Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
696 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #62360 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33775 | C2 | 1 | 2 | yes | maximal |
| 3 | #28737 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #12162 | C2 | 1 | 8 | yes | maximal |
| 5 | #11743 | C2 | 1 | 2 | yes | maximal |
| 6 | #11493 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #3502 | — | 2 | 1 | no | — |
| 8 | #3395 | — | 2 | 2 | no | — |
| 9 | Q8×Q16 | — | 2 | 2 | no | — |
| 10 | #3363 | — | 2 | 4 | no | — |
| 11 | #3249 | — | 2 | 1 | no | — |
| 12 | #3249 | C22 | 1 | 4 | yes | — |
| 13 | #3170 | — | 2 | 2 | no | — |
| 14 | #3170 | C22 | 1 | 2 | yes | — |
| 15 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2568 | — | 2 | 1 | no | — |
| 18 | #2247 | — | 2 | 2 | no | — |
| 19 | #2241 | — | 2 | 2 | no | — |
| 20 | #2209 | — | 2 | 4 | no | — |
| 21 | #2188 | C22 | 1 | 8 | yes | — |
| 22 | #2162 | C22 | 1 | 2 | yes | — |
| 23 | #2056 | C22 | 1 | 4 | yes | — |
| 24 | #1981 | — | 2 | 1 | no | — |
| 25 | #1969 | — | 2 | 2 | no | — |
| 26 | C4×#107 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1348 | C22 | 1 | 8 | yes | — |
| 28 | #1322 | C22 | 1 | 4 | yes | — |
| 29 | C22×Q16 | — | 2 | 1 | no | — |
| 30 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 31 | Q82 | — | 2 | 4 | no | — |
| 32 | #557 | — | 2 | 4 | no | — |
| 33 | #557 | — | 4 | 4 | no | — |
| 34 | #544 | — | 4 | 1 | no | — |
| 35 | #544 | — | 4 | 2 | no | — |
| 36 | #533 | — | 2 | 4 | no | — |
| 37 | #533 | — | 2 | 2 | no | — |
| 38 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 39 | #519 | — | 4 | 1 | no | — |
| 40 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×Q32 | — | 2 | 8 | no | — |
| 42 | #500 | — | 2 | 2 | no | — |
| 43 | #500 | D4 | 1 | 2 | yes | — |
| 44 | #494 | — | 2 | 2 | no | — |
| 45 | #494 | — | 2 | 2 | no | — |
| 46 | #494 | — | 2 | 2 | no | — |
| 47 | #494 | D4 | 1 | 2 | yes | — |
| 48 | #487 | — | 2 | 4 | no | — |
| 49 | #487 | — | 2 | 2 | no | — |
| 50 | #477 | — | 2 | 4 | no | — |
| 51 | #477 | — | 2 | 8 | no | — |
| 52 | #467 | — | 2 | 2 | no | — |
| 53 | #467 | — | 2 | 4 | no | — |
| 54 | #467 | C23 | 1 | 4 | yes | — |
| 55 | #462 | — | 2 | 4 | no | — |
| 56 | #462 | — | 2 | 8 | no | — |
| 57 | C8×Q8 | — | 2 | 2 | no | — |
| 58 | C8×Q8 | — | 2 | 2 | no | — |
| 59 | C4×Q16 | — | 2 | 4 | no | — |
| 60 | C4×Q16 | — | 2 | 4 | no | — |
| 61 | C4×Q16 | — | 2 | 2 | no | — |
| 62 | C4×Q16 | D4 | 1 | 4 | yes | — |
| 63 | C2×#107 | — | 2 | 1 | no | — |
| 64 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 65 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 66 | C2×#103 | — | 2 | 4 | no | — |
| 67 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 68 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 69 | #400 | — | 4 | 2 | no | — |
| 70 | #391 | — | 2 | 4 | no | — |
| 71 | #391 | — | 4 | 1 | no | — |
| 72 | #382 | — | 4 | 1 | no | — |
| 73 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 74 | C4⋊C16 | — | 2 | 2 | no | — |
| 75 | #358 | — | 2 | 8 | no | — |
| 76 | C22⋊C16 | C23 | 1 | 4 | yes | — |
| 77 | C4×C16 | — | 2 | 2 | no | — |
| 78 | #340 | C23 | 1 | 2 | yes | — |
| 79 | C22×Q8 | — | 2 | 2 | no | — |
| 80 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 81 | C2×Q16 | — | 2 | 4 | no | — |
| 82 | C2×Q16 | — | 2 | 2 | no | — |
| 83 | C2×Q16 | — | 2 | 4 | no | — |
