Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
540 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 186 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 | #28836 | C2 | 1 | 2 | yes | maximal |
| 2 | #28817 | C2 | 1 | 2 | yes | maximal |
| 3 | #10580 | C2 | 1 | 2 | yes | maximal |
| 4 | #10577 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10574 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9618 | C2 | 1 | 2 | yes | maximal |
| 7 | #9617 | C2 | 1 | 2 | yes | maximal |
| 8 | #8813 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #8282 | C2 | 1 | 2 | yes | maximal |
| 10 | C2×#501 | — | 2 | 2 | no | — |
| 11 | #2571 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2552 | C22 | 1 | 2 | yes | — |
| 13 | #2551 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2085 | — | 2 | 2 | no | — |
| 15 | #2084 | — | 2 | 2 | no | — |
| 16 | #2059 | — | 2 | 2 | no | — |
| 17 | #2058 | — | 2 | 2 | no | — |
| 18 | #1948 | C22 | 1 | 2 | yes | — |
| 19 | #1948 | C22 | 1 | 2 | yes | — |
| 20 | #1947 | C22 | 1 | 2 | yes | — |
| 21 | #1947 | C22 | 1 | 2 | yes | — |
| 22 | #1944 | — | 2 | 2 | no | — |
| 23 | #1943 | — | 2 | 2 | no | — |
| 24 | #1847 | — | 2 | 2 | no | — |
| 25 | #1844 | C22 | 1 | 2 | yes | — |
| 26 | #1844 | C22 | 1 | 2 | yes | — |
| 27 | #1844 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1844 | C22 | 1 | 1 | yes | characteristic |
| 29 | C4×#107 | C22 | 1 | 2 | yes | — |
| 30 | #1770 | C22 | 1 | 2 | yes | — |
| 31 | #1770 | C22 | 1 | 2 | yes | — |
| 32 | C2×#335 | C22 | 1 | 2 | yes | — |
| 33 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1296 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1296 | C22 | 1 | 1 | yes | characteristic |
| 36 | #1296 | C22 | 1 | 2 | yes | — |
| 37 | #1279 | C22 | 1 | 2 | yes | — |
| 38 | #1279 | C22 | 1 | 2 | yes | — |
| 39 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 40 | C2×#125 | — | 2 | 2 | no | — |
| 41 | #501 | — | 4 | 8 | no | — |
| 42 | C2×#107 | — | 2 | 2 | no | — |
| 43 | C2×#107 | — | 2 | 2 | no | — |
| 44 | C2×#107 | D4 | 1 | 2 | yes | — |
| 45 | C2×#107 | D4 | 1 | 2 | yes | — |
| 46 | C2×#106 | — | 2 | 2 | no | — |
| 47 | C2×#106 | — | 2 | 2 | no | — |
| 48 | C2×#106 | — | 2 | 2 | no | — |
| 49 | C2×#106 | — | 2 | 2 | no | — |
| 50 | C2×#106 | — | 2 | 2 | no | — |
| 51 | C2×#106 | — | 2 | 2 | no | — |
| 52 | C2×#106 | — | 4 | 1 | no | — |
| 53 | C2×#106 | — | 4 | 1 | no | — |
| 54 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 55 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 56 | C2×#97 | — | 2 | 2 | no | — |
| 57 | C2×#97 | — | 2 | 2 | no | — |
| 58 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 60 | #400 | — | 4 | 1 | no | — |
| 61 | #398 | — | 4 | 2 | no | — |
| 62 | #395 | — | 4 | 1 | no | — |
| 63 | #389 | — | 2 | 2 | no | — |
| 64 | #389 | — | 2 | 2 | no | — |
| 65 | #389 | — | 2 | 2 | no | — |
| 66 | #389 | — | 2 | 2 | no | — |
| 67 | #384 | — | 2 | 2 | no | — |
| 68 | #384 | C23 | 1 | 1 | yes | characteristic |
| 69 | #384 | C23 | 1 | 1 | yes | characteristic |
| 70 | #384 | C23 | 1 | 2 | yes | — |
| 71 | #382 | — | 2 | 2 | no | — |
| 72 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 73 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 74 | #340 | — | 2 | 2 | no | — |
| 75 | #340 | — | 2 | 2 | no | — |
| 76 | #340 | — | 2 | 2 | no | — |
| 77 | #340 | — | 2 | 2 | no | — |
| 78 | #340 | D4 | 1 | 2 | yes | — |
| 79 | #340 | D4 | 1 | 2 | yes | — |
| 80 | #336 | — | 2 | 2 | no | — |
| 81 | #336 | — | 2 | 2 | no | — |
| 82 | #335 | Q8 | 1 | 8 | yes | complemented |
| 83 | C82 | Q8 | 1 | 4 | yes | complemented |
