Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
37739 conjugacy classes of subgroups.
The subgroup lattice has 302 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10587001 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10585848 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10574096 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #7622507 | C2 | 1 | 30 | yes | maximal, complemented |
| 5 | C2×#33388 | C2 | 1 | 30 | yes | maximal, complemented |
| 6 | #63103 | — | 2 | 1 | no | — |
| 7 | #63093 | C4 | 1 | 16 | yes | complemented |
| 8 | #63092 | — | 2 | 16 | no | — |
| 9 | #63091 | C4 | 1 | 16 | yes | complemented |
| 10 | #63090 | — | 2 | 15 | no | — |
| 11 | C2×#3584 | C22 | 1 | 30 | yes | — |
| 12 | C2×#3573 | C22 | 1 | 1 | yes | characteristic |
| 13 | #62654 | — | 2 | 1 | no | — |
| 14 | #62624 | — | 2 | 16 | no | — |
| 15 | C2×#3407 | C22 | 1 | 30 | yes | complemented |
| 16 | #60109 | — | 2 | 15 | no | — |
| 17 | #60108 | C22 | 1 | 15 | yes | complemented |
| 18 | C2×#2912 | C22 | 1 | 15 | yes | complemented |
| 19 | #33388 | C22 | 1 | 240 | yes | complemented |
| 20 | #33388 | C22 | 1 | 80 | yes | complemented |
| 21 | C2×#2145 | C22 | 1 | 180 | yes | complemented |
| 22 | C22×#360 | C22 | 1 | 60 | yes | complemented |
| 23 | 2−1+6 | — | 2 | 16 | no | — |
| 24 | 2−1+6 | — | 4 | 6 | no | — |
| 25 | H(7,𝔽2) | — | 2 | 16 | no | — |
| 26 | H(7,𝔽2) | — | 4 | 10 | no | — |
| 27 | C2×#585 | — | 2 | 30 | no | — |
| 28 | C2×#585 | — | 4 | 16 | no | — |
| 29 | C2×#585 | D4 | 1 | 1 | yes | characteristic |
| 30 | #3586 | — | 2 | 240 | no | — |
| 31 | #3585 | — | 2 | 240 | no | — |
| 32 | #3584 | — | 2 | 120 | no | — |
| 33 | #3584 | — | 2 | 120 | no | — |
| 34 | #3584 | C2×C4 | 1 | 240 | yes | complemented |
| 35 | #3584 | C23 | 1 | 80 | yes | — |
| 36 | C2×#578 | C2×C4 | 1 | 120 | yes | complemented |
| 37 | C2×#577 | — | 2 | 120 | no | — |
| 38 | C2×#576 | C2×C4 | 1 | 120 | yes | complemented |
| 39 | C2×#575 | — | 2 | 90 | no | — |
| 40 | C22×#135 | C23 | 1 | 60 | yes | — |
| 41 | #3573 | C2×C4 | 1 | 16 | yes | — |
| 42 | #3573 | D4 | 1 | 16 | yes | — |
| 43 | C2×#568 | D4 | 1 | 15 | yes | — |
| 44 | C22×#131 | C23 | 1 | 15 | yes | — |
| 45 | C4×2−1+4 | — | 4 | 6 | no | — |
| 46 | C4×H(5,𝔽2) | — | 4 | 10 | no | — |
| 47 | #3428 | — | 2 | 15 | no | — |
| 48 | C2×#520 | — | 2 | 15 | no | — |
