Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1338 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C2×#5923 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#568⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#5423 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #85453 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #85452 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #85448 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #60182 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | #5923 | C4 | 1 | 2 | yes | complemented |
| 10 | #5923 | C4 | 1 | 2 | yes | complemented |
| 11 | #5923 | C22 | 1 | 2 | yes | — |
| 12 | #5923 | C22 | 1 | 2 | yes | — |
| 13 | C2×#986 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×#985 | C22 | 1 | 2 | yes | — |
| 15 | C2×#984 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#983 | C4 | 1 | 2 | yes | complemented |
| 17 | #5745 | — | 4 | 1 | no | maximal |
| 18 | #568⋊C3 | C22 | 1 | 2 | yes | complemented |
| 19 | #568⋊C3 | C22 | 1 | 2 | yes | complemented |
| 20 | C2×#131⋊C3 | C22 | 1 | 2 | yes | complemented |
| 21 | #5423 | C22 | 1 | 2 | yes | complemented |
| 22 | #5423 | C22 | 1 | 2 | yes | complemented |
| 23 | C2×GU(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 24 | #5405 | C22 | 1 | 8 | yes | complemented |
| 25 | #5404 | C22 | 1 | 8 | yes | complemented |
| 26 | C2×#575 | — | 3 | 1 | no | — |
| 27 | C2×#568 | — | 3 | 1 | no | — |
| 28 | C2×#568 | D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | #2957 | — | 3 | 4 | no | — |
| 30 | #2957 | — | 3 | 4 | no | — |
| 31 | #2956 | — | 3 | 4 | no | — |
| 32 | #2956 | — | 3 | 4 | no | — |
| 33 | #2955 | — | 3 | 4 | no | — |
| 34 | #2955 | — | 3 | 4 | no | — |
| 35 | C2×#430 | — | 3 | 1 | no | — |
| 36 | C2×#421 | — | 3 | 1 | no | — |
| 37 | C2×#421 | — | 3 | 1 | no | — |
| 38 | #2873 | — | 3 | 1 | no | — |
| 39 | C2×#235 | — | 4 | 1 | no | — |
| 40 | C2×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 46 | C22×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #986 | C2×C4 | 1 | 2 | yes | — |
| 48 | #986 | C2×C4 | 1 | 2 | yes | complemented |
| 49 | #985 | C2×C4 | 1 | 4 | yes | complemented |
| 50 | #985 | C23 | 1 | 4 | yes | — |
| 51 | #984 | C2×C4 | 1 | 2 | yes | — |
| 52 | #984 | C2×C4 | 1 | 2 | yes | complemented |
| 53 | #983 | C2×C4 | 1 | 4 | yes | — |
| 54 | #983 | C2×C4 | 1 | 4 | yes | complemented |
| 55 | C2×GL(2,𝔽3) | C2×C4 | 1 | 2 | yes | complemented |
| 56 | C2×#226 | C2×C4 | 1 | 2 | yes | complemented |
| 57 | C2×C3×M16 | — | 4 | 1 | no | — |
| 58 | C2×#208 | — | 4 | 1 | no | — |
| 59 | #907 | — | 4 | 4 | no | — |
| 60 | D3×M16 | — | 4 | 4 | no | — |
| 61 | #901 | — | 4 | 4 | no | — |
| 62 | #131⋊C3 | C23 | 1 | 4 | yes | complemented |
| 63 | GU(2,𝔽3) | C23 | 1 | 4 | yes | complemented |
| 64 | C22×#41 | — | 3 | 1 | no | — |
| 65 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | #575 | — | 3 | 2 | no | — |
| 67 | #575 | — | 3 | 2 | no | — |
| 68 | #575 | — | 3 | 2 | no | — |
| 69 | #575 | — | 3 | 2 | no | — |
| 70 | #575 | — | 3 | 2 | no | — |
| 71 | #575 | — | 3 | 2 | no | — |
