Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1306 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 251 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 | #87296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #84125 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #78656 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C3×#1791 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #70136 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #70133 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #70002 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #21484 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#585 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5828 | C22 | 1 | 3 | yes | — |
| 11 | #5819 | — | 2 | 6 | no | — |
| 12 | #5819 | C22 | 1 | 3 | yes | — |
| 13 | C2×#948 | — | 2 | 6 | no | — |
| 14 | C2×#948 | C22 | 1 | 3 | yes | — |
| 15 | C22×ΓL(1,𝔽25) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×#413 | C22 | 1 | 3 | yes | — |
| 17 | C3×#408 | C22 | 1 | 3 | yes | — |
| 18 | #5244 | C4 | 1 | 4 | yes | — |
| 19 | #5243 | C4 | 1 | 4 | yes | — |
| 20 | #5240 | C22 | 1 | 3 | yes | — |
| 21 | #5126 | C22 | 1 | 3 | yes | complemented |
| 22 | #5122 | C22 | 1 | 3 | yes | complemented |
| 23 | #5102 | C22 | 1 | 3 | yes | complemented |
| 24 | #4730 | — | 2 | 6 | no | — |
| 25 | #4729 | — | 2 | 6 | no | — |
| 26 | #4559 | C22 | 1 | 6 | yes | complemented |
| 27 | #3573 | — | 3 | 1 | no | — |
| 28 | #3025 | — | 3 | 1 | no | — |
| 29 | #2930 | — | 3 | 3 | no | — |
| 30 | #1891 | — | 3 | 3 | no | — |
| 31 | #1868 | — | 3 | 3 | no | — |
| 32 | #1791 | D3 | 1 | 1 | yes | characteristic, complemented |
| 33 | #1757 | — | 3 | 3 | no | — |
| 34 | C3×2−1+4 | — | 2 | 3 | no | — |
| 35 | C3×H(5,𝔽2) | — | 2 | 3 | no | — |
| 36 | C3×H(5,𝔽2) | C2×C4 | 1 | 4 | yes | — |
| 37 | C2×C3×#41 | — | 2 | 6 | no | — |
| 38 | C2×C3×#41 | C23 | 1 | 3 | yes | — |
| 39 | C22×C4×D3 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C3×M16 | C23 | 1 | 3 | yes | — |
| 41 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | #948 | — | 2 | 4 | no | — |
| 43 | #948 | — | 2 | 12 | no | — |
| 44 | #948 | — | 2 | 6 | no | — |
| 45 | #948 | — | 2 | 12 | no | — |
| 46 | #948 | — | 2 | 6 | no | — |
| 47 | #946 | — | 2 | 3 | no | — |
| 48 | #938 | — | 2 | 3 | no | — |
| 49 | #937 | C2×C4 | 1 | 6 | yes | — |
| 50 | #935 | C2×C4 | 1 | 6 | yes | — |
| 51 | C4×#67 | — | 2 | 6 | no | — |
| 52 | #926 | C23 | 1 | 3 | yes | — |
| 53 | C2×#208 | — | 2 | 12 | no | — |