| 49 | #3407 | — | 2 | 120 | no | — |
| 50 | #3407 | C23 | 1 | 80 | yes | complemented |
| 51 | C2×#504 | — | 2 | 120 | no | — |
| 52 | C22×M32 | C23 | 1 | 60 | yes | complemented |
| 53 | #2913 | — | 2 | 60 | no | — |
| 54 | #2912 | — | 2 | 60 | no | — |
| 55 | #2912 | C23 | 1 | 60 | yes | complemented |
| 56 | #2912 | C23 | 1 | 20 | yes | complemented |
| 57 | #2911 | — | 2 | 45 | no | — |
| 58 | C2×#428 | — | 2 | 45 | no | — |
| 59 | C2×#427 | C23 | 1 | 45 | yes | complemented |
| 60 | C22×#106 | C23 | 1 | 15 | yes | complemented |
| 61 | #2145 | C23 | 1 | 480 | yes | complemented |
| 62 | C2×#360 | C23 | 1 | 360 | yes | complemented |
| 63 | C2×#360 | C23 | 1 | 120 | yes | complemented |
| 64 | #585 | — | 2 | 240 | no | — |
| 65 | #585 | — | 2 | 80 | no | — |
| 66 | #585 | — | 4 | 96 | no | — |
| 67 | #585 | — | 4 | 160 | no | — |
| 68 | #585 | — | 4 | 120 | no | — |
| 69 | #585 | — | 4 | 120 | no | — |
| 70 | #585 | C22⋊C4 | 1 | 16 | yes | — |
| 71 | #585 | SD16 | 1 | 16 | yes | — |
| 72 | C2×2−1+4 | — | 2 | 120 | no | — |
| 73 | C2×2−1+4 | — | 4 | 96 | no | — |
| 74 | C2×2−1+4 | — | 4 | 30 | no | — |
| 75 | C2×2−1+4 | D8 | 1 | 6 | yes | — |
| 76 | C2×H(5,𝔽2) | — | 2 | 120 | no | — |
| 77 | C2×H(5,𝔽2) | — | 4 | 160 | no | — |
| 78 | C2×H(5,𝔽2) | — | 4 | 90 | no | — |
| 79 | C2×H(5,𝔽2) | D8 | 1 | 10 | yes | — |
| 80 | C22×#41 | — | 2 | 60 | no | — |
| 81 | C22×#41 | — | 4 | 120 | no | — |
| 82 | C22×#41 | C2×D4 | 1 | 15 | yes | — |
| 83 | #578 | — | 2 | 480 | no | — |
| 84 | #578 | C22×C4 | 1 | 320 | yes | complemented |
| 85 | #577 | — | 2 | 960 | no | — |
| 86 | #577 | — | 2 | 320 | no | — |
| 87 | #576 | — | 2 | 480 | no | — |
| 88 | #576 | C22×C4 | 1 | 320 | yes | complemented |
| 89 | #575 | — | 2 | 720 | no | — |
| 90 | #575 | — | 2 | 240 | no | — |
| 91 | C2×#137 | — | 2 | 360 | no | — |
| 92 | C2×AGL(1,ℤ/8) | — | 2 | 360 | no | — |
| 93 | C2×#135 | — | 2 | 180 | no | — |
| 94 | C2×#135 | — | 2 | 180 | no | — |
| 95 | C2×#135 | C22×C4 | 1 | 360 | yes | complemented |
| 96 | C2×#135 | C24 | 1 | 120 | yes | — |
| 97 | C22×Q16 | C22×C4 | 1 | 60 | yes | complemented |
| 98 | C22×SD16 | — | 2 | 60 | no | — |
| 99 | C22×D8 | C22×C4 | 1 | 60 | yes | complemented |
| 100 | #568 | C22⋊C4 | 1 | 240 | yes | — |
| 101 | C2×#131 | C22×C4 | 1 | 120 | yes | — |
| 102 | C2×#131 | C2×D4 | 1 | 120 | yes | — |
| 103 | C2×#131 | C2×D4 | 1 | 60 | yes | — |
