Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
360 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #21965 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12923 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×#1327 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11243 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7255 | C2 | 1 | 2 | yes | maximal |
| 6 | #2911 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#423 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2283 | — | 2 | 1 | no | — |
| 9 | #2159 | — | 2 | 2 | no | — |
| 10 | C2×#349 | C22 | 1 | 2 | yes | — |
| 11 | #1949 | — | 2 | 2 | no | — |
| 12 | #1949 | — | 2 | 2 | no | — |
| 13 | #1809 | — | 2 | 1 | no | — |
| 14 | #1809 | C22 | 1 | 2 | yes | — |
| 15 | #1742 | — | 2 | 1 | no | — |
| 16 | C2×#339 | — | 2 | 1 | no | — |
| 17 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1386 | — | 2 | 2 | no | — |
| 19 | #1327 | C4 | 1 | 4 | yes | complemented |
| 20 | C22×M16 | — | 2 | 1 | no | — |
| 21 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 22 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 23 | C2×#117 | — | 2 | 1 | no | — |
| 24 | C22×C42 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×M32 | — | 2 | 2 | no | — |
| 26 | #428 | — | 4 | 2 | no | — |
| 27 | #427 | — | 2 | 2 | no | — |
| 28 | C2×#107 | — | 2 | 1 | no | — |
| 29 | C2×#107 | D4 | 1 | 2 | yes | — |
| 30 | C2×#106 | — | 2 | 1 | no | — |
| 31 | C2×#106 | D4 | 1 | 2 | yes | — |
| 32 | #423 | — | 2 | 2 | no | — |
| 33 | #423 | C2×C4 | 1 | 2 | yes | — |
| 34 | #423 | Q8 | 1 | 2 | yes | — |
| 35 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 36 | #383 | — | 4 | 1 | no | — |
| 37 | #382 | — | 2 | 1 | no | — |
| 38 | #382 | — | 2 | 1 | no | — |
| 39 | #381 | — | 4 | 1 | no | — |
| 40 | #380 | — | 2 | 2 | no | — |
| 41 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 42 | C4×C22⋊C4 | — | 4 | 1 | no | — |
| 43 | #376 | — | 4 | 1 | no | — |
| 44 | #367 | — | 4 | 2 | no | — |
| 45 | #366 | — | 4 | 2 | no | — |
| 46 | #349 | — | 2 | 2 | no | — |
| 47 | #349 | C2×C4 | 1 | 4 | yes | — |
| 48 | #343 | — | 4 | 2 | no | — |
| 49 | #340 | — | 2 | 2 | no | — |
| 50 | #340 | — | 2 | 2 | no | — |
| 51 | #339 | — | 4 | 1 | no | — |
| 52 | #339 | — | 4 | 1 | no | — |
| 53 | #338 | — | 2 | 2 | no | — |
| 54 | C23×C4 | — | 2 | 1 | no | — |
| 55 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | C2×M16 | — | 2 | 1 | no | — |
| 57 | C2×M16 | — | 2 | 2 | no | — |
| 58 | C2×M16 | — | 4 | 1 | no | — |
| 59 | C2×M16 | — | 4 | 1 | no | — |
| 60 | C2×M16 | — | 4 | 1 | no | — |
| 61 | C2×M16 | — | 4 | 1 | no | — |
| 62 | C2×M16 | — | 4 | 1 | no | — |
| 63 | C2×M16 | — | 4 | 1 | no | — |
| 64 | C2×M16 | — | 4 | 1 | no | — |
| 65 | C2×M16 | — | 4 | 1 | no | — |
| 66 | C2×M16 | — | 4 | 1 | no | — |
| 67 | C2×M16 | C4⋊C4 | 1 | 2 | yes | — |
| 68 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C2×M16 | C2×Q8 | 1 | 1 | yes | characteristic |
| 70 | C2×M16 | #41 | 1 | 2 | yes | — |
| 71 | C22×C8 | — | 2 | 1 | no | — |
| 72 | C22×C8 | — | 4 | 1 | no | — |
| 73 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 74 | #117 | — | 4 | 1 | no | — |
| 75 | #117 | — | 4 | 1 | no | — |
