Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
10698 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #87572 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #87299 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | #87297 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #87296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#5838 | C2 | 1 | 15 | yes | maximal, complemented |
| 7 | #87287 | C2 | 1 | 15 | yes | maximal, complemented |
| 8 | #63093 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#585 | — | 2 | 10 | no | — |
| 10 | C3×#585 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#585 | C22 | 1 | 15 | yes | complemented |
| 12 | C2×C3×2−1+4 | — | 2 | 6 | no | — |
| 13 | C2×C3×2−1+4 | C22 | 1 | 15 | yes | complemented |
| 14 | D3×2−1+4 | — | 2 | 6 | no | — |
| 15 | #5966 | — | 2 | 10 | no | — |
| 16 | #5964 | C22 | 1 | 15 | yes | complemented |
| 17 | C2×#1010 | C22 | 1 | 15 | yes | complemented |
| 18 | #5838 | C22 | 1 | 60 | yes | complemented |
| 19 | #5838 | C22 | 1 | 120 | yes | complemented |
| 20 | #5837 | C22 | 1 | 60 | yes | complemented |
| 21 | #5836 | C22 | 1 | 60 | yes | complemented |
| 22 | #5835 | C22 | 1 | 20 | yes | complemented |
| 23 | C2×#951 | C22 | 1 | 45 | yes | complemented |
| 24 | #5822 | C22 | 1 | 90 | yes | complemented |
| 25 | C2×#950 | C22 | 1 | 45 | yes | complemented |
| 26 | #5819 | C22 | 1 | 15 | yes | complemented |
| 27 | C2×#948 | C22 | 1 | 15 | yes | complemented |
| 28 | C22×#216 | C22 | 1 | 15 | yes | complemented |
| 29 | C2×#942 | C22 | 1 | 45 | yes | complemented |
| 30 | 2−1+6 | — | 3 | 1 | no | — |
| 31 | 2−1+6 | D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | #3586 | — | 3 | 15 | no | — |
| 33 | #3586 | — | 3 | 15 | no | — |
| 34 | #3584 | — | 3 | 15 | no | — |
| 35 | C2×#578 | — | 3 | 15 | no | — |
| 36 | #3573 | — | 3 | 1 | no | — |
| 37 | C3×2−1+4 | — | 2 | 60 | no | — |
| 38 | C3×2−1+4 | — | 2 | 15 | no | — |
| 39 | C3×2−1+4 | D4 | 1 | 6 | yes | — |
| 40 | C3×2−1+4 | C23 | 1 | 60 | yes | — |
| 41 | C3×H(5,𝔽2) | — | 2 | 45 | no | — |
| 42 | C3×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 43 | C3×H(5,𝔽2) | C23 | 1 | 20 | yes | — |
| 44 | C2×C3×#41 | — | 2 | 60 | no | — |
| 45 | C2×C3×#41 | — | 2 | 45 | no | — |
| 46 | C2×C3×#41 | C23 | 1 | 45 | yes | — |
| 47 | C2×C3×#41 | C23 | 1 | 15 | yes | — |
| 48 | C22×C3×Q8 | — | 2 | 15 | no | — |
| 49 | C22×C3×Q8 | C23 | 1 | 15 | yes | — |
| 50 | #1010 | — | 2 | 60 | no | — |
| 51 | #1010 | C23 | 1 | 60 | yes | — |
| 52 | #1009 | C23 | 1 | 20 | yes | — |
| 53 | D3×#41 | — | 2 | 60 | no | — |
| 54 | #1007 | — | 2 | 15 | no | — |