| 104 | C2×#131 | C24 | 1 | 20 | yes | — |
| 105 | C22×M16 | C2×D4 | 1 | 45 | yes | — |
| 106 | C23×C8 | C24 | 1 | 15 | yes | — |
| 107 | #520 | — | 2 | 60 | no | — |
| 108 | #520 | — | 2 | 20 | no | — |
| 109 | #520 | — | 4 | 60 | no | — |
| 110 | #519 | — | 4 | 15 | no | — |
| 111 | #518 | — | 4 | 45 | no | — |
| 112 | C4×#41 | — | 4 | 60 | no | — |
| 113 | C2×C4×Q8 | — | 4 | 15 | no | — |
| 114 | C2×C4×D4 | — | 4 | 45 | no | — |
| 115 | C2×#117 | — | 2 | 45 | no | — |
| 116 | C22×C4⋊C4 | — | 2 | 15 | no | — |
| 117 | #504 | — | 2 | 320 | no | — |
| 118 | C2×M32 | — | 2 | 180 | no | — |
| 119 | C2×M32 | C24 | 1 | 120 | yes | — |
| 120 | C22×C16 | — | 2 | 60 | no | — |
| 121 | #428 | — | 2 | 180 | no | — |
| 122 | #428 | — | 2 | 60 | no | — |
| 123 | #427 | — | 2 | 90 | no | — |
| 124 | #427 | C24 | 1 | 60 | yes | — |
| 125 | C2×#107 | — | 2 | 45 | no | — |
| 126 | C2×#106 | — | 2 | 45 | no | — |
| 127 | C2×#106 | C24 | 1 | 45 | yes | — |
| 128 | C2×#106 | C24 | 1 | 15 | yes | — |
| 129 | #360 | C24 | 1 | 160 | yes | — |
| 130 | #360 | C24 | 1 | 96 | yes | — |
| 131 | 2−1+4 | — | 2 | 320 | no | — |
| 132 | 2−1+4 | — | 4 | 480 | no | — |
| 133 | 2−1+4 | — | 4 | 80 | no | — |
| 134 | 2−1+4 | — | 4 | 240 | no | — |
| 135 | 2−1+4 | #102 | 1 | 96 | yes | — |
| 136 | H(5,𝔽2) | — | 2 | 320 | no | — |
| 137 | H(5,𝔽2) | — | 4 | 480 | no | — |
| 138 | H(5,𝔽2) | — | 4 | 240 | no | — |
| 139 | H(5,𝔽2) | — | 4 | 720 | no | — |
| 140 | H(5,𝔽2) | #102 | 1 | 160 | yes | — |
| 141 | C2×#41 | — | 2 | 360 | no | — |
| 142 | C2×#41 | — | 2 | 120 | no | — |
| 143 | C2×#41 | — | 4 | 320 | no | — |
| 144 | C2×#41 | — | 4 | 960 | no | — |
| 145 | C2×#41 | — | 4 | 120 | no | — |
| 146 | C2×#41 | — | 4 | 360 | no | — |
| 147 | C2×#41 | — | 4 | 180 | no | — |
| 148 | C2×#41 | — | 4 | 180 | no | — |
| 149 | C2×#41 | C2×C22⋊C4 | 1 | 120 | yes | — |
| 150 | C2×#41 | C2×D8 | 1 | 60 | yes | — |
| 151 | C2×#41 | C2×SD16 | 1 | 120 | yes | — |
| 152 | C2×#41 | C22×D4 | 1 | 20 | yes | — |
| 153 | C22×Q8 | — | 2 | 60 | no | — |
| 154 | C22×Q8 | — | 4 | 120 | no | — |
| 155 | C22×Q8 | C2×D8 | 1 | 15 | yes | — |
| 156 | C22×D4 | — | 2 | 60 | no | — |
| 157 | C22×D4 | — | 4 | 60 | no | — |
| 158 | C22×D4 | — | 4 | 360 | no | — |
| 159 | C22×D4 | C2×D8 | 1 | 45 | yes | — |
| 160 | C23×C4 | — | 4 | 60 | no | — |
| 161 | C23×C4 | — | 4 | 15 | no | — |