| 84 | C2×Q16 | — | 4 | 4 | no | — |
| 85 | C2×Q16 | — | 4 | 1 | no | — |
| 86 | C2×Q16 | — | 4 | 1 | no | — |
| 87 | C2×Q16 | — | 8 | 2 | no | — |
| 88 | C2×Q16 | C2×D4 | 1 | 4 | yes | — |
| 89 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 91 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 92 | C4⋊Q8 | — | 2 | 4 | no | — |
| 93 | C4⋊Q8 | — | 2 | 4 | no | — |
| 94 | C4⋊Q8 | — | 2 | 2 | no | — |
| 95 | C4⋊Q8 | — | 2 | 2 | no | — |
| 96 | C4⋊Q8 | — | 4 | 8 | no | — |
| 97 | C4⋊Q8 | — | 4 | 4 | no | — |
| 98 | C4⋊Q8 | — | 4 | 2 | no | — |
| 99 | C4⋊Q8 | — | 4 | 2 | no | — |
| 100 | #125 | — | 4 | 4 | no | — |
| 101 | #125 | — | 4 | 8 | no | — |
| 102 | #125 | — | 4 | 4 | no | — |
| 103 | #125 | — | 4 | 2 | no | — |
| 104 | C22⋊Q8 | — | 2 | 4 | no | — |
| 105 | C22⋊Q8 | — | 4 | 4 | no | — |
| 106 | C22⋊Q8 | — | 4 | 2 | no | — |
| 107 | C4×Q8 | — | 2 | 2 | no | — |
| 108 | C4×Q8 | — | 2 | 2 | no | — |
| 109 | C4×Q8 | — | 2 | 8 | no | — |
| 110 | C4×Q8 | — | 2 | 4 | no | — |
| 111 | C4×Q8 | — | 4 | 4 | no | — |
| 112 | C4×Q8 | — | 4 | 4 | no | — |
| 113 | C4×Q8 | — | 4 | 4 | no | — |
| 114 | C4×Q8 | — | 4 | 2 | no | — |
| 115 | #117 | — | 4 | 4 | no | — |
| 116 | #117 | — | 4 | 2 | no | — |
| 117 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 118 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 119 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 120 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 121 | C2×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 122 | C2×C42 | — | 4 | 1 | no | — |
| 123 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 124 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | Q32 | — | 8 | 8 | no | — |
| 126 | C2×C16 | — | 2 | 4 | no | — |
| 127 | #107 | — | 2 | 4 | no | — |
| 128 | #107 | — | 2 | 2 | no | — |
| 129 | #107 | — | 4 | 2 | no | — |
| 130 | #107 | C2×D4 | 1 | 4 | yes | — |
| 131 | C4⋊C8 | — | 2 | 4 | no | — |
| 132 | C4⋊C8 | — | 2 | 2 | no | — |
| 133 | #103 | — | 2 | 8 | no | — |
| 134 | #103 | — | 2 | 4 | no | — |
| 135 | #103 | — | 2 | 4 | no | — |
| 136 | #103 | — | 8 | 4 | no | — |
| 137 | C4×C8 | — | 2 | 2 | no | — |
| 138 | C4×C8 | — | 2 | 2 | no | — |
| 139 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 140 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 141 | #95 | — | 4 | 1 | no | — |
| 142 | #95 | — | 4 | 4 | no | — |
| 143 | #95 | — | 4 | 4 | no | — |
| 144 | C2×Q8 | — | 2 | 2 | no | — |
| 145 | C2×Q8 | — | 2 | 2 | no | — |
| 146 | C2×Q8 | — | 2 | 4 | no | — |
| 147 | C2×Q8 | — | 4 | 4 | no | — |
| 148 | C2×Q8 | — | 4 | 4 | no | — |
| 149 | C2×Q8 | — | 4 | 4 | no | — |
| 150 | C2×Q8 | — | 4 | 2 | no | — |
| 151 | C2×Q8 | — | 4 | 2 | no | — |
| 152 | C2×Q8 | — | 4 | 2 | no | — |
| 153 | C2×Q8 | Q32 | 1 | 4 | yes | complemented |
| 154 | C22×C4 | — | 4 | 2 | no | — |
| 155 | C22×C4 | — | 4 | 2 | no | — |
| 156 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 157 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 158 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 159 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 160 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 161 | Q16 | — | 4 | 4 | no | — |
| 162 | Q16 | — | 4 | 8 | no | — |
| 163 | Q16 | — | 4 | 4 | no | — |
| 164 | Q16 | — | 8 | 2 | no | — |
| 165 | Q16 | — | 8 | 4 | no | — |
| 166 | Q16 | — | 8 | 2 | no | — |
| 167 | C2×C8 | — | 2 | 2 | no | — |
