Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5225 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 5225 conjugacy classes of subgroups — too many to draw. Only its 511 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×#5945 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #87499 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#5927 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C2×#5924 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#5923 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C22×#984 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#3586 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C2×#1010 | — | 4 | 1 | no | maximal |
| 9 | #5945 | C22 | 1 | 8 | yes | complemented |
| 10 | C2×2−1+4⋊C3 | C22 | 1 | 3 | yes | complemented |
| 11 | C2×#994 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C22×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 13 | #5927 | C22 | 1 | 48 | yes | complemented |
| 14 | #5924 | C22 | 1 | 16 | yes | complemented |
| 15 | #5923 | C22 | 1 | 24 | yes | complemented |
| 16 | C2×#986 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#985 | C22 | 1 | 6 | yes | complemented |
| 18 | C2×#984 | C22 | 1 | 3 | yes | complemented |
| 19 | C2×#984 | C22 | 1 | 6 | yes | complemented |
| 20 | C2×#984 | C22 | 1 | 24 | yes | complemented |
| 21 | C2×#983 | C22 | 1 | 6 | yes | complemented |
| 22 | C22×#226 | C22 | 1 | 6 | yes | complemented |
| 23 | C2×#585 | D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C22×2−1+4 | — | 3 | 1 | no | — |
| 25 | #3586 | — | 3 | 16 | no | — |
| 26 | #3586 | — | 3 | 16 | no | — |
| 27 | C2×#578 | — | 3 | 2 | no | — |
| 28 | C2×#578 | — | 3 | 6 | no | — |
| 29 | C2×#577 | — | 3 | 6 | no | — |
| 30 | C2×#577 | — | 3 | 2 | no | — |
| 31 | C2×#575 | — | 3 | 3 | no | — |
| 32 | C2×#575 | — | 3 | 3 | no | — |
| 33 | C22×#137 | — | 3 | 3 | no | — |
| 34 | C22×#137 | — | 3 | 3 | no | — |
| 35 | C2×#568 | — | 3 | 1 | no | — |
| 36 | C2×C3×#41 | — | 4 | 1 | no | — |
| 37 | #1010 | — | 4 | 16 | no | — |
| 38 | C2×D3×Q8 | — | 4 | 2 | no | — |
| 39 | C2×#237 | — | 4 | 6 | no | — |
| 40 | C2×#235 | — | 4 | 3 | no | — |
| 41 | C22×Dic6 | — | 4 | 3 | no | — |
| 42 | 2−1+4⋊C3 | C23 | 1 | 12 | yes | complemented |
| 43 | #994 | C23 | 1 | 4 | yes | complemented |
| 44 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×#41⋊C3 | C23 | 1 | 3 | yes | — |
| 46 | C2×#41⋊C3 | C23 | 1 | 12 | yes | — |
| 47 | C22×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 48 | #986 | C23 | 1 | 4 | yes | complemented |
| 49 | #985 | C23 | 1 | 24 | yes | complemented |
| 50 | #984 | C23 | 1 | 12 | yes | complemented |
| 51 | #984 | C23 | 1 | 24 | yes | complemented |
| 52 | #983 | C23 | 1 | 24 | yes | complemented |
| 53 | C2×GL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 54 | C2×#226 | C23 | 1 | 6 | yes | — |
| 55 | C2×#226 | C23 | 1 | 24 | yes | complemented |
| 56 | #585 | — | 3 | 8 | no | — |
| 57 | #585 | C2×D3 | 1 | 8 | yes | complemented |
| 58 | C2×2−1+4 | — | 3 | 1 | no | — |
| 59 | C2×2−1+4 | — | 3 | 16 | no | — |
| 60 | C2×2−1+4 | — | 3 | 6 | no | — |
| 61 | C2×2−1+4 | — | 3 | 2 | no | — |
| 62 | C2×2−1+4 | — | 6 | 8 | no | — |