| 162 | C23×C4 | C22×D4 | 1 | 15 | yes | — |
| 163 | #137 | — | 2 | 960 | no | — |
| 164 | AGL(1,ℤ/8) | — | 2 | 960 | no | — |
| 165 | #135 | — | 2 | 720 | no | — |
| 166 | #135 | — | 2 | 120 | no | — |
| 167 | #135 | — | 2 | 120 | no | — |
| 168 | #135 | C23×C4 | 1 | 240 | yes | — |
| 169 | #135 | C25 | 1 | 16 | yes | — |
| 170 | C2×Q16 | — | 2 | 180 | no | — |
| 171 | C2×Q16 | C23×C4 | 1 | 120 | yes | — |
| 172 | C2×SD16 | — | 2 | 360 | no | — |
| 173 | C2×SD16 | — | 2 | 120 | no | — |
| 174 | C2×D8 | — | 2 | 180 | no | — |
| 175 | C2×D8 | C23×C4 | 1 | 120 | yes | — |
| 176 | #131 | C2×C22⋊C4 | 1 | 480 | yes | — |
| 177 | #131 | C23×C4 | 1 | 80 | yes | — |
| 178 | #131 | C22×D4 | 1 | 80 | yes | — |
| 179 | C2×M16 | C2×C22⋊C4 | 1 | 360 | yes | — |
| 180 | C2×M16 | C22×D4 | 1 | 60 | yes | — |
| 181 | C22×C8 | C23×C4 | 1 | 60 | yes | — |
| 182 | C22×C8 | C22×D4 | 1 | 45 | yes | — |
| 183 | C22×C8 | C22×D4 | 1 | 60 | yes | — |
| 184 | C22×C8 | C25 | 1 | 15 | yes | — |
| 185 | C4×Q8 | — | 4 | 60 | no | — |
| 186 | C4×Q8 | — | 4 | 20 | no | — |
| 187 | C4×D4 | — | 4 | 180 | no | — |
| 188 | C4×D4 | — | 4 | 60 | no | — |
| 189 | #117 | — | 2 | 60 | no | — |
| 190 | #117 | — | 4 | 90 | no | — |
| 191 | C2×C4⋊C4 | — | 2 | 45 | no | — |
| 192 | C2×C4⋊C4 | — | 2 | 15 | no | — |
| 193 | C2×C4⋊C4 | — | 4 | 45 | no | — |
| 194 | C2×C22⋊C4 | — | 4 | 45 | no | — |
| 195 | C2×C42 | — | 4 | 45 | no | — |
| 196 | M32 | — | 2 | 120 | no | — |
| 197 | M32 | C25 | 1 | 16 | yes | — |
| 198 | C2×C16 | — | 2 | 120 | no | — |
| 199 | #107 | — | 2 | 60 | no | — |
| 200 | #106 | — | 2 | 45 | no | — |
| 201 | #106 | — | 2 | 15 | no | — |
| 202 | #106 | C25 | 1 | 10 | yes | — |
| 203 | #106 | C25 | 1 | 6 | yes | — |
| 204 | C24 | — | 4 | 60 | no | — |
| 205 | C24 | C22×D8 | 1 | 15 | yes | — |
| 206 | #41 | — | 2 | 16 | no | — |
| 207 | #41 | — | 2 | 240 | no | — |
| 208 | #41 | — | 4 | 320 | no | — |
| 209 | #41 | — | 4 | 960 | no | — |
| 210 | #41 | — | 4 | 720 | no | — |
| 211 | #41 | — | 4 | 120 | no | — |
| 212 | #41 | — | 4 | 120 | no | — |
| 213 | #41 | C2×#102 | 1 | 480 | yes | — |
| 214 | #41 | C22×C22⋊C4 | 1 | 80 | yes | — |
| 215 | #41 | C22×SD16 | 1 | 80 | yes | — |
| 216 | C2×Q8 | — | 2 | 120 | no | — |
| 217 | C2×Q8 | — | 4 | 320 | no | — |
| 218 | C2×Q8 | — | 4 | 180 | no | — |
| 219 | C2×Q8 | C2×#102 | 1 | 120 | yes | — |