| 72 | #575 | — | 3 | 2 | no | — |
| 73 | #575 | — | 3 | 2 | no | — |
| 74 | C2×#137 | — | 3 | 1 | no | — |
| 75 | C2×#137 | — | 3 | 1 | no | — |
| 76 | C2×#137 | — | 3 | 2 | no | — |
| 77 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 78 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 79 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 80 | C2×#135 | — | 3 | 2 | no | — |
| 81 | C2×#135 | — | 3 | 2 | no | — |
| 82 | #568 | — | 3 | 2 | no | — |
| 83 | #568 | — | 3 | 2 | no | — |
| 84 | #568 | — | 3 | 4 | no | — |
| 85 | #568 | — | 3 | 4 | no | — |
| 86 | #568 | — | 6 | 1 | no | — |
| 87 | #568 | — | 6 | 2 | no | — |
| 88 | #568 | — | 6 | 1 | no | — |
| 89 | #568 | C2×D3 | 1 | 2 | yes | — |
| 90 | #568 | C2×D3 | 1 | 2 | yes | complemented |
| 91 | C2×#131 | — | 3 | 2 | no | — |
| 92 | C2×#131 | — | 3 | 4 | no | — |
| 93 | C2×#131 | — | 6 | 2 | no | — |
| 94 | C2×#131 | C2×D3 | 1 | 2 | yes | complemented |
| 95 | C22×M16 | — | 3 | 1 | no | — |
| 96 | C22×M16 | — | 3 | 1 | no | — |
| 97 | C22×M16 | — | 3 | 1 | no | — |
| 98 | C22×M16 | — | 3 | 1 | no | — |
| 99 | C22×M16 | — | 6 | 2 | no | — |
| 100 | C2×#117 | — | 3 | 1 | no | — |
| 101 | #444 | — | 3 | 8 | no | — |
| 102 | #444 | — | 3 | 8 | no | — |
| 103 | #444 | — | 3 | 8 | no | — |
| 104 | #444 | — | 3 | 8 | no | — |
| 105 | #443 | — | 3 | 8 | no | — |
| 106 | #443 | — | 3 | 8 | no | — |
| 107 | #443 | — | 3 | 8 | no | — |
| 108 | #443 | — | 3 | 8 | no | — |
| 109 | #430 | — | 3 | 4 | no | — |
| 110 | #430 | — | 3 | 4 | no | — |
| 111 | C2×#108 | — | 3 | 4 | no | — |
| 112 | #421 | — | 3 | 4 | no | — |
| 113 | #421 | — | 3 | 2 | no | — |
| 114 | #421 | — | 3 | 4 | no | — |
| 115 | #421 | — | 3 | 2 | no | — |
| 116 | #421 | — | 3 | 2 | no | — |
| 117 | #421 | — | 3 | 2 | no | — |
| 118 | C2×#104 | — | 3 | 4 | no | — |
| 119 | C2×#104 | — | 3 | 2 | no | — |
| 120 | C2×#104 | — | 3 | 2 | no | — |
| 121 | #405 | — | 3 | 4 | no | — |
| 122 | #405 | — | 3 | 4 | no | — |
| 123 | C4×M16 | — | 3 | 4 | no | — |
| 124 | C22×C3×C4 | — | 4 | 1 | no | — |
| 125 | C2×#67 | — | 4 | 2 | no | — |
| 126 | #235 | — | 4 | 2 | no | — |
| 127 | #235 | — | 4 | 2 | no | — |
| 128 | #235 | — | 4 | 2 | no | — |
| 129 | #235 | — | 4 | 2 | no | — |
| 130 | C2×D12 | — | 4 | 1 | no | — |
| 131 | C2×C4×D3 | — | 4 | 2 | no | — |
| 132 | C2×Dic6 | — | 4 | 1 | no | — |
| 133 | #41⋊C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 134 | #41⋊C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 135 | #41⋊C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 136 | #41⋊C3 | C24 | 1 | 1 | yes | characteristic |
| 137 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 138 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 139 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 140 | GL(2,𝔽3) | C22×C4 | 1 | 4 | yes | — |
| 141 | #226 | C22×C4 | 1 | 4 | yes | — |
| 142 | C3×M16 | — | 4 | 2 | no | — |
| 143 | C3×M16 | — | 4 | 2 | no | — |
| 144 | C2×C3×C8 | — | 4 | 2 | no | — |
| 145 | #208 | — | 4 | 2 | no | — |
| 146 | #208 | — | 4 | 2 | no | — |