| 55 | C2×D3×Q8 | — | 2 | 15 | no | — |
| 56 | #1004 | — | 2 | 45 | no | — |
| 57 | C2×#237 | — | 2 | 45 | no | — |
| 58 | C2×#235 | C23 | 1 | 45 | yes | — |
| 59 | C22×Dic6 | C23 | 1 | 15 | yes | — |
| 60 | #951 | C23 | 1 | 180 | yes | — |
| 61 | #950 | C23 | 1 | 60 | yes | — |
| 62 | #950 | C23 | 1 | 180 | yes | — |
| 63 | #949 | C23 | 1 | 60 | yes | — |
| 64 | #948 | C23 | 1 | 20 | yes | — |
| 65 | #948 | C23 | 1 | 60 | yes | — |
| 66 | C2×#216 | C23 | 1 | 90 | yes | — |
| 67 | C2×#216 | C23 | 1 | 45 | yes | — |
| 68 | #942 | C23 | 1 | 180 | yes | — |
| 69 | C2×#215 | C23 | 1 | 45 | yes | — |
| 70 | C2×#214 | C23 | 1 | 45 | yes | — |
| 71 | #932 | C23 | 1 | 60 | yes | — |
| 72 | C2×#213 | C23 | 1 | 15 | yes | — |
| 73 | C2×#208 | C23 | 1 | 45 | yes | — |
| 74 | C22×C3⋊C8 | C23 | 1 | 15 | yes | — |
| 75 | #585 | — | 2 | 10 | no | — |
| 76 | #585 | — | 3 | 15 | no | — |
| 77 | #585 | — | 6 | 10 | no | — |
| 78 | #585 | C2×D3 | 1 | 1 | yes | characteristic |
| 79 | #585 | C2×D3 | 1 | 15 | yes | complemented |
| 80 | C2×2−1+4 | — | 2 | 6 | no | — |
| 81 | C2×2−1+4 | — | 3 | 15 | no | — |
| 82 | C2×2−1+4 | — | 6 | 6 | no | — |
| 83 | C2×2−1+4 | C2×D3 | 1 | 15 | yes | complemented |
| 84 | #578 | — | 3 | 60 | no | — |
| 85 | #578 | — | 3 | 120 | no | — |
| 86 | #577 | — | 3 | 60 | no | — |
| 87 | #577 | — | 3 | 60 | no | — |
| 88 | #576 | — | 3 | 20 | no | — |
| 89 | #575 | — | 3 | 90 | no | — |
| 90 | C2×#137 | — | 3 | 45 | no | — |
| 91 | C2×#137 | — | 3 | 45 | no | — |
| 92 | C2×#135 | — | 3 | 45 | no | — |
| 93 | C22×Q16 | — | 3 | 15 | no | — |
| 94 | #568 | — | 3 | 15 | no | — |
| 95 | C2×#131 | — | 3 | 15 | no | — |
| 96 | C3×#41 | — | 2 | 20 | no | — |
| 97 | C3×#41 | — | 2 | 180 | no | — |
| 98 | C3×#41 | — | 2 | 90 | no | — |
| 99 | C3×#41 | C2×D4 | 1 | 60 | yes | — |
| 100 | C3×#41 | C24 | 1 | 60 | yes | — |
| 101 | C3×#41 | C24 | 1 | 20 | yes | — |
| 102 | C2×C3×Q8 | — | 2 | 60 | no | — |
| 103 | C2×C3×Q8 | — | 2 | 45 | no | — |
| 104 | C2×C3×Q8 | C2×D4 | 1 | 15 | yes | — |
| 105 | C2×C3×Q8 | C24 | 1 | 45 | yes | — |
| 106 | C2×C3×D4 | — | 2 | 45 | no | — |
| 107 | C2×C3×D4 | — | 2 | 60 | no | — |
| 108 | C2×C3×D4 | C2×D4 | 1 | 45 | yes | — |
| 109 | C2×C3×D4 | C24 | 1 | 15 | yes | — |
| 110 | C22×C3×C4 | — | 2 | 15 | no | — |
| 111 | C22×C3×C4 | — | 2 | 45 | no | — |
| 112 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 113 | C2×#67 | — | 2 | 45 | no | — |
| 114 | C22×Dic3 | — | 2 | 15 | no | — |
| 115 | #239 | — | 2 | 20 | no | — |
| 116 | D3×Q8 | — | 2 | 60 | no | — |
| 117 | #237 | — | 2 | 180 | no | — |
| 118 | D3×D4 | — | 2 | 60 | no | — |
| 119 | #235 | — | 2 | 90 | no | — |