| 63 | C2×2−1+4 | — | 6 | 3 | no | — |
| 64 | C2×2−1+4 | — | 6 | 1 | no | — |
| 65 | C2×2−1+4 | C2×D3 | 1 | 3 | yes | — |
| 66 | C2×H(5,𝔽2) | — | 3 | 3 | no | — |
| 67 | C2×H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic |
| 68 | C22×#41 | — | 3 | 1 | no | — |
| 69 | C22×#41 | — | 3 | 3 | no | — |
| 70 | C22×#41 | — | 3 | 3 | no | — |
| 71 | C22×#41 | — | 6 | 3 | no | — |
| 72 | C22×#41 | C2×D3 | 1 | 3 | yes | — |
| 73 | C23×Q8 | — | 3 | 3 | no | — |
| 74 | C23×Q8 | — | 6 | 1 | no | — |
| 75 | #578 | — | 3 | 32 | no | — |
| 76 | #578 | — | 3 | 48 | no | — |
| 77 | #578 | — | 3 | 48 | no | — |
| 78 | #577 | — | 3 | 96 | no | — |
| 79 | #577 | — | 3 | 16 | no | — |
| 80 | #577 | — | 3 | 16 | no | — |
| 81 | #575 | — | 3 | 24 | no | — |
| 82 | #575 | — | 3 | 48 | no | — |
| 83 | #575 | — | 3 | 24 | no | — |
| 84 | C2×#137 | — | 3 | 24 | no | — |
| 85 | C2×#137 | — | 3 | 48 | no | — |
| 86 | C2×#137 | — | 3 | 3 | no | — |
| 87 | C2×#137 | — | 3 | 6 | no | — |
| 88 | C2×#137 | — | 3 | 6 | no | — |
| 89 | C2×#137 | — | 3 | 12 | no | — |
| 90 | C2×#137 | — | 3 | 6 | no | — |
| 91 | C2×#137 | — | 3 | 24 | no | — |
| 92 | C2×#137 | — | 3 | 3 | no | — |
| 93 | C2×#137 | — | 3 | 6 | no | — |
| 94 | C2×#137 | — | 3 | 6 | no | — |
| 95 | C2×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 96 | C2×AGL(1,ℤ/8) | — | 3 | 6 | no | — |
| 97 | C2×AGL(1,ℤ/8) | — | 3 | 6 | no | — |
| 98 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 99 | C2×#135 | — | 3 | 6 | no | — |
| 100 | C2×#135 | — | 3 | 12 | no | — |
| 101 | C2×#135 | — | 3 | 6 | no | — |
| 102 | C22×Q16 | — | 3 | 6 | no | — |
| 103 | C22×Q16 | — | 3 | 6 | no | — |
| 104 | C22×SD16 | — | 3 | 6 | no | — |
| 105 | C22×SD16 | — | 3 | 6 | no | — |
| 106 | #568 | — | 3 | 16 | no | — |
| 107 | C2×#131 | — | 3 | 2 | no | — |
| 108 | C2×#131 | — | 3 | 6 | no | — |
| 109 | C22×M16 | — | 3 | 3 | no | — |
| 110 | C22×M16 | — | 3 | 3 | no | — |
| 111 | C3×#41 | — | 4 | 8 | no | — |
| 112 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 113 | C2×C3×D4 | — | 4 | 3 | no | — |
| 114 | C22×C3×C4 | — | 4 | 3 | no | — |
| 115 | C2×#67 | — | 4 | 6 | no | — |
| 116 | C22×Dic3 | — | 4 | 6 | no | — |
| 117 | D3×Q8 | — | 4 | 16 | no | — |
| 118 | #237 | — | 4 | 48 | no | — |
| 119 | #235 | — | 4 | 24 | no | — |
| 120 | C2×D12 | — | 4 | 1 | no | — |
| 121 | C2×C4×D3 | — | 4 | 6 | no | — |
| 122 | C2×Dic6 | — | 4 | 3 | no | — |
| 123 | C2×Dic6 | — | 4 | 6 | no | — |
| 124 | C2×Dic6 | — | 4 | 24 | no | — |
| 125 | #41⋊C3 | C24 | 1 | 2 | yes | — |
| 126 | #41⋊C3 | C24 | 1 | 6 | yes | — |
| 127 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 128 | C2×SL(2,𝔽3) | C24 | 1 | 6 | yes | — |
| 129 | GL(2,𝔽3) | C24 | 1 | 4 | yes | — |
| 130 | #226 | C24 | 1 | 12 | yes | — |
| 131 | 2−1+4 | — | 3 | 8 | no | — |
| 132 | 2−1+4 | — | 3 | 12 | no | — |
| 133 | 2−1+4 | — | 3 | 48 | no | — |
| 134 | 2−1+4 | — | 3 | 16 | no | — |
| 135 | 2−1+4 | — | 6 | 4 | no | — |
| 136 | 2−1+4 | — | 6 | 24 | no | — |
| 137 | 2−1+4 | — | 6 | 8 | no | — |
| 138 | 2−1+4 | — | 6 | 48 | no | — |
| 139 | 2−1+4 | — | 6 | 16 | no | — |
| 140 | 2−1+4 | C22×D3 | 1 | 12 | yes | — |