| 54 | C2×#208 | — | 2 | 6 | no | — |
| 55 | C2×#208 | D4 | 1 | 6 | yes | complemented |
| 56 | C2×#208 | C23 | 1 | 3 | yes | — |
| 57 | C22×C3⋊C8 | — | 2 | 4 | no | — |
| 58 | C22×C3⋊C8 | D4 | 1 | 6 | yes | complemented |
| 59 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×ΓL(1,𝔽25) | — | 2 | 3 | no | — |
| 61 | C2×ΓL(1,𝔽25) | — | 2 | 3 | no | — |
| 62 | C3×#101 | — | 2 | 3 | no | — |
| 63 | C3×#100 | — | 2 | 3 | no | — |
| 64 | C3×C22⋊C8 | — | 2 | 6 | no | — |
| 65 | #834 | C2×C4 | 1 | 12 | yes | — |
| 66 | (C2×D3)⋊C8 | — | 2 | 6 | no | — |
| 67 | #815 | — | 2 | 3 | no | — |
| 68 | Dic3⋊C8 | — | 2 | 6 | no | — |
| 69 | C8×Dic3 | — | 2 | 3 | no | — |
| 70 | #585 | C2×D3 | 1 | 1 | yes | characteristic |
| 71 | #568 | — | 3 | 3 | no | — |
| 72 | #568 | — | 6 | 6 | no | — |
| 73 | C2×#131 | — | 3 | 3 | no | — |
| 74 | C2×#131 | — | 6 | 6 | no | — |
| 75 | C22×M16 | — | 3 | 1 | no | — |
| 76 | #525 | — | 3 | 3 | no | — |
| 77 | #458 | — | 3 | 4 | no | — |
| 78 | B(4,𝔽2) | — | 3 | 4 | no | — |
| 79 | #436 | — | 6 | 6 | no | — |
| 80 | #435 | — | 6 | 6 | no | — |
| 81 | #424 | — | 3 | 3 | no | — |
| 82 | #413 | C2×D3 | 1 | 3 | yes | — |
| 83 | #410 | — | 3 | 3 | no | — |
| 84 | #408 | C2×D3 | 1 | 3 | yes | — |
| 85 | #407 | — | 3 | 3 | no | — |
| 86 | #405 | — | 3 | 3 | no | — |
| 87 | #344 | — | 3 | 6 | no | — |
| 88 | C23×D3 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C3×#41 | — | 2 | 4 | no | — |
| 90 | C3×#41 | — | 2 | 12 | no | — |
| 91 | C3×#41 | — | 2 | 12 | no | — |
| 92 | C3×#41 | — | 2 | 6 | no | — |
| 93 | C3×#41 | — | 2 | 6 | no | — |
| 94 | C2×C3×Q8 | — | 2 | 6 | no | — |
| 95 | C2×C3×Q8 | #41 | 1 | 3 | yes | — |
| 96 | C2×C3×D4 | — | 2 | 6 | no | — |
| 97 | C2×C3×D4 | — | 2 | 12 | no | — |
| 98 | C2×C3×D4 | C22×C4 | 1 | 6 | yes | — |
| 99 | C2×C3×D4 | #41 | 1 | 3 | yes | — |
| 100 | C22×C3×C4 | — | 2 | 4 | no | — |
| 101 | C22×C3×C4 | C2×D4 | 1 | 6 | yes | — |
| 102 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 103 | C2×#67 | — | 2 | 6 | no | — |
| 104 | C22×Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×D3 | — | 2 | 3 | no | — |
| 106 | C2×C4×D3 | — | 2 | 3 | no | — |
| 107 | C3×M16 | — | 2 | 6 | no | — |
| 108 | C2×C3×C8 | — | 2 | 3 | no | — |
| 109 | C2×C3×C8 | — | 2 | 3 | no | — |
| 110 | #217 | — | 2 | 6 | no | — |
| 111 | #217 | C22×C4 | 1 | 6 | yes | — |
| 112 | (C2×D3)⋊C4 | — | 2 | 6 | no | — |
| 113 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 114 | Dic3⋊C4 | — | 2 | 6 | no | — |