| 120 | #235 | C24 | 1 | 60 | yes | — |
| 121 | C2×D12 | C24 | 1 | 15 | yes | — |
| 122 | C2×C4×D3 | — | 2 | 45 | no | — |
| 123 | C2×Dic6 | — | 2 | 45 | no | — |
| 124 | C2×Dic6 | C24 | 1 | 45 | yes | — |
| 125 | #216 | C24 | 1 | 90 | yes | — |
| 126 | #216 | C24 | 1 | 10 | yes | — |
| 127 | #215 | C24 | 1 | 60 | yes | — |
| 128 | #214 | C24 | 1 | 60 | yes | — |
| 129 | #213 | C24 | 1 | 6 | yes | — |
| 130 | #213 | C24 | 1 | 30 | yes | — |
| 131 | #208 | C24 | 1 | 60 | yes | — |
| 132 | C2×C3⋊C8 | C24 | 1 | 15 | yes | — |
| 133 | C2×C3⋊C8 | C24 | 1 | 45 | yes | — |
| 134 | 2−1+4 | — | 2 | 60 | no | — |
| 135 | 2−1+4 | — | 2 | 15 | no | — |
| 136 | 2−1+4 | — | 3 | 60 | no | — |
| 137 | 2−1+4 | — | 6 | 60 | no | — |
| 138 | 2−1+4 | — | 6 | 15 | no | — |
| 139 | 2−1+4 | #67 | 1 | 6 | yes | — |
| 140 | 2−1+4 | C22×D3 | 1 | 60 | yes | — |
| 141 | H(5,𝔽2) | — | 2 | 45 | no | — |
| 142 | H(5,𝔽2) | — | 3 | 20 | no | — |
| 143 | H(5,𝔽2) | — | 6 | 45 | no | — |
| 144 | H(5,𝔽2) | #67 | 1 | 10 | yes | — |
| 145 | H(5,𝔽2) | C22×D3 | 1 | 20 | yes | — |
| 146 | C2×#41 | — | 2 | 60 | no | — |
| 147 | C2×#41 | — | 2 | 45 | no | — |
| 148 | C2×#41 | — | 3 | 45 | no | — |
| 149 | C2×#41 | — | 6 | 60 | no | — |
| 150 | C2×#41 | — | 6 | 45 | no | — |
| 151 | C2×#41 | C22×D3 | 1 | 45 | yes | — |
| 152 | C2×#41 | C22×D3 | 1 | 15 | yes | — |
| 153 | C22×Q8 | — | 2 | 15 | no | — |
| 154 | C22×Q8 | — | 3 | 15 | no | — |
| 155 | C22×Q8 | — | 6 | 15 | no | — |
| 156 | C22×Q8 | C22×D3 | 1 | 15 | yes | — |
| 157 | #137 | — | 3 | 180 | no | — |
| 158 | #137 | — | 3 | 180 | no | — |
| 159 | AGL(1,ℤ/8) | — | 3 | 60 | no | — |
| 160 | AGL(1,ℤ/8) | — | 3 | 60 | no | — |
| 161 | #135 | — | 3 | 60 | no | — |
| 162 | #135 | — | 3 | 180 | no | — |
| 163 | C2×Q16 | — | 3 | 45 | no | — |
| 164 | C2×Q16 | — | 3 | 90 | no | — |
| 165 | C2×SD16 | — | 3 | 45 | no | — |
| 166 | C2×SD16 | — | 3 | 45 | no | — |
| 167 | C2×D8 | — | 3 | 15 | no | — |
| 168 | #131 | — | 3 | 20 | no | — |
| 169 | #131 | — | 3 | 60 | no | — |
| 170 | C2×M16 | — | 3 | 45 | no | — |
| 171 | C22×C8 | — | 3 | 15 | no | — |
| 172 | C23×C3 | — | 2 | 15 | no | — |
| 173 | C23×C3 | C22×D4 | 1 | 15 | yes | — |
| 174 | C22×D3 | — | 2 | 15 | no | — |
| 175 | C3×Q8 | — | 2 | 45 | no | — |
| 176 | C3×Q8 | C22×D4 | 1 | 20 | yes | — |
| 177 | C3×Q8 | C25 | 1 | 10 | yes | — |
| 178 | C3×D4 | — | 2 | 60 | no | — |
| 179 | C3×D4 | — | 2 | 15 | no | — |
| 180 | C3×D4 | C22×D4 | 1 | 60 | yes | — |
| 181 | C3×D4 | C25 | 1 | 6 | yes | — |
| 182 | C2×C3×C4 | — | 2 | 45 | no | — |
| 183 | C2×C3×C4 | — | 2 | 60 | no | — |
| 184 | C2×C3×C4 | C22×D4 | 1 | 45 | yes | — |