| 220 | C2×Q8 | C22×D8 | 1 | 20 | yes | — |
| 221 | C2×D4 | — | 2 | 120 | no | — |
| 222 | C2×D4 | — | 4 | 360 | no | — |
| 223 | C2×D4 | — | 4 | 120 | no | — |
| 224 | C2×D4 | — | 4 | 960 | no | — |
| 225 | C2×D4 | — | 4 | 180 | no | — |
| 226 | C2×D4 | C2×#102 | 1 | 360 | yes | — |
| 227 | C2×D4 | C22×D8 | 1 | 60 | yes | — |
| 228 | C22×C4 | — | 4 | 60 | no | — |
| 229 | C22×C4 | — | 4 | 360 | no | — |
| 230 | C22×C4 | — | 4 | 120 | no | — |
| 231 | C22×C4 | — | 4 | 15 | no | — |
| 232 | C22×C4 | — | 4 | 45 | no | — |
| 233 | C22×C4 | — | 32 | 15 | no | — |
| 234 | C22×C4 | C22×C22⋊C4 | 1 | 60 | yes | — |
| 235 | C22×C4 | C22×D8 | 1 | 45 | yes | — |
| 236 | C22×C4 | C22×SD16 | 1 | 60 | yes | — |
| 237 | C22×C4 | C23×D4 | 1 | 15 | yes | — |
| 238 | Q16 | — | 2 | 120 | no | — |
| 239 | Q16 | C24×C4 | 1 | 16 | yes | — |
| 240 | SD16 | — | 2 | 16 | no | — |
| 241 | SD16 | — | 2 | 240 | no | — |
| 242 | D8 | — | 2 | 120 | no | — |
| 243 | D8 | C24×C4 | 1 | 16 | yes | — |
| 244 | M16 | C22×C22⋊C4 | 1 | 240 | yes | — |
| 245 | C2×C8 | C22×C22⋊C4 | 1 | 180 | yes | — |
| 246 | C2×C8 | C24×C4 | 1 | 30 | yes | — |
| 247 | C2×C8 | C23×D4 | 1 | 30 | yes | — |
| 248 | C2×C8 | C23×D4 | 1 | 15 | yes | — |
| 249 | C2×C8 | C26 | 1 | 1 | yes | characteristic |
| 250 | C4⋊C4 | — | 2 | 10 | no | — |
| 251 | C4⋊C4 | — | 2 | 6 | no | — |
| 252 | C4⋊C4 | — | 4 | 45 | no | — |
| 253 | C4⋊C4 | — | 4 | 15 | no | — |
| 254 | C22⋊C4 | — | 4 | 60 | no | — |
| 255 | C42 | — | 4 | 60 | no | — |
| 256 | C16 | — | 2 | 16 | no | — |
| 257 | C23 | — | 4 | 120 | no | — |
| 258 | C23 | — | 8 | 15 | no | — |
| 259 | C23 | — | 32 | 60 | no | — |
| 260 | C23 | C22×#102 | 1 | 60 | yes | — |
| 261 | C23 | C23×D8 | 1 | 15 | yes | — |
| 262 | Q8 | — | 2 | 16 | no | — |
| 263 | Q8 | — | 4 | 120 | no | — |
| 264 | Q8 | C22×#102 | 1 | 80 | yes | — |
| 265 | D4 | — | 2 | 16 | no | — |
| 266 | D4 | — | 4 | 16 | no | — |
| 267 | D4 | — | 4 | 240 | no | — |
| 268 | D4 | — | 4 | 120 | no | — |
| 269 | D4 | C22×#102 | 1 | 240 | yes | — |
| 270 | C2×C4 | — | 4 | 120 | no | — |
| 271 | C2×C4 | — | 4 | 16 | no | — |
| 272 | C2×C4 | — | 4 | 240 | no | — |
| 273 | C2×C4 | — | 4 | 6 | no | — |
| 274 | C2×C4 | — | 4 | 10 | no | — |
| 275 | C2×C4 | — | 16 | 15 | no | — |
| 276 | C2×C4 | — | 16 | 45 | no | — |
| 277 | C2×C4 | C22×#102 | 1 | 180 | yes | — |
| 278 | C2×C4 | C23×C22⋊C4 | 1 | 30 | yes | — |