| 147 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 148 | ΓL(1,𝔽25) | — | 4 | 8 | no | — |
| 149 | D3×C8 | — | 4 | 8 | no | — |
| 150 | C2×#41 | — | 3 | 1 | no | — |
| 151 | C2×#41 | — | 3 | 1 | no | — |
| 152 | C2×#41 | — | 3 | 1 | no | — |
| 153 | C2×#41 | — | 3 | 1 | no | — |
| 154 | C2×#41 | — | 3 | 1 | no | — |
| 155 | C2×#41 | — | 3 | 1 | no | — |
| 156 | C2×#41 | — | 3 | 2 | no | — |
| 157 | C2×#41 | — | 3 | 2 | no | — |
| 158 | C2×#41 | — | 3 | 2 | no | — |
| 159 | C2×#41 | — | 3 | 2 | no | — |
| 160 | C2×#41 | — | 3 | 2 | no | — |
| 161 | C2×#41 | — | 3 | 2 | no | — |
| 162 | C2×#41 | — | 6 | 1 | no | — |
| 163 | C2×#41 | — | 6 | 1 | no | — |
| 164 | C2×#41 | — | 6 | 1 | no | — |
| 165 | C2×#41 | — | 6 | 1 | no | — |
| 166 | C2×#41 | — | 6 | 1 | no | — |
| 167 | C2×#41 | — | 6 | 1 | no | — |
| 168 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 169 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 170 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 171 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 172 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 173 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 174 | C22×Q8 | — | 3 | 1 | no | — |
| 175 | C22×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 176 | C22×D4 | — | 3 | 1 | no | — |
| 177 | C22×D4 | — | 3 | 1 | no | — |
| 178 | C22×D4 | — | 6 | 1 | no | — |
| 179 | C23×C4 | — | 3 | 1 | no | — |
| 180 | C23×C4 | — | 6 | 1 | no | — |
| 181 | #137 | — | 3 | 4 | no | — |
| 182 | #137 | — | 3 | 4 | no | — |
| 183 | #137 | — | 3 | 2 | no | — |
| 184 | #137 | — | 3 | 2 | no | — |
| 185 | #137 | — | 3 | 2 | no | — |
| 186 | #137 | — | 3 | 2 | no | — |
| 187 | #137 | — | 3 | 2 | no | — |
| 188 | #137 | — | 3 | 2 | no | — |
| 189 | #137 | — | 3 | 2 | no | — |
| 190 | #137 | — | 3 | 2 | no | — |
| 191 | #137 | — | 3 | 4 | no | — |
| 192 | #137 | — | 3 | 4 | no | — |
| 193 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 194 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 195 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 196 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 197 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 198 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 199 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 200 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 201 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 202 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 203 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 204 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 205 | #135 | — | 3 | 4 | no | — |
| 206 | #135 | — | 3 | 4 | no | — |
| 207 | #135 | — | 3 | 4 | no | — |
| 208 | #135 | — | 3 | 4 | no | — |
| 209 | #135 | — | 3 | 4 | no | — |
| 210 | #135 | — | 3 | 4 | no | — |
| 211 | #135 | — | 3 | 4 | no | — |
| 212 | #135 | — | 3 | 4 | no | — |
| 213 | C2×Q16 | — | 3 | 2 | no | — |
| 214 | C2×Q16 | — | 3 | 2 | no | — |
| 215 | C2×SD16 | — | 3 | 2 | no | — |
| 216 | C2×SD16 | — | 3 | 2 | no | — |
| 217 | C2×SD16 | — | 3 | 2 | no | — |
| 218 | C2×SD16 | — | 3 | 2 | no | — |