| 76 | #117 | — | 4 | 1 | no | — |
| 77 | #117 | — | 4 | 1 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 80 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 82 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 83 | C2×C42 | — | 2 | 1 | no | — |
| 84 | C2×C42 | — | 2 | 1 | no | — |
| 85 | C2×C42 | — | 2 | 1 | no | — |
| 86 | C2×C42 | — | 4 | 1 | no | — |
| 87 | C2×C42 | — | 4 | 2 | no | — |
| 88 | C2×C42 | C4⋊C4 | 1 | 2 | yes | — |
| 89 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 90 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 91 | C2×C42 | Q16 | 1 | 2 | yes | — |
| 92 | M32 | — | 2 | 4 | no | — |
| 93 | M32 | — | 4 | 2 | no | — |
| 94 | C2×C16 | — | 4 | 2 | no | — |
| 95 | #107 | — | 2 | 2 | no | — |
| 96 | #107 | — | 2 | 2 | no | — |
| 97 | #107 | — | 4 | 2 | no | — |
| 98 | #107 | — | 8 | 1 | no | — |
| 99 | #106 | — | 4 | 2 | no | — |
| 100 | #106 | — | 4 | 2 | no | — |
| 101 | #106 | — | 8 | 1 | no | — |
| 102 | C4⋊C8 | — | 2 | 4 | no | — |
| 103 | C4⋊C8 | — | 2 | 4 | no | — |
| 104 | #95 | — | 4 | 1 | no | — |
| 105 | #95 | — | 4 | 1 | no | — |
| 106 | #95 | — | 4 | 1 | no | — |
| 107 | #95 | — | 4 | 1 | no | — |
| 108 | #95 | — | 4 | 2 | no | — |
| 109 | #95 | — | 4 | 2 | no | — |
| 110 | C24 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 111 | C22×C4 | — | 2 | 1 | no | — |
| 112 | C22×C4 | — | 2 | 1 | no | — |
| 113 | C22×C4 | — | 2 | 1 | no | — |
| 114 | C22×C4 | — | 2 | 1 | no | — |
| 115 | C22×C4 | — | 2 | 1 | no | — |
| 116 | C22×C4 | — | 2 | 1 | no | — |
| 117 | C22×C4 | — | 2 | 2 | no | — |
| 118 | C22×C4 | — | 2 | 2 | no | — |
| 119 | C22×C4 | — | 4 | 2 | no | — |
| 120 | C22×C4 | — | 4 | 2 | no | — |
| 121 | C22×C4 | — | 4 | 2 | no | — |
| 122 | C22×C4 | — | 8 | 2 | no | — |
| 123 | C22×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 125 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 126 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 127 | C22×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 128 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 129 | M16 | — | 4 | 1 | no | — |
| 130 | M16 | — | 4 | 1 | no | — |
| 131 | M16 | — | 4 | 2 | no | — |
| 132 | M16 | — | 4 | 2 | no | — |
| 133 | M16 | — | 4 | 2 | no | — |
| 134 | M16 | — | 4 | 2 | no | — |
| 135 | M16 | — | 8 | 1 | no | — |
| 136 | M16 | — | 8 | 1 | no | — |
| 137 | M16 | — | 8 | 1 | no | — |
| 138 | C2×C8 | — | 2 | 2 | no | — |
| 139 | C2×C8 | — | 2 | 2 | no | — |
| 140 | C2×C8 | — | 2 | 2 | no | — |
| 141 | C2×C8 | — | 4 | 1 | no | — |
| 142 | C2×C8 | — | 4 | 1 | no | — |
| 143 | C2×C8 | — | 4 | 2 | no | — |
| 144 | C2×C8 | — | 8 | 1 | no | — |
| 145 | C2×C8 | — | 8 | 1 | no | — |
| 146 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 147 | C2×C8 | C22⋊Q8 | 1 | 2 | yes | — |
| 148 | C4⋊C4 | — | 4 | 2 | no | — |
| 149 | C4⋊C4 | — | 4 | 2 | no | — |
| 150 | C4⋊C4 | — | 4 | 2 | no | — |
| 151 | C4⋊C4 | — | 4 | 2 | no | — |
| 152 | C4⋊C4 | — | 4 | 2 | no | — |
| 153 | C4⋊C4 | — | 4 | 2 | no | — |
| 154 | C22⋊C4 | — | 8 | 1 | no | — |
| 155 | C22⋊C4 | — | 8 | 4 | no | — |
| 156 | C22⋊C4 | — | 8 | 1 | no | — |
| 157 | C42 | — | 2 | 2 | no | — |
| 158 | C42 | — | 2 | 2 | no | — |