| 84 | C22×Q8 | — | 4 | 1 | no | — |
| 85 | C22×C8 | — | 2 | 1 | no | — |
| 86 | C22×C8 | — | 2 | 2 | no | — |
| 87 | C22×C8 | — | 2 | 2 | no | — |
| 88 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 89 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 90 | C4⋊Q8 | — | 4 | 4 | no | — |
| 91 | C4⋊Q8 | — | 8 | 2 | no | — |
| 92 | #125 | — | 4 | 4 | no | — |
| 93 | #125 | — | 8 | 2 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 96 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 97 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 98 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 99 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 100 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 101 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 102 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 103 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 105 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 107 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 108 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 109 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 110 | C2×C42 | — | 2 | 2 | no | — |
| 111 | C2×C42 | — | 2 | 2 | no | — |
| 112 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 113 | #107 | — | 2 | 4 | no | — |
| 114 | #107 | — | 2 | 4 | no | — |
| 115 | #107 | — | 4 | 4 | no | — |
| 116 | #107 | — | 8 | 2 | no | — |
| 117 | #106 | — | 4 | 4 | no | — |
| 118 | #106 | — | 4 | 4 | no | — |
| 119 | #106 | — | 4 | 4 | no | — |
| 120 | #106 | — | 8 | 2 | no | — |
| 121 | #106 | — | 8 | 2 | no | — |
| 122 | #106 | — | 8 | 2 | no | — |
| 123 | #106 | — | 8 | 2 | no | — |
| 124 | #106 | — | 8 | 2 | no | — |
| 125 | C4⋊C8 | — | 2 | 2 | no | — |
| 126 | C4⋊C8 | — | 2 | 2 | no | — |
| 127 | C4⋊C8 | C2×Q8 | 1 | 4 | yes | — |
| 128 | C4⋊C8 | C2×Q8 | 1 | 4 | yes | — |
| 129 | #97 | — | 4 | 4 | no | — |
| 130 | #97 | — | 4 | 4 | no | — |
| 131 | C4×C8 | — | 2 | 1 | no | — |
| 132 | C4×C8 | C2×Q8 | 1 | 2 | yes | — |
| 133 | C4×C8 | C2×Q8 | 1 | 4 | yes | — |
| 134 | C4×C8 | C2×Q8 | 1 | 4 | yes | — |
| 135 | #95 | — | 2 | 2 | no | — |
| 136 | #95 | — | 2 | 2 | no | — |
| 137 | #95 | — | 4 | 1 | no | — |
| 138 | #95 | — | 4 | 2 | no | — |
| 139 | #95 | — | 4 | 1 | no | — |
| 140 | #95 | — | 4 | 2 | no | — |
| 141 | C2×Q8 | — | 8 | 4 | no | — |
| 142 | C2×Q8 | — | 16 | 1 | no | — |
| 143 | C22×C4 | — | 2 | 2 | no | — |
| 144 | C22×C4 | — | 2 | 2 | no | — |
| 145 | C22×C4 | — | 2 | 2 | no | — |
| 146 | C22×C4 | — | 2 | 2 | no | — |
| 147 | C22×C4 | — | 4 | 1 | no | — |
| 148 | C22×C4 | — | 4 | 2 | no | — |
| 149 | C22×C4 | — | 4 | 1 | no | — |
| 150 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 151 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 152 | C2×C8 | — | 2 | 2 | no | — |
| 153 | C2×C8 | — | 2 | 2 | no | — |
| 154 | C2×C8 | — | 2 | 2 | no | — |
| 155 | C2×C8 | — | 2 | 2 | no | — |
| 156 | C2×C8 | — | 2 | 2 | no | — |
| 157 | C2×C8 | — | 2 | 4 | no | — |
| 158 | C2×C8 | — | 2 | 4 | no | — |
| 159 | C2×C8 | — | 4 | 2 | no | — |
| 160 | C2×C8 | — | 4 | 2 | no | — |
| 161 | C2×C8 | — | 4 | 2 | no | — |
| 162 | C2×C8 | — | 4 | 2 | no | — |
| 163 | C2×C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 164 | C2×C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 165 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 166 | C2×C8 | #137 | 1 | 4 | yes | — |
| 167 | C4⋊C4 | — | 2 | 2 | no | — |
| 168 | C4⋊C4 | — | 2 | 2 | no | — |
| 169 | C4⋊C4 | — | 2 | 2 | no | — |
| 170 | C4⋊C4 | — | 2 | 2 | no | — |
| 171 | C4⋊C4 | — | 2 | 2 | no | — |
| 172 | C4⋊C4 | — | 2 | 2 | no | — |