| 219 | C2×D8 | — | 3 | 2 | no | — |
| 220 | C2×D8 | — | 3 | 2 | no | — |
| 221 | #131 | — | 3 | 4 | no | — |
| 222 | #131 | — | 3 | 4 | no | — |
| 223 | #131 | — | 3 | 4 | no | — |
| 224 | #131 | — | 3 | 8 | no | — |
| 225 | #131 | — | 3 | 8 | no | — |
| 226 | #131 | — | 6 | 4 | no | — |
| 227 | #131 | — | 6 | 4 | no | — |
| 228 | #131 | — | 6 | 4 | no | — |
| 229 | #131 | — | 6 | 8 | no | — |
| 230 | #131 | — | 6 | 4 | no | — |
| 231 | #131 | — | 6 | 4 | no | — |
| 232 | #131 | C22×D3 | 1 | 4 | yes | — |
| 233 | C2×M16 | — | 3 | 2 | no | — |
| 234 | C2×M16 | — | 3 | 1 | no | — |
| 235 | C2×M16 | — | 3 | 1 | no | — |
| 236 | C2×M16 | — | 3 | 2 | no | — |
| 237 | C2×M16 | — | 3 | 2 | no | — |
| 238 | C2×M16 | — | 3 | 2 | no | — |
| 239 | C2×M16 | — | 3 | 2 | no | — |
| 240 | C2×M16 | — | 3 | 1 | no | — |
| 241 | C2×M16 | — | 3 | 2 | no | — |
| 242 | C2×M16 | — | 3 | 2 | no | — |
| 243 | C2×M16 | — | 3 | 2 | no | — |
| 244 | C2×M16 | — | 3 | 2 | no | — |
| 245 | C2×M16 | — | 3 | 2 | no | — |
| 246 | C2×M16 | — | 3 | 2 | no | — |
| 247 | C2×M16 | — | 3 | 2 | no | — |
| 248 | C2×M16 | — | 3 | 2 | no | — |
| 249 | C2×M16 | — | 3 | 2 | no | — |
| 250 | C2×M16 | — | 3 | 2 | no | — |
| 251 | C2×M16 | — | 3 | 2 | no | — |
| 252 | C2×M16 | — | 6 | 4 | no | — |
| 253 | C2×M16 | — | 6 | 1 | no | — |
| 254 | C2×M16 | — | 6 | 1 | no | — |
| 255 | C2×M16 | — | 6 | 4 | no | — |
| 256 | C2×M16 | — | 6 | 8 | no | — |
| 257 | C2×M16 | — | 6 | 1 | no | — |
| 258 | C2×M16 | — | 6 | 1 | no | — |
| 259 | C2×M16 | — | 6 | 1 | no | — |
| 260 | C2×M16 | — | 6 | 4 | no | — |
| 261 | C2×M16 | S4 | 1 | 1 | yes | characteristic |
| 262 | C22×C8 | — | 3 | 2 | no | — |
| 263 | C22×C8 | — | 3 | 2 | no | — |
| 264 | C22×C8 | — | 6 | 1 | no | — |
| 265 | C22×C8 | — | 6 | 4 | no | — |
| 266 | C22×C8 | — | 6 | 1 | no | — |
| 267 | #117 | — | 3 | 2 | no | — |
| 268 | #117 | — | 3 | 2 | no | — |
| 269 | #117 | — | 6 | 1 | no | — |
| 270 | #117 | — | 6 | 1 | no | — |
| 271 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 272 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 273 | C2×C42 | — | 3 | 2 | no | — |
| 274 | #108 | — | 3 | 8 | no | — |
| 275 | #108 | — | 3 | 8 | no | — |
| 276 | #104 | — | 3 | 8 | no | — |
| 277 | #104 | — | 3 | 8 | no | — |
| 278 | #104 | — | 3 | 4 | no | — |
| 279 | #104 | — | 3 | 4 | no | — |
| 280 | #104 | — | 3 | 4 | no | — |
| 281 | #104 | — | 3 | 4 | no | — |
| 282 | #97 | — | 3 | 8 | no | — |
| 283 | C4×C8 | — | 3 | 8 | no | — |
| 284 | C23×C3 | — | 4 | 1 | no | — |
| 285 | C22×D3 | — | 4 | 2 | no | — |
| 286 | C2×C3×C4 | — | 4 | 1 | no | — |
| 287 | C2×C3×C4 | — | 4 | 1 | no | — |
| 288 | C2×C3×C4 | — | 4 | 1 | no | — |
| 289 | C2×C3×C4 | — | 4 | 1 | no | — |
| 290 | C2×C3×C4 | — | 4 | 1 | no | — |
| 291 | C2×C3×C4 | — | 4 | 1 | no | — |
| 292 | #67 | — | 4 | 4 | no | — |
| 293 | #67 | — | 4 | 4 | no | — |
| 294 | C2×Dic3 | — | 4 | 2 | no | — |
| 295 | D12 | — | 4 | 2 | no | — |
| 296 | D12 | — | 4 | 2 | no | — |
| 297 | C4×D3 | — | 4 | 4 | no | — |
| 298 | C4×D3 | — | 4 | 4 | no | — |
| 299 | Dic6 | — | 4 | 2 | no | — |
| 300 | Dic6 | — | 4 | 2 | no | — |