| 185 | C2×C3×C4 | C25 | 1 | 15 | yes | — |
| 186 | #67 | — | 2 | 60 | no | — |
| 187 | C2×Dic3 | — | 2 | 45 | no | — |
| 188 | D12 | — | 2 | 15 | no | — |
| 189 | D12 | C25 | 1 | 6 | yes | — |
| 190 | C4×D3 | — | 2 | 60 | no | — |
| 191 | Dic6 | — | 2 | 45 | no | — |
| 192 | Dic6 | C25 | 1 | 10 | yes | — |
| 193 | C3⋊C8 | C25 | 1 | 1 | yes | characteristic |
| 194 | C3⋊C8 | C25 | 1 | 15 | yes | — |
| 195 | #41 | — | 2 | 20 | no | — |
| 196 | #41 | — | 2 | 180 | no | — |
| 197 | #41 | — | 2 | 90 | no | — |
| 198 | #41 | — | 3 | 60 | no | — |
| 199 | #41 | — | 6 | 20 | no | — |
| 200 | #41 | — | 6 | 180 | no | — |
| 201 | #41 | — | 6 | 90 | no | — |
| 202 | #41 | C2×#67 | 1 | 60 | yes | — |
| 203 | #41 | C23×D3 | 1 | 60 | yes | — |
| 204 | #41 | C23×D3 | 1 | 20 | yes | — |
| 205 | C2×Q8 | — | 2 | 60 | no | — |
| 206 | C2×Q8 | — | 2 | 45 | no | — |
| 207 | C2×Q8 | — | 3 | 45 | no | — |
| 208 | C2×Q8 | — | 6 | 60 | no | — |
| 209 | C2×Q8 | — | 6 | 45 | no | — |
| 210 | C2×Q8 | C2×#67 | 1 | 15 | yes | — |
| 211 | C2×Q8 | C23×D3 | 1 | 45 | yes | — |
| 212 | C2×D4 | — | 2 | 45 | no | — |
| 213 | C2×D4 | — | 2 | 60 | no | — |
| 214 | C2×D4 | — | 3 | 15 | no | — |
| 215 | C2×D4 | — | 6 | 45 | no | — |
| 216 | C2×D4 | — | 6 | 60 | no | — |
| 217 | C2×D4 | C2×#67 | 1 | 45 | yes | — |
| 218 | C2×D4 | C23×D3 | 1 | 15 | yes | — |
| 219 | C22×C4 | — | 2 | 15 | no | — |
| 220 | C22×C4 | — | 2 | 45 | no | — |
| 221 | C22×C4 | — | 6 | 15 | no | — |
| 222 | C22×C4 | — | 6 | 45 | no | — |
| 223 | C22×C4 | C23×D3 | 1 | 15 | yes | — |
| 224 | Q16 | — | 3 | 10 | no | — |
| 225 | Q16 | — | 3 | 90 | no | — |
| 226 | SD16 | — | 3 | 60 | no | — |
| 227 | SD16 | — | 3 | 60 | no | — |
| 228 | D8 | — | 3 | 6 | no | — |
| 229 | D8 | — | 3 | 30 | no | — |
| 230 | M16 | — | 3 | 60 | no | — |
| 231 | C2×C8 | — | 3 | 15 | no | — |
| 232 | C2×C8 | — | 3 | 45 | no | — |
| 233 | C22×C3 | — | 2 | 6 | no | — |
| 234 | C22×C3 | — | 4 | 15 | no | — |
| 235 | C22×C3 | — | 8 | 15 | no | — |
| 236 | C22×C3 | C23×D4 | 1 | 15 | yes | — |
| 237 | C2×D3 | — | 2 | 6 | no | — |
| 238 | C2×D3 | — | 8 | 15 | no | — |
| 239 | C3×C4 | — | 2 | 10 | no | — |
| 240 | C3×C4 | C23×D4 | 1 | 15 | yes | — |
| 241 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 242 | Dic3 | — | 2 | 10 | no | — |
| 243 | C23 | — | 2 | 15 | no | — |
| 244 | C23 | — | 6 | 15 | no | — |
| 245 | C23 | C22×#67 | 1 | 15 | yes | — |
| 246 | Q8 | — | 2 | 45 | no | — |
| 247 | Q8 | — | 3 | 10 | no | — |
| 248 | Q8 | — | 6 | 45 | no | — |
| 249 | Q8 | C22×#67 | 1 | 20 | yes | — |
| 250 | Q8 | C24×D3 | 1 | 10 | yes | — |
| 251 | D4 | — | 2 | 60 | no | — |
| 252 | D4 | — | 2 | 15 | no | — |