| 168 | C2×C8 | — | 2 | 2 | no | — |
| 169 | C2×C8 | — | 4 | 1 | no | — |
| 170 | C2×C8 | — | 4 | 2 | no | — |
| 171 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 174 | C4⋊C4 | — | 2 | 4 | no | — |
| 175 | C4⋊C4 | — | 2 | 2 | no | — |
| 176 | C4⋊C4 | — | 2 | 4 | no | — |
| 177 | C4⋊C4 | — | 4 | 4 | no | — |
| 178 | C4⋊C4 | — | 4 | 4 | no | — |
| 179 | C4⋊C4 | — | 4 | 8 | no | — |
| 180 | C4⋊C4 | — | 4 | 4 | no | — |
| 181 | C4⋊C4 | — | 4 | 2 | no | — |
| 182 | C4⋊C4 | — | 4 | 2 | no | — |
| 183 | C4⋊C4 | — | 4 | 4 | no | — |
| 184 | C4⋊C4 | — | 4 | 2 | no | — |
| 185 | C4⋊C4 | — | 4 | 2 | no | — |
| 186 | C4⋊C4 | — | 4 | 4 | no | — |
| 187 | C4⋊C4 | — | 4 | 2 | no | — |
| 188 | C4⋊C4 | — | 8 | 2 | no | — |
| 189 | C4⋊C4 | — | 8 | 2 | no | — |
| 190 | C22⋊C4 | — | 4 | 4 | no | — |
| 191 | C42 | — | 2 | 2 | no | — |
| 192 | C42 | — | 2 | 2 | no | — |
| 193 | C42 | — | 4 | 8 | no | — |
| 194 | C42 | — | 4 | 4 | no | — |
| 195 | C42 | — | 8 | 2 | no | — |
| 196 | C42 | #120 | 1 | 2 | yes | — |
| 197 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 198 | C16 | — | 4 | 4 | no | — |
| 199 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 200 | Q8 | — | 2 | 8 | no | — |
| 201 | Q8 | — | 4 | 4 | no | — |
| 202 | Q8 | — | 4 | 4 | no | — |
| 203 | Q8 | — | 8 | 4 | no | — |
| 204 | Q8 | — | 8 | 2 | no | — |
| 205 | Q8 | — | 8 | 2 | no | — |
| 206 | Q8 | — | 16 | 2 | no | — |
| 207 | Q8 | — | 16 | 4 | no | — |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 2 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 2 | 2 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 4 | 2 | no | — |
| 215 | C2×C4 | — | 4 | 4 | no | — |
| 216 | C2×C4 | — | 4 | 4 | no | — |
| 217 | C2×C4 | — | 8 | 2 | no | — |
| 218 | C2×C4 | — | 8 | 2 | no | — |
| 219 | C2×C4 | — | 8 | 4 | no | — |
| 220 | C2×C4 | — | 8 | 2 | no | — |
| 221 | C2×C4 | — | 8 | 2 | no | — |
| 222 | C2×C4 | #447 | 1 | 2 | yes | — |
| 223 | C2×C4 | #447 | 1 | 2 | yes | — |
| 224 | C2×C4 | C2×Q32 | 1 | 4 | yes | — |
| 225 | C2×C4 | C2×Q32 | 1 | 2 | yes | — |
| 226 | C2×C4 | #510 | 1 | 2 | yes | — |
| 227 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 229 | C8 | — | 2 | 1 | no | — |
| 230 | C8 | — | 2 | 2 | no | — |
| 231 | C8 | — | 4 | 2 | no | — |
| 232 | C8 | #454 | 1 | 2 | yes | — |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | — | 2 | 1 | no | — |
| 235 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 236 | C22 | C22×Q32 | 1 | 1 | yes | characteristic |
| 237 | C22 | C2×#510 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 2 | 4 | no | — |
| 240 | C4 | — | 2 | 2 | no | — |
| 241 | C4 | — | 4 | 2 | no | — |
| 242 | C4 | — | 8 | 4 | no | — |
| 243 | C4 | — | 8 | 4 | no | — |
| 244 | C4 | — | 16 | 2 | no | — |
| 245 | C4 | — | 16 | 2 | no | — |
| 246 | C4 | #2188 | 1 | 4 | yes | — |
| 247 | C4 | C2×#454 | 1 | 1 | yes | characteristic |
| 248 | C4 | #3027 | 1 | 1 | yes | characteristic |
| 249 | C2 | — | 2 | 1 | no | — |
| 250 | C2 | — | 2 | 1 | no | — |
| 251 | C2 | #18401 | 1 | 1 | yes | characteristic |
| 252 | C2 | C2×#2188 | 1 | 1 | yes | characteristic |
| 253 | C2 | #33532 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C22×D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 153. C2×Q8 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62360 | 2 |
| #33775 | 2 |
| #28737 | 2 |
| #12162 | 2 |
| #11743 | 2 |
| #11493 | 2 |