| 301 | SL(2,𝔽3) | C23×C4 | 1 | 1 | yes | characteristic |
| 302 | C3×C8 | — | 4 | 4 | no | — |
| 303 | C3⋊C8 | — | 4 | 4 | no | — |
| 304 | C24 | — | 3 | 1 | no | — |
| 305 | C24 | — | 6 | 1 | no | — |
| 306 | #41 | — | 3 | 1 | no | — |
| 307 | #41 | — | 3 | 1 | no | — |
| 308 | #41 | — | 3 | 1 | no | — |
| 309 | #41 | — | 3 | 1 | no | — |
| 310 | #41 | — | 3 | 1 | no | — |
| 311 | #41 | — | 3 | 1 | no | — |
| 312 | #41 | — | 3 | 1 | no | — |
| 313 | #41 | — | 3 | 1 | no | — |
| 314 | #41 | — | 3 | 1 | no | — |
| 315 | #41 | — | 3 | 1 | no | — |
| 316 | #41 | — | 3 | 1 | no | — |
| 317 | #41 | — | 3 | 1 | no | — |
| 318 | #41 | — | 3 | 4 | no | — |
| 319 | #41 | — | 3 | 4 | no | — |
| 320 | #41 | — | 3 | 4 | no | — |
| 321 | #41 | — | 3 | 4 | no | — |
| 322 | #41 | — | 6 | 1 | no | — |
| 323 | #41 | — | 6 | 1 | no | — |
| 324 | #41 | — | 6 | 1 | no | — |
| 325 | #41 | — | 6 | 1 | no | — |
| 326 | #41 | — | 6 | 2 | no | — |
| 327 | #41 | — | 6 | 2 | no | — |
| 328 | #41 | — | 6 | 2 | no | — |
| 329 | #41 | — | 6 | 2 | no | — |
| 330 | #41 | — | 6 | 2 | no | — |
| 331 | #41 | — | 6 | 2 | no | — |
| 332 | #41 | — | 6 | 2 | no | — |
| 333 | #41 | — | 6 | 2 | no | — |
| 334 | #41 | — | 6 | 2 | no | — |
| 335 | #41 | — | 6 | 2 | no | — |
| 336 | #41 | — | 6 | 2 | no | — |
| 337 | #41 | — | 6 | 2 | no | — |
| 338 | #41 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 339 | #41 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 340 | #41 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 341 | #41 | C23×D3 | 1 | 1 | yes | characteristic |
| 342 | C2×Q8 | — | 3 | 1 | no | — |
| 343 | C2×Q8 | — | 3 | 1 | no | — |
| 344 | C2×Q8 | — | 3 | 1 | no | — |
| 345 | C2×Q8 | — | 3 | 2 | no | — |
| 346 | C2×Q8 | — | 3 | 2 | no | — |
| 347 | C2×Q8 | — | 3 | 2 | no | — |
| 348 | C2×Q8 | — | 6 | 1 | no | — |
| 349 | C2×Q8 | — | 6 | 1 | no | — |
| 350 | C2×Q8 | — | 6 | 1 | no | — |
| 351 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 352 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 353 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 354 | C2×D4 | — | 3 | 1 | no | — |
| 355 | C2×D4 | — | 3 | 1 | no | — |
| 356 | C2×D4 | — | 3 | 1 | no | — |
| 357 | C2×D4 | — | 3 | 1 | no | — |
| 358 | C2×D4 | — | 3 | 1 | no | — |
| 359 | C2×D4 | — | 3 | 1 | no | — |
| 360 | C2×D4 | — | 3 | 2 | no | — |
| 361 | C2×D4 | — | 3 | 2 | no | — |
| 362 | C2×D4 | — | 3 | 2 | no | — |
| 363 | C2×D4 | — | 6 | 1 | no | — |
| 364 | C2×D4 | — | 6 | 1 | no | — |
| 365 | C2×D4 | — | 6 | 1 | no | — |
| 366 | C2×D4 | — | 6 | 1 | no | — |
| 367 | C2×D4 | — | 6 | 1 | no | — |
| 368 | C2×D4 | — | 6 | 2 | no | — |
| 369 | C2×D4 | — | 6 | 2 | no | — |
| 370 | C2×D4 | — | 6 | 2 | no | — |
| 371 | C2×D4 | — | 6 | 2 | no | — |
| 372 | C2×D4 | — | 6 | 2 | no | — |
| 373 | C2×D4 | — | 6 | 2 | no | — |
| 374 | C2×D4 | — | 6 | 1 | no | — |
| 375 | C22×C4 | — | 3 | 1 | no | — |
| 376 | C22×C4 | — | 3 | 1 | no | — |
| 377 | C22×C4 | — | 3 | 1 | no | — |
| 378 | C22×C4 | — | 3 | 1 | no | — |
| 379 | C22×C4 | — | 3 | 1 | no | — |
| 380 | C22×C4 | — | 3 | 1 | no | — |