| 253 | D4 | — | 3 | 6 | no | — |
| 254 | D4 | — | 6 | 60 | no | — |
| 255 | D4 | — | 6 | 15 | no | — |
| 256 | D4 | C22×#67 | 1 | 60 | yes | — |
| 257 | D4 | C24×D3 | 1 | 6 | yes | — |
| 258 | C2×C4 | — | 2 | 45 | no | — |
| 259 | C2×C4 | — | 2 | 60 | no | — |
| 260 | C2×C4 | — | 6 | 45 | no | — |
| 261 | C2×C4 | — | 6 | 60 | no | — |
| 262 | C2×C4 | C22×#67 | 1 | 45 | yes | — |
| 263 | C2×C4 | C24×D3 | 1 | 15 | yes | — |
| 264 | C8 | — | 3 | 1 | no | — |
| 265 | C8 | — | 3 | 15 | no | — |
| 266 | C2×C3 | — | 2 | 15 | no | — |
| 267 | C2×C3 | — | 4 | 6 | no | — |
| 268 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 269 | D3 | — | 4 | 6 | no | — |
| 270 | C22 | — | 2 | 6 | no | — |
| 271 | C22 | — | 4 | 15 | no | — |
| 272 | C22 | — | 6 | 6 | no | — |
| 273 | C22 | — | 8 | 15 | no | — |
| 274 | C22 | — | 24 | 15 | no | — |
| 275 | C22 | C23×#67 | 1 | 15 | yes | — |
| 276 | C4 | — | 2 | 10 | no | — |
| 277 | C4 | — | 6 | 10 | no | — |
| 278 | C4 | C23×#67 | 1 | 15 | yes | — |
| 279 | C4 | C25×D3 | 1 | 1 | yes | characteristic |
| 280 | C3 | #63093 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 281 | C2 | — | 2 | 15 | no | — |
| 282 | C2 | — | 4 | 6 | no | — |
| 283 | C2 | — | 12 | 6 | no | — |
| 284 | C2 | C24×#67 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×#67 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×2−1+6 | C2 |
| socle | C2×C3 | C24×D4 |
| hypercenter | 2−1+6 | D3 |
| hypocenter | C3 | #63093 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ 2−1+6 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×2−1+6 | C2 | semidirect |
| 2. #87572 | C2 | semidirect |
| 3. #87299 | C2 | semidirect |
| 4. #87297 | C2 | semidirect |
| 5. #87296 | C2 | semidirect |
| 6. C2×#5838 | C2 | semidirect |
| 7. #87287 | C2 | semidirect |
| 11. C3×#585 | C22 | semidirect |
| 13. C2×C3×2−1+4 | C22 | semidirect |
| 16. #5964 | C22 | semidirect |
| 17. C2×#1010 | C22 | semidirect |
| 18. #5838 | C22 | semidirect |
| 19. #5838 | C22 | semidirect |
| 20. #5837 | C22 | semidirect |
| 21. #5836 | C22 | semidirect |
| 22. #5835 | C22 | semidirect |
| 23. C2×#951 | C22 | semidirect |
| 24. #5822 | C22 | semidirect |
| 25. C2×#950 | C22 | semidirect |
| 26. #5819 | C22 | semidirect |
| 27. C2×#948 | C22 | semidirect |
| 28. C22×#216 | C22 | semidirect |
| 29. C2×#942 | C22 | semidirect |
| 31. 2−1+6 | D3 | semidirect |
| 79. #585 | C2×D3 | semidirect |
| 83. C2×2−1+4 | C2×D3 | semidirect |
| 280. C3 | #63093 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×2−1+6 | 2 |
| #87572 | 2 |
| #87299 | 2 |
| #87297 | 2 |
| #87296 | 2 |
| C2×#5838 | 2 |
| #87287 | 2 |
| #63093 | 3 |