| 115 | C4×Dic3 | — | 2 | 3 | no | — |
| 116 | #208 | — | 2 | 12 | no | — |
| 117 | #208 | — | 2 | 6 | no | — |
| 118 | #208 | — | 2 | 12 | no | — |
| 119 | C2×C3⋊C8 | — | 2 | 12 | no | — |
| 120 | C2×C3⋊C8 | — | 2 | 6 | no | — |
| 121 | C2×C3⋊C8 | — | 2 | 6 | no | — |
| 122 | C2×C3⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 123 | C2×C3⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 124 | C2×C3⋊C8 | C2×D4 | 1 | 3 | yes | — |
| 125 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 126 | ΓL(1,𝔽25) | — | 4 | 3 | no | — |
| 127 | 2−1+4 | — | 2 | 3 | no | — |
| 128 | H(5,𝔽2) | — | 2 | 3 | no | — |
| 129 | H(5,𝔽2) | C4×D3 | 1 | 4 | yes | — |
| 130 | C2×#41 | — | 2 | 6 | no | — |
| 131 | C2×#41 | C22×D3 | 1 | 3 | yes | — |
| 132 | C23×C4 | — | 3 | 1 | no | — |
| 133 | #131 | — | 6 | 6 | no | — |
| 134 | #131 | — | 6 | 6 | no | — |
| 135 | #131 | — | 6 | 12 | no | — |
| 136 | #131 | — | 6 | 4 | no | — |
| 137 | #131 | — | 6 | 12 | no | — |
| 138 | C2×M16 | — | 3 | 6 | no | — |
| 139 | C2×M16 | — | 3 | 3 | no | — |
| 140 | C2×M16 | — | 6 | 3 | no | — |
| 141 | C2×M16 | — | 6 | 3 | no | — |
| 142 | C2×M16 | — | 6 | 12 | no | — |
| 143 | C2×M16 | — | 6 | 6 | no | — |
| 144 | C2×M16 | C22×D3 | 1 | 3 | yes | — |
| 145 | C22×C8 | — | 3 | 1 | no | — |
| 146 | C22×C8 | — | 3 | 6 | no | — |
| 147 | C22×C8 | — | 6 | 4 | no | — |
| 148 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 149 | #123 | — | 3 | 6 | no | — |
| 150 | C22⋊Q8 | — | 6 | 3 | no | — |
| 151 | #121 | — | 6 | 3 | no | — |
| 152 | #120 | — | 3 | 6 | no | — |
| 153 | C4×D4 | — | 6 | 6 | no | — |
| 154 | #117 | — | 3 | 3 | no | — |
| 155 | C4⋊C8 | — | 6 | 6 | no | — |
| 156 | #101 | — | 2 | 3 | no | — |
| 157 | #100 | — | 2 | 3 | no | — |
| 158 | #99 | — | 3 | 12 | no | — |
| 159 | C22⋊C8 | — | 2 | 6 | no | — |
| 160 | C22⋊C8 | — | 6 | 6 | no | — |
| 161 | #97 | — | 6 | 3 | no | — |
| 162 | C4×C8 | — | 6 | 3 | no | — |
| 163 | C23×C3 | — | 2 | 4 | no | — |
| 164 | C23×C3 | C4×D4 | 1 | 6 | yes | — |
| 165 | C23×C3 | C23×C4 | 1 | 1 | yes | characteristic |
| 166 | C22×D3 | — | 2 | 3 | no | — |
| 167 | C22×D3 | — | 8 | 1 | no | — |
| 168 | C3×Q8 | — | 2 | 4 | no | — |
| 169 | C3×Q8 | — | 2 | 6 | no | — |
| 170 | C3×D4 | — | 2 | 12 | no | — |
| 171 | C3×D4 | — | 2 | 12 | no | — |
| 172 | C3×D4 | — | 2 | 6 | no | — |
| 173 | C2×C3×C4 | — | 2 | 12 | no | — |
| 174 | C2×C3×C4 | — | 2 | 6 | no | — |
| 175 | C2×C3×C4 | — | 2 | 6 | no | — |
| 176 | C2×C3×C4 | C4×D4 | 1 | 6 | yes | — |