| 381 | C22×C4 | — | 3 | 1 | no | — |
| 382 | C22×C4 | — | 3 | 1 | no | — |
| 383 | C22×C4 | — | 3 | 1 | no | — |
| 384 | C22×C4 | — | 3 | 1 | no | — |
| 385 | C22×C4 | — | 3 | 1 | no | — |
| 386 | C22×C4 | — | 3 | 1 | no | — |
| 387 | C22×C4 | — | 3 | 1 | no | — |
| 388 | C22×C4 | — | 6 | 2 | no | — |
| 389 | C22×C4 | — | 6 | 2 | no | — |
| 390 | C22×C4 | — | 6 | 2 | no | — |
| 391 | C22×C4 | — | 6 | 2 | no | — |
| 392 | C22×C4 | — | 6 | 2 | no | — |
| 393 | C22×C4 | — | 6 | 1 | no | — |
| 394 | C22×C4 | — | 6 | 2 | no | — |
| 395 | C22×C4 | — | 6 | 2 | no | — |
| 396 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 397 | Q16 | — | 3 | 4 | no | — |
| 398 | Q16 | — | 3 | 4 | no | — |
| 399 | Q16 | — | 3 | 4 | no | — |
| 400 | Q16 | — | 3 | 4 | no | — |
| 401 | SD16 | — | 3 | 4 | no | — |
| 402 | SD16 | — | 3 | 4 | no | — |
| 403 | SD16 | — | 3 | 4 | no | — |
| 404 | SD16 | — | 3 | 4 | no | — |
| 405 | SD16 | — | 3 | 4 | no | — |
| 406 | SD16 | — | 3 | 4 | no | — |
| 407 | SD16 | — | 3 | 4 | no | — |
| 408 | SD16 | — | 3 | 4 | no | — |
| 409 | D8 | — | 3 | 4 | no | — |
| 410 | D8 | — | 3 | 4 | no | — |
| 411 | D8 | — | 3 | 4 | no | — |
| 412 | D8 | — | 3 | 4 | no | — |
| 413 | M16 | — | 3 | 4 | no | — |
| 414 | M16 | — | 3 | 4 | no | — |
| 415 | M16 | — | 3 | 2 | no | — |
| 416 | M16 | — | 3 | 2 | no | — |
| 417 | M16 | — | 3 | 2 | no | — |
| 418 | M16 | — | 3 | 2 | no | — |
| 419 | M16 | — | 3 | 2 | no | — |
| 420 | M16 | — | 3 | 2 | no | — |
| 421 | M16 | — | 3 | 4 | no | — |
| 422 | M16 | — | 3 | 4 | no | — |
| 423 | M16 | — | 3 | 4 | no | — |
| 424 | M16 | — | 3 | 4 | no | — |
| 425 | M16 | — | 6 | 2 | no | — |
| 426 | M16 | — | 6 | 8 | no | — |
| 427 | M16 | — | 6 | 2 | no | — |
| 428 | M16 | — | 6 | 2 | no | — |
| 429 | M16 | — | 6 | 2 | no | — |
| 430 | M16 | — | 6 | 8 | no | — |
| 431 | M16 | C2×S4 | 1 | 2 | yes | — |
| 432 | M16 | C2×S4 | 1 | 2 | yes | — |
| 433 | C2×C8 | — | 3 | 4 | no | — |
| 434 | C2×C8 | — | 3 | 2 | no | — |
| 435 | C2×C8 | — | 3 | 2 | no | — |
| 436 | C2×C8 | — | 3 | 4 | no | — |
| 437 | C2×C8 | — | 3 | 2 | no | — |
| 438 | C2×C8 | — | 3 | 4 | no | — |
| 439 | C2×C8 | — | 3 | 4 | no | — |
| 440 | C2×C8 | — | 6 | 2 | no | — |
| 441 | C2×C8 | — | 6 | 2 | no | — |
| 442 | C2×C8 | — | 6 | 8 | no | — |
| 443 | C2×C8 | — | 6 | 2 | no | — |
| 444 | C2×C8 | — | 6 | 2 | no | — |
| 445 | C2×C8 | — | 6 | 2 | no | — |
| 446 | C2×C8 | — | 6 | 8 | no | — |
| 447 | C2×C8 | C2×S4 | 1 | 2 | yes | — |
| 448 | C4⋊C4 | — | 6 | 2 | no | — |
| 449 | C4⋊C4 | — | 6 | 2 | no | — |
| 450 | C22⋊C4 | — | 6 | 2 | no | — |
| 451 | C22⋊C4 | — | 6 | 2 | no | — |
| 452 | C42 | — | 3 | 4 | no | — |
| 453 | C42 | — | 6 | 2 | no | — |
| 454 | C22×C3 | — | 4 | 1 | no | — |
| 455 | C22×C3 | — | 4 | 1 | no | — |
| 456 | C22×C3 | — | 4 | 1 | no | — |
| 457 | C22×C3 | — | 16 | 1 | no | — |
| 458 | C2×D3 | — | 4 | 4 | no | — |
| 459 | C2×D3 | — | 16 | 2 | no | — |
| 460 | C3×C4 | — | 4 | 1 | no | — |
| 461 | C3×C4 | — | 4 | 1 | no | — |
| 462 | C3×C4 | — | 4 | 1 | no | — |
| 463 | C3×C4 | — | 4 | 1 | no | — |
| 464 | Dic3 | — | 4 | 4 | no | — |