| 173 | C4⋊C4 | — | 2 | 2 | no | — |
| 174 | C4⋊C4 | — | 2 | 2 | no | — |
| 175 | C4⋊C4 | — | 4 | 2 | no | — |
| 176 | C4⋊C4 | — | 4 | 2 | no | — |
| 177 | C4⋊C4 | — | 4 | 2 | no | — |
| 178 | C4⋊C4 | — | 4 | 2 | no | — |
| 179 | C4⋊C4 | — | 4 | 2 | no | — |
| 180 | C4⋊C4 | — | 4 | 2 | no | — |
| 181 | C4⋊C4 | — | 4 | 2 | no | — |
| 182 | C4⋊C4 | — | 4 | 2 | no | — |
| 183 | C4⋊C4 | — | 8 | 2 | no | — |
| 184 | C4⋊C4 | — | 8 | 1 | no | — |
| 185 | C4⋊C4 | — | 8 | 1 | no | — |
| 186 | C4⋊C4 | — | 8 | 4 | no | — |
| 187 | C4⋊C4 | — | 8 | 4 | no | — |
| 188 | C4⋊C4 | — | 8 | 4 | no | — |
| 189 | C4⋊C4 | — | 16 | 2 | no | — |
| 190 | C4⋊C4 | — | 16 | 1 | no | — |
| 191 | C42 | — | 4 | 2 | no | — |
| 192 | C42 | — | 4 | 2 | no | — |
| 193 | C42 | — | 4 | 2 | no | — |
| 194 | C42 | — | 4 | 2 | no | — |
| 195 | C42 | C22×Q8 | 1 | 1 | yes | characteristic |
| 196 | C42 | C22×Q8 | 1 | 2 | yes | — |
| 197 | C42 | 2−1+4 | 1 | 1 | yes | characteristic |
| 198 | C23 | D42 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 16 | 4 | no | — |
| 200 | C2×C4 | — | 2 | 1 | no | — |
| 201 | C2×C4 | — | 2 | 1 | no | — |
| 202 | C2×C4 | — | 4 | 2 | no | — |
| 203 | C2×C4 | — | 4 | 2 | no | — |
| 204 | C2×C4 | — | 4 | 2 | no | — |
| 205 | C2×C4 | — | 4 | 2 | no | — |
| 206 | C2×C4 | — | 4 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 2 | no | — |
| 208 | C2×C4 | — | 4 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 2 | no | — |
| 210 | C2×C4 | — | 4 | 2 | no | — |
| 211 | C2×C4 | — | 4 | 2 | no | — |
| 212 | C2×C4 | — | 4 | 2 | no | — |
| 213 | C2×C4 | — | 4 | 2 | no | — |
| 214 | C2×C4 | — | 8 | 2 | no | — |
| 215 | C2×C4 | — | 8 | 2 | no | — |
| 216 | C2×C4 | — | 8 | 2 | no | — |
| 217 | C2×C4 | — | 8 | 2 | no | — |
| 218 | C2×C4 | — | 8 | 2 | no | — |
| 219 | C2×C4 | — | 8 | 1 | no | — |
| 220 | C2×C4 | — | 8 | 1 | no | — |
| 221 | C2×C4 | C2×C4⋊Q8 | 1 | 2 | yes | — |
| 222 | C2×C4 | #544 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | D4×Q8 | 1 | 2 | yes | — |
| 224 | C2×C4 | Q82 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 226 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 227 | C2×C4 | #577 | 1 | 2 | yes | — |
| 228 | C2×C4 | #577 | 1 | 2 | yes | — |
| 229 | C8 | — | 2 | 4 | no | — |
| 230 | C8 | — | 2 | 4 | no | — |
| 231 | C8 | — | 4 | 4 | no | — |
| 232 | C8 | — | 4 | 4 | no | — |
| 233 | C8 | #501 | 1 | 8 | yes | complemented |
| 234 | C22 | #2551 | 1 | 1 | yes | characteristic |
| 235 | C22 | #3272 | 1 | 2 | yes | — |
| 236 | C22 | #3278 | 1 | 2 | yes | — |
| 237 | C22 | #3291 | 1 | 1 | yes | characteristic |
| 238 | C22 | #3291 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 2 | 1 | no | — |
| 240 | C4 | — | 2 | 1 | no | — |
| 241 | C4 | — | 8 | 2 | no | — |
| 242 | C4 | — | 8 | 2 | no | — |
| 243 | C4 | — | 8 | 2 | no | — |
| 244 | C4 | — | 8 | 2 | no | — |
| 245 | C4 | — | 16 | 1 | no | — |
| 246 | C4 | — | 16 | 2 | no | — |
| 247 | C4 | — | 16 | 1 | no | — |
| 248 | C4 | C2×#501 | 1 | 2 | yes | — |
| 249 | C4 | #3163 | 1 | 2 | yes | — |
| 250 | C4 | #3384 | 1 | 2 | yes | — |
| 251 | C4 | #3387 | 1 | 2 | yes | — |
| 252 | C2 | #20314 | 1 | 1 | yes | characteristic |
| 253 | C2 | #20317 | 1 | 2 | yes | — |
| 254 | C2 | #20323 | 1 | 1 | yes | characteristic |
| 255 | C2 | #28817 | 1 | 2 | yes | — |
| 256 | C2 | #28824 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 82. #335 | Q8 | semidirect |
| 83. C82 | Q8 | semidirect |
| 233. C8 | #501 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28836 | 2 |
| #28817 | 2 |
| #10580 | 2 |
| #10577 | 2 |
| #10574 | 2 |
| #9618 | 2 |
| #9617 | 2 |
| #8813 | 2 |
| #8282 | 2 |