| 177 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 178 | C2×C3×C4 | C2×#41 | 1 | 3 | yes | — |
| 179 | #67 | — | 8 | 6 | no | — |
| 180 | C2×Dic3 | — | 2 | 6 | no | — |
| 181 | C2×Dic3 | — | 2 | 3 | no | — |
| 182 | C4×D3 | — | 4 | 1 | no | — |
| 183 | C4×D3 | — | 4 | 3 | no | — |
| 184 | C3×C8 | — | 4 | 1 | no | — |
| 185 | C3×C8 | — | 4 | 3 | no | — |
| 186 | C3⋊C8 | — | 2 | 6 | no | — |
| 187 | C3⋊C8 | #120 | 1 | 1 | yes | characteristic |
| 188 | C3⋊C8 | #120 | 1 | 3 | yes | — |
| 189 | C24 | — | 3 | 1 | no | — |
| 190 | #41 | — | 2 | 4 | no | — |
| 191 | #41 | — | 2 | 12 | no | — |
| 192 | #41 | — | 2 | 12 | no | — |
| 193 | #41 | — | 2 | 6 | no | — |
| 194 | #41 | — | 2 | 6 | no | — |
| 195 | C2×Q8 | — | 2 | 6 | no | — |
| 196 | C2×Q8 | #237 | 1 | 3 | yes | — |
| 197 | C2×D4 | — | 2 | 6 | no | — |
| 198 | C2×D4 | — | 2 | 12 | no | — |
| 199 | C2×D4 | — | 6 | 6 | no | — |
| 200 | C2×D4 | C2×C4×D3 | 1 | 6 | yes | — |
| 201 | C2×D4 | #237 | 1 | 3 | yes | — |
| 202 | C22×C4 | — | 2 | 4 | no | — |
| 203 | C22×C4 | — | 3 | 1 | no | — |
| 204 | C22×C4 | — | 6 | 3 | no | — |
| 205 | C22×C4 | — | 6 | 3 | no | — |
| 206 | C22×C4 | D3×D4 | 1 | 6 | yes | — |
| 207 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 208 | M16 | — | 2 | 6 | no | — |
| 209 | M16 | — | 6 | 12 | no | — |
| 210 | M16 | — | 6 | 12 | no | — |
| 211 | M16 | — | 6 | 6 | no | — |
| 212 | M16 | — | 12 | 1 | no | — |
| 213 | M16 | — | 12 | 3 | no | — |
| 214 | C2×C8 | — | 2 | 3 | no | — |
| 215 | C2×C8 | — | 2 | 3 | no | — |
| 216 | C2×C8 | — | 3 | 3 | no | — |
| 217 | C2×C8 | — | 3 | 3 | no | — |
| 218 | C2×C8 | — | 3 | 6 | no | — |
| 219 | C2×C8 | — | 6 | 6 | no | — |
| 220 | C2×C8 | — | 6 | 6 | no | — |
| 221 | C2×C8 | — | 6 | 12 | no | — |
| 222 | C4⋊C4 | — | 6 | 6 | no | — |
| 223 | C4⋊C4 | — | 6 | 3 | no | — |
| 224 | C22⋊C4 | — | 3 | 6 | no | — |
| 225 | C22⋊C4 | — | 6 | 6 | no | — |
| 226 | C22⋊C4 | — | 6 | 6 | no | — |
| 227 | C42 | — | 6 | 3 | no | — |
| 228 | C22×C3 | — | 2 | 6 | no | — |
| 229 | C22×C3 | — | 4 | 1 | no | — |
| 230 | C22×C3 | — | 4 | 6 | no | — |
| 231 | C22×C3 | — | 8 | 4 | no | — |
| 232 | C22×C3 | C2×C4×D4 | 1 | 3 | yes | — |
| 233 | C2×D3 | — | 4 | 3 | no | — |
| 234 | C2×D3 | — | 4 | 3 | no | — |
| 235 | C2×D3 | — | 4 | 1 | no | — |
| 236 | C3×C4 | — | 2 | 6 | no | — |
| 237 | C3×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 238 | C3×C4 | #525 | 1 | 3 | yes | — |
| 239 | Dic3 | — | 4 | 6 | no | — |
| 240 | Dic3 | — | 4 | 1 | no | — |
| 241 | C23 | — | 2 | 4 | no | — |