| 465 | C23 | — | 3 | 1 | no | — |
| 466 | C23 | — | 3 | 1 | no | — |
| 467 | C23 | — | 3 | 1 | no | — |
| 468 | C23 | — | 3 | 1 | no | — |
| 469 | C23 | — | 3 | 1 | no | — |
| 470 | C23 | — | 3 | 1 | no | — |
| 471 | C23 | — | 6 | 2 | no | — |
| 472 | C23 | — | 6 | 2 | no | — |
| 473 | C23 | — | 6 | 2 | no | — |
| 474 | C23 | — | 24 | 1 | no | — |
| 475 | C23 | — | 24 | 2 | no | — |
| 476 | C23 | C4×S4 | 1 | 1 | yes | characteristic |
| 477 | Q8 | — | 3 | 1 | no | — |
| 478 | Q8 | — | 3 | 1 | no | — |
| 479 | Q8 | — | 3 | 1 | no | — |
| 480 | Q8 | — | 3 | 4 | no | — |
| 481 | Q8 | — | 6 | 1 | no | — |
| 482 | Q8 | — | 6 | 2 | no | — |
| 483 | Q8 | — | 6 | 2 | no | — |
| 484 | Q8 | — | 6 | 2 | no | — |
| 485 | Q8 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 486 | D4 | — | 3 | 1 | no | — |
| 487 | D4 | — | 3 | 1 | no | — |
| 488 | D4 | — | 3 | 1 | no | — |
| 489 | D4 | — | 3 | 1 | no | — |
| 490 | D4 | — | 3 | 4 | no | — |
| 491 | D4 | — | 6 | 1 | no | — |
| 492 | D4 | — | 6 | 1 | no | — |
| 493 | D4 | — | 6 | 1 | no | — |
| 494 | D4 | — | 6 | 1 | no | — |
| 495 | D4 | — | 6 | 1 | no | — |
| 496 | D4 | — | 6 | 1 | no | — |
| 497 | D4 | — | 6 | 2 | no | — |
| 498 | D4 | — | 6 | 4 | no | — |
| 499 | D4 | — | 6 | 4 | no | — |
| 500 | D4 | — | 6 | 4 | no | — |
| 501 | D4 | — | 6 | 4 | no | — |
| 502 | D4 | — | 6 | 2 | no | — |
| 503 | D4 | — | 6 | 2 | no | — |
| 504 | C2×C4 | — | 3 | 1 | no | — |
| 505 | C2×C4 | — | 3 | 1 | no | — |
| 506 | C2×C4 | — | 3 | 1 | no | — |
| 507 | C2×C4 | — | 3 | 1 | no | — |
| 508 | C2×C4 | — | 3 | 1 | no | — |
| 509 | C2×C4 | — | 3 | 1 | no | — |
| 510 | C2×C4 | — | 3 | 1 | no | — |
| 511 | C2×C4 | — | 3 | 1 | no | — |
| 512 | C2×C4 | — | 3 | 1 | no | — |
| 513 | C2×C4 | — | 3 | 1 | no | — |
| 514 | C2×C4 | — | 3 | 1 | no | — |
| 515 | C2×C4 | — | 3 | 1 | no | — |
| 516 | C2×C4 | — | 3 | 1 | no | — |
| 517 | C2×C4 | — | 3 | 1 | no | — |
| 518 | C2×C4 | — | 3 | 1 | no | — |
| 519 | C2×C4 | — | 3 | 1 | no | — |
| 520 | C2×C4 | — | 3 | 1 | no | — |
| 521 | C2×C4 | — | 3 | 1 | no | — |
| 522 | C2×C4 | — | 3 | 1 | no | — |
| 523 | C2×C4 | — | 3 | 1 | no | — |
| 524 | C2×C4 | — | 3 | 1 | no | — |
| 525 | C2×C4 | — | 3 | 1 | no | — |
| 526 | C2×C4 | — | 6 | 4 | no | — |
| 527 | C2×C4 | — | 6 | 4 | no | — |
| 528 | C2×C4 | — | 6 | 2 | no | — |
| 529 | C2×C4 | — | 6 | 2 | no | — |
| 530 | C2×C4 | — | 6 | 2 | no | — |
| 531 | C2×C4 | — | 6 | 4 | no | — |
| 532 | C2×C4 | — | 6 | 4 | no | — |
| 533 | C2×C4 | — | 6 | 4 | no | — |
| 534 | C2×C4 | — | 24 | 2 | no | — |
| 535 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 536 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 537 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 538 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 539 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 540 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 541 | C8 | — | 3 | 4 | no | — |
| 542 | C8 | — | 3 | 4 | no | — |
| 543 | C8 | — | 3 | 4 | no | — |
| 544 | C8 | — | 6 | 4 | no | — |
| 545 | C8 | C22×S4 | 1 | 4 | yes | — |
| 546 | C2×C3 | — | 4 | 1 | no | — |