| 279 | C2×C4 | C23×D8 | 1 | 15 | yes | — |
| 280 | C2×C4 | C23×SD16 | 1 | 30 | yes | — |
| 281 | C2×C4 | C24×D4 | 1 | 1 | yes | characteristic |
| 282 | C8 | C23×C22⋊C4 | 1 | 30 | yes | — |
| 283 | C8 | C25×C4 | 1 | 1 | yes | characteristic |
| 284 | C8 | C24×D4 | 1 | 1 | yes | characteristic |
| 285 | C22 | — | 4 | 15 | no | — |
| 286 | C22 | — | 4 | 60 | no | — |
| 287 | C22 | — | 4 | 16 | no | — |
| 288 | C22 | — | 16 | 120 | no | — |
| 289 | C22 | C23×#102 | 1 | 30 | yes | — |
| 290 | C22 | C24×D8 | 1 | 1 | yes | characteristic |
| 291 | C4 | — | 4 | 16 | no | — |
| 292 | C4 | — | 8 | 6 | no | — |
| 293 | C4 | — | 8 | 10 | no | — |
| 294 | C4 | C23×#102 | 1 | 30 | yes | — |
| 295 | C4 | C24×C22⋊C4 | 1 | 1 | yes | characteristic |
| 296 | C4 | C24×SD16 | 1 | 1 | yes | characteristic |
| 297 | C2 | — | 2 | 1 | no | — |
| 298 | C2 | — | 2 | 30 | no | — |
| 299 | C2 | — | 8 | 16 | no | — |
| 300 | C2 | C24×#102 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C24×#102 |
| commutator | C8 | C25×C4 |
| Frattini, cosocle | C2×C8 | C26 |
| prime radical | #10587001 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#585 ◁ C2×#3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10585848 | C2 | semidirect |
| 3. #10574096 | C2 | semidirect |
| 4. #7622507 | C2 | semidirect |
| 5. C2×#33388 | C2 | semidirect |
| 7. #63093 | C4 | semidirect |
| 9. #63091 | C4 | semidirect |
| 15. C2×#3407 | C22 | semidirect |
| 17. #60108 | C22 | semidirect |
| 18. C2×#2912 | C22 | semidirect |
| 19. #33388 | C22 | semidirect |
| 20. #33388 | C22 | semidirect |
| 21. C2×#2145 | C22 | semidirect |
| 22. C22×#360 | C22 | semidirect |
| 34. #3584 | C2×C4 | semidirect |
| 36. C2×#578 | C2×C4 | semidirect |
| 38. C2×#576 | C2×C4 | semidirect |
| 50. #3407 | C23 | semidirect |
| 52. C22×M32 | C23 | semidirect |
| 55. #2912 | C23 | semidirect |
| 56. #2912 | C23 | semidirect |
| 59. C2×#427 | C23 | semidirect |
| 60. C22×#106 | C23 | semidirect |
| 61. #2145 | C23 | semidirect |
| 62. C2×#360 | C23 | semidirect |
| 63. C2×#360 | C23 | semidirect |
| 84. #578 | C22×C4 | semidirect |
| 88. #576 | C22×C4 | semidirect |
| 95. C2×#135 | C22×C4 | semidirect |
| 97. C22×Q16 | C22×C4 | semidirect |
| 99. C22×D8 | C22×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10587001 | 2 |
| #10585848 | 2 |
| #10574096 | 2 |
| #7622507 | 2 |
| C2×#33388 | 2 |