| 242 | C23 | — | 6 | 3 | no | — |
| 243 | C23 | — | 24 | 1 | no | — |
| 244 | C23 | #67⋊C4 | 1 | 6 | yes | — |
| 245 | C23 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 246 | Q8 | — | 2 | 4 | no | — |
| 247 | Q8 | — | 2 | 6 | no | — |
| 248 | D4 | — | 2 | 12 | no | — |
| 249 | D4 | — | 2 | 12 | no | — |
| 250 | D4 | — | 2 | 6 | no | — |
| 251 | D4 | — | 24 | 6 | no | — |
| 252 | C2×C4 | — | 2 | 12 | no | — |
| 253 | C2×C4 | — | 2 | 6 | no | — |
| 254 | C2×C4 | — | 2 | 6 | no | — |
| 255 | C2×C4 | — | 6 | 6 | no | — |
| 256 | C2×C4 | — | 6 | 3 | no | — |
| 257 | C2×C4 | — | 12 | 3 | no | — |
| 258 | C2×C4 | — | 12 | 1 | no | — |
| 259 | C2×C4 | #67⋊C4 | 1 | 6 | yes | — |
| 260 | C2×C4 | C2×D3×D4 | 1 | 3 | yes | — |
| 261 | C2×C4 | C2×#237 | 1 | 3 | yes | — |
| 262 | C8 | — | 3 | 1 | no | — |
| 263 | C8 | — | 3 | 3 | no | — |
| 264 | C8 | — | 4 | 3 | no | — |
| 265 | C8 | — | 4 | 1 | no | — |
| 266 | C8 | — | 6 | 6 | no | — |
| 267 | C2×C3 | — | 2 | 3 | no | — |
| 268 | C2×C3 | — | 4 | 6 | no | — |
| 269 | C2×C3 | C4×#120 | 1 | 1 | yes | characteristic |
| 270 | D3 | — | 8 | 1 | no | — |
| 271 | C22 | — | 2 | 6 | no | — |
| 272 | C22 | — | 4 | 1 | no | — |
| 273 | C22 | — | 4 | 6 | no | — |
| 274 | C22 | — | 8 | 4 | no | — |
| 275 | C22 | — | 12 | 3 | no | — |
| 276 | C22 | — | 12 | 3 | no | — |
| 277 | C22 | — | 12 | 1 | no | — |
| 278 | C22 | C2×#67⋊C4 | 1 | 3 | yes | — |
| 279 | C4 | — | 2 | 6 | no | — |
| 280 | C4 | — | 12 | 6 | no | — |
| 281 | C4 | — | 12 | 1 | no | — |
| 282 | C4 | #5586 | 1 | 3 | yes | — |
| 283 | C4 | D3×#120 | 1 | 1 | yes | characteristic |
| 284 | C3 | #21484 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 285 | C2 | — | 2 | 3 | no | — |
| 286 | C2 | — | 4 | 6 | no | — |
| 287 | C2 | — | 24 | 1 | no | — |
| 288 | C2 | #73818 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #87296 | C2 | semidirect |
| 3. #78656 | C2 | semidirect |
| 4. C3×#1791 | C2 | semidirect |
| 5. #70136 | C2 | semidirect |
| 6. #70133 | C2 | semidirect |
| 7. #70002 | C2 | semidirect |
| 15. C22×ΓL(1,𝔽25) | C22 | semidirect |
| 21. #5126 | C22 | semidirect |
| 22. #5122 | C22 | semidirect |
| 23. #5102 | C22 | semidirect |
| 26. #4559 | C22 | semidirect |
| 32. #1791 | D3 | semidirect |
| 55. C2×#208 | D4 | semidirect |
| 58. C22×C3⋊C8 | D4 | semidirect |
| 284. C3 | #21484 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #87296 | 2 |
| #84125 | 2 |
| #78656 | 2 |
| C3×#1791 | 2 |
| #70136 | 2 |
| #70133 | 2 |
| #70002 | 2 |
| #21484 | 3 |