
| 156 | C2×Q8 | — | 4 | 1 | no | — |
| 157 | C2×Q8 | — | 4 | 1 | no | — |
| 158 | C2×Q8 | — | 8 | 1 | no | — |
| 159 | C2×Q8 | — | 8 | 1 | no | — |
| 160 | C2×Q8 | — | 16 | 2 | no | — |
| 161 | C22×C4 | — | 4 | 2 | no | — |
| 162 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 163 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 165 | Q16 | — | 4 | 2 | no | — |
| 166 | Q16 | — | 4 | 2 | no | — |
| 167 | Q16 | — | 8 | 4 | no | — |
| 168 | M16 | — | 4 | 2 | no | — |
| 169 | M16 | — | 8 | 2 | no | — |
| 170 | C2×C8 | — | 2 | 2 | no | — |
| 171 | C2×C8 | — | 2 | 2 | no | — |
| 172 | C2×C8 | — | 2 | 2 | no | — |
| 173 | C2×C8 | — | 2 | 2 | no | — |
| 174 | C2×C8 | — | 8 | 1 | no | — |
| 175 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 176 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 177 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 178 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 179 | C4⋊C4 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 4 | 1 | no | — |
| 181 | C4⋊C4 | — | 4 | 1 | no | — |
| 182 | C4⋊C4 | — | 4 | 2 | no | — |
| 183 | C4⋊C4 | — | 4 | 1 | no | — |
| 184 | C4⋊C4 | — | 4 | 1 | no | — |
| 185 | C4⋊C4 | — | 4 | 1 | no | — |
| 186 | C4⋊C4 | — | 4 | 1 | no | — |
| 187 | C4⋊C4 | — | 4 | 1 | no | — |
| 188 | C4⋊C4 | — | 4 | 2 | no | — |
| 189 | C4⋊C4 | — | 8 | 1 | no | — |
| 190 | C4⋊C4 | — | 8 | 1 | no | — |
| 191 | C4⋊C4 | — | 8 | 1 | no | — |
| 192 | C4⋊C4 | — | 8 | 1 | no | — |
| 193 | C4⋊C4 | — | 8 | 1 | no | — |
| 194 | C4⋊C4 | — | 8 | 1 | no | — |
| 195 | C22⋊C4 | — | 8 | 1 | no | — |
| 196 | C22⋊C4 | — | 8 | 1 | no | — |
| 197 | C42 | — | 8 | 1 | no | — |
| 198 | C42 | — | 8 | 1 | no | — |
| 199 | C42 | #120 | 1 | 1 | yes | characteristic |
| 200 | C42 | #127 | 1 | 1 | yes | characteristic |
| 201 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 202 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 203 | C16 | — | 2 | 4 | no | — |
| 204 | C16 | — | 4 | 2 | no | — |
| 205 | C23 | #459 | 1 | 1 | yes | characteristic |
| 206 | Q8 | — | 8 | 2 | no | — |
| 207 | Q8 | — | 8 | 2 | no | — |
| 208 | Q8 | — | 8 | 2 | no | — |
| 209 | Q8 | — | 8 | 1 | no | — |
| 210 | Q8 | — | 8 | 1 | no | — |
| 211 | Q8 | — | 16 | 2 | no | — |
| 212 | Q8 | — | 16 | 2 | no | — |
| 213 | C2×C4 | — | 2 | 2 | no | — |
| 214 | C2×C4 | — | 2 | 2 | no | — |
| 215 | C2×C4 | — | 2 | 1 | no | — |
| 216 | C2×C4 | — | 2 | 1 | no | — |
| 217 | C2×C4 | — | 8 | 1 | no | — |
| 218 | C2×C4 | — | 8 | 1 | no | — |
| 219 | C2×C4 | — | 8 | 1 | no | — |
| 220 | C2×C4 | — | 8 | 1 | no | — |
| 221 | C2×C4 | — | 8 | 2 | no | — |
| 222 | C2×C4 | — | 16 | 2 | no | — |
| 223 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #496 | 1 | 1 | yes | characteristic |
| 229 | C8 | — | 2 | 1 | no | — |
| 230 | C8 | — | 2 | 1 | no | — |
| 231 | C8 | — | 4 | 2 | no | — |
| 232 | C8 | — | 4 | 2 | no | — |
| 233 | C8 | — | 16 | 1 | no | — |
| 234 | C8 | #456 | 1 | 2 | yes | — |
| 235 | C8 | #473 | 1 | 2 | yes | — |
| 236 | C22 | — | 2 | 1 | no | — |
| 237 | C22 | — | 2 | 1 | no | — |
| 238 | C22 | — | 2 | 1 | no | — |
| 239 | C22 | #2011 | 1 | 1 | yes | characteristic |
| 240 | C4 | — | 2 | 1 | no | — |
| 241 | C4 | — | 2 | 1 | no | — |
| 242 | C4 | — | 2 | 1 | no | — |
| 243 | C4 | — | 16 | 2 | no | — |
| 244 | C4 | — | 16 | 1 | no | — |
| 245 | C4 | — | 16 | 1 | no | — |
| 246 | C4 | — | 16 | 1 | no | — |
| 247 | C4 | — | 16 | 1 | no | — |
| 248 | C4 | #2005 | 1 | 1 | yes | characteristic |
| 249 | C4 | #2020 | 1 | 1 | yes | characteristic |
| 250 | C4 | #2191 | 1 | 2 | yes | — |
| 251 | C4 | #2193 | 1 | 2 | yes | — |
| 252 | C2 | — | 4 | 1 | no | — |
| 253 | C2 | #10174 | 1 | 1 | yes | characteristic |
| 254 | C2 | #13337 | 1 | 1 | yes | characteristic |
| 255 | C2 | #13342 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #459 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33795 | 2 |
| #33731 | 2 |
| #19690 | 2 |
| #11968 | 2 |
| #11681 | 2 |
| #9810 | 2 |
| #7166 | 2 |