| 159 | C42 | — | 2 | 1 | no | — |
| 160 | C42 | — | 2 | 1 | no | — |
| 161 | C42 | — | 8 | 1 | no | — |
| 162 | C42 | — | 8 | 4 | no | — |
| 163 | C42 | — | 8 | 1 | no | — |
| 164 | C42 | #107 | 1 | 2 | yes | — |
| 165 | C42 | #107 | 1 | 2 | yes | — |
| 166 | C16 | — | 4 | 4 | no | — |
| 167 | C23 | — | 2 | 1 | no | — |
| 168 | C23 | — | 2 | 1 | no | — |
| 169 | C23 | — | 4 | 2 | no | — |
| 170 | C23 | #428 | 1 | 1 | yes | characteristic |
| 171 | C23 | #447 | 1 | 1 | yes | characteristic |
| 172 | C23 | #484 | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | — | 2 | 1 | no | — |
| 174 | C2×C4 | — | 2 | 1 | no | — |
| 175 | C2×C4 | — | 2 | 1 | no | — |
| 176 | C2×C4 | — | 2 | 1 | no | — |
| 177 | C2×C4 | — | 2 | 1 | no | — |
| 178 | C2×C4 | — | 2 | 1 | no | — |
| 179 | C2×C4 | — | 2 | 1 | no | — |
| 180 | C2×C4 | — | 2 | 1 | no | — |
| 181 | C2×C4 | — | 2 | 1 | no | — |
| 182 | C2×C4 | — | 2 | 1 | no | — |
| 183 | C2×C4 | — | 2 | 2 | no | — |
| 184 | C2×C4 | — | 2 | 2 | no | — |
| 185 | C2×C4 | — | 4 | 2 | no | — |
| 186 | C2×C4 | — | 4 | 2 | no | — |
| 187 | C2×C4 | — | 8 | 2 | no | — |
| 188 | C2×C4 | — | 8 | 2 | no | — |
| 189 | C2×C4 | — | 8 | 2 | no | — |
| 190 | C2×C4 | — | 8 | 2 | no | — |
| 191 | C2×C4 | — | 8 | 2 | no | — |
| 192 | C2×C4 | — | 8 | 2 | no | — |
| 193 | C2×C4 | — | 8 | 2 | no | — |
| 194 | C2×C4 | — | 8 | 2 | no | — |
| 195 | C2×C4 | — | 8 | 2 | no | — |
| 196 | C2×C4 | — | 16 | 2 | no | — |
| 197 | C2×C4 | #385 | 1 | 1 | yes | characteristic |
| 198 | C2×C4 | C2×#107 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #451 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | #454 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | #456 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | #480 | 1 | 1 | yes | characteristic |
| 205 | C8 | — | 2 | 2 | no | — |
| 206 | C8 | — | 2 | 2 | no | — |
| 207 | C8 | — | 4 | 2 | no | — |
| 208 | C8 | — | 8 | 2 | no | — |
| 209 | C22 | — | 2 | 2 | no | — |
| 210 | C22 | — | 2 | 1 | no | — |
| 211 | C22 | — | 2 | 1 | no | — |
| 212 | C22 | — | 4 | 1 | no | — |
| 213 | C22 | — | 4 | 1 | no | — |
| 214 | C22 | — | 4 | 2 | no | — |
| 215 | C22 | — | 8 | 1 | no | — |
| 216 | C22 | #1598 | 1 | 1 | yes | characteristic |
| 217 | C22 | #1610 | 1 | 1 | yes | characteristic |
| 218 | C22 | #1852 | 1 | 1 | yes | characteristic |
| 219 | C4 | — | 2 | 1 | no | — |
| 220 | C4 | — | 2 | 1 | no | — |
| 221 | C4 | — | 4 | 2 | no | — |
| 222 | C4 | — | 4 | 2 | no | — |
| 223 | C4 | — | 16 | 2 | no | — |
| 224 | C4 | — | 16 | 2 | no | — |
| 225 | C4 | — | 16 | 2 | no | — |
| 226 | C4 | #1605 | 1 | 1 | yes | characteristic |
| 227 | C4 | #1617 | 1 | 1 | yes | characteristic |
| 228 | C4 | #1857 | 1 | 1 | yes | characteristic |
| 229 | C4 | #1867 | 1 | 1 | yes | characteristic |
| 230 | C2 | — | 2 | 1 | no | — |
| 231 | C2 | — | 2 | 1 | no | — |
| 232 | C2 | — | 4 | 1 | no | — |
| 233 | C2 | — | 4 | 1 | no | — |
| 234 | C2 | #8856 | 1 | 1 | yes | characteristic |
| 235 | C2 | #12175 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 19. #1327 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21965 | 2 |
| #12923 | 2 |
| C2×#1327 | 2 |
| #11243 | 2 |
| #7255 | 2 |