| 547 | C2×C3 | — | 8 | 1 | no | — |
| 548 | C2×C3 | — | 8 | 1 | no | — |
| 549 | C2×C3 | — | 8 | 1 | no | — |
| 550 | D3 | — | 8 | 4 | no | — |
| 551 | C22 | — | 3 | 1 | no | — |
| 552 | C22 | — | 3 | 1 | no | — |
| 553 | C22 | — | 3 | 1 | no | — |
| 554 | C22 | — | 3 | 1 | no | — |
| 555 | C22 | — | 4 | 1 | no | — |
| 556 | C22 | — | 6 | 4 | no | — |
| 557 | C22 | — | 12 | 1 | no | — |
| 558 | C22 | — | 12 | 1 | no | — |
| 559 | C22 | — | 12 | 1 | no | — |
| 560 | C22 | — | 12 | 1 | no | — |
| 561 | C22 | — | 12 | 1 | no | — |
| 562 | C22 | — | 12 | 1 | no | — |
| 563 | C22 | — | 12 | 4 | no | — |
| 564 | C22 | — | 24 | 2 | no | — |
| 565 | C22 | — | 24 | 2 | no | — |
| 566 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 567 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 568 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 569 | C4 | — | 3 | 1 | no | — |
| 570 | C4 | — | 3 | 1 | no | — |
| 571 | C4 | — | 3 | 1 | no | — |
| 572 | C4 | — | 3 | 1 | no | — |
| 573 | C4 | — | 6 | 4 | no | — |
| 574 | C4 | — | 12 | 4 | no | — |
| 575 | C4 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 576 | C4 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 577 | C4 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 578 | C4 | C23×S4 | 1 | 1 | yes | characteristic |
| 579 | C3 | — | 4 | 1 | no | Sylow |
| 580 | C2 | — | 2 | 1 | no | — |
| 581 | C2 | — | 2 | 1 | no | — |
| 582 | C2 | — | 2 | 1 | no | — |
| 583 | C2 | — | 6 | 1 | no | — |
| 584 | C2 | — | 6 | 1 | no | — |
| 585 | C2 | — | 6 | 1 | no | — |
| 586 | C2 | — | 6 | 1 | no | — |
| 587 | C2 | — | 12 | 4 | no | — |
| 588 | C2 | C22×C4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C23×C4 |
| Fitting | C2×#568 | D3 |
| socle | C2 | C22×C4×S4 |
| prime radical | C2×#5923 | C2 |
| cosocle | #41⋊C3 | C24 |
| hypercenter | C2×M16 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C2×M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#568⋊C3 | C2 | semidirect |
| 3. C2×#5423 | C2 | semidirect |
| 4. #85453 | C2 | semidirect |
| 5. #85452 | C2 | semidirect |
| 6. #85448 | C2 | semidirect |
| 9. #5923 | C4 | semidirect |
| 10. #5923 | C4 | semidirect |
| 13. C2×#986 | C4 | semidirect |
| 15. C2×#984 | C4 | semidirect |
| 16. C2×#983 | C4 | semidirect |
| 18. #568⋊C3 | C22 | semidirect |
| 19. #568⋊C3 | C22 | semidirect |
| 20. C2×#131⋊C3 | C22 | semidirect |
| 21. #5423 | C22 | semidirect |
| 22. #5423 | C22 | semidirect |
| 23. C2×GU(2,𝔽3) | C22 | semidirect |
| 24. #5405 | C22 | semidirect |
| 25. #5404 | C22 | semidirect |
| 28. C2×#568 | D3 | semidirect |
| 48. #986 | C2×C4 | semidirect |
| 49. #985 | C2×C4 | semidirect |
| 52. #984 | C2×C4 | semidirect |
| 54. #983 | C2×C4 | semidirect |
| 55. C2×GL(2,𝔽3) | C2×C4 | semidirect |
| 56. C2×#226 | C2×C4 | semidirect |
| 62. #131⋊C3 | C23 | semidirect |
| 63. GU(2,𝔽3) | C23 | semidirect |
| 90. #568 | C2×D3 | semidirect |
| 94. C2×#131 | C2×D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5923 | 2 |
| C2×#568⋊C3 | 2 |
| C2×#5423 | 2 |
| #85453 | 2 |
| #85452 | 2 |
| #85448 | 2 |
| #60182 | 3 |
| #5745 | 4 |