| 141 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 142 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 143 | H(5,𝔽2) | — | 6 | 12 | no | — |
| 144 | H(5,𝔽2) | C22×D3 | 1 | 4 | yes | — |
| 145 | C2×#41 | — | 3 | 8 | no | — |
| 146 | C2×#41 | — | 3 | 8 | no | — |
| 147 | C2×#41 | — | 3 | 1 | no | — |
| 148 | C2×#41 | — | 3 | 3 | no | — |
| 149 | C2×#41 | — | 3 | 3 | no | — |
| 150 | C2×#41 | — | 3 | 3 | no | — |
| 151 | C2×#41 | — | 3 | 3 | no | — |
| 152 | C2×#41 | — | 3 | 12 | no | — |
| 153 | C2×#41 | — | 3 | 6 | no | — |
| 154 | C2×#41 | — | 3 | 24 | no | — |
| 155 | C2×#41 | — | 3 | 6 | no | — |
| 156 | C2×#41 | — | 3 | 6 | no | — |
| 157 | C2×#41 | — | 3 | 24 | no | — |
| 158 | C2×#41 | — | 6 | 3 | no | — |
| 159 | C2×#41 | — | 6 | 12 | no | — |
| 160 | C2×#41 | — | 6 | 3 | no | — |
| 161 | C2×#41 | — | 6 | 12 | no | — |
| 162 | C2×#41 | — | 6 | 48 | no | — |
| 163 | C2×#41 | — | 6 | 12 | no | — |
| 164 | C2×#41 | — | 6 | 6 | no | — |
| 165 | C2×#41 | — | 6 | 6 | no | — |
| 166 | C2×#41 | — | 6 | 3 | no | — |
| 167 | C2×#41 | S4 | 1 | 1 | yes | characteristic |
| 168 | C2×#41 | C22×D3 | 1 | 3 | yes | — |
| 169 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 170 | C2×#41 | C22×D3 | 1 | 12 | yes | — |
| 171 | C22×Q8 | — | 3 | 1 | no | — |
| 172 | C22×Q8 | — | 3 | 3 | no | — |
| 173 | C22×Q8 | — | 3 | 6 | no | — |
| 174 | C22×Q8 | — | 3 | 6 | no | — |
| 175 | C22×Q8 | — | 3 | 3 | no | — |
| 176 | C22×Q8 | — | 3 | 24 | no | — |
| 177 | C22×Q8 | — | 6 | 3 | no | — |
| 178 | C22×Q8 | — | 6 | 3 | no | — |
| 179 | C22×Q8 | — | 6 | 12 | no | — |
| 180 | C22×Q8 | — | 6 | 16 | no | — |
| 181 | C22×Q8 | — | 6 | 6 | no | — |
| 182 | C22×Q8 | — | 6 | 2 | no | — |
| 183 | C22×Q8 | — | 6 | 3 | no | — |
| 184 | C22×Q8 | C22×D3 | 1 | 3 | yes | — |
| 185 | C22×D4 | — | 3 | 3 | no | — |
| 186 | C22×D4 | — | 3 | 3 | no | — |
| 187 | C22×D4 | — | 3 | 3 | no | — |
| 188 | C22×D4 | — | 3 | 1 | no | — |
| 189 | C22×D4 | — | 6 | 3 | no | — |
| 190 | C22×D4 | — | 6 | 3 | no | — |
| 191 | C23×C4 | — | 3 | 3 | no | — |
| 192 | C23×C4 | — | 6 | 1 | no | — |
| 193 | C23×C4 | — | 6 | 3 | no | — |
| 194 | C23×C4 | — | 6 | 3 | no | — |
| 195 | #137 | — | 3 | 24 | no | — |
| 196 | #137 | — | 3 | 12 | no | — |
| 197 | #137 | — | 3 | 48 | no | — |
| 198 | #137 | — | 3 | 24 | no | — |
| 199 | #137 | — | 3 | 48 | no | — |
| 200 | #137 | — | 3 | 96 | no | — |
| 201 | #137 | — | 3 | 24 | no | — |
| 202 | #137 | — | 3 | 48 | no | — |
| 203 | #137 | — | 3 | 12 | no | — |
| 204 | #137 | — | 3 | 24 | no | — |
| 205 | #137 | — | 3 | 24 | no | — |
| 206 | AGL(1,ℤ/8) | — | 3 | 24 | no | — |
| 207 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 208 | AGL(1,ℤ/8) | — | 3 | 48 | no | — |
| 209 | AGL(1,ℤ/8) | — | 3 | 48 | no | — |
| 210 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 211 | #135 | — | 3 | 48 | no | — |
| 212 | #135 | — | 3 | 24 | no | — |
| 213 | #135 | — | 3 | 96 | no | — |
| 214 | #135 | — | 3 | 24 | no | — |
| 215 | C2×Q16 | — | 3 | 6 | no | — |
| 216 | C2×Q16 | — | 3 | 12 | no | — |
| 217 | C2×Q16 | — | 3 | 24 | no | — |
| 218 | C2×Q16 | — | 3 | 48 | no | — |
| 219 | C2×Q16 | — | 3 | 6 | no | — |
| 220 | C2×Q16 | — | 3 | 24 | no | — |
| 221 | C2×SD16 | — | 3 | 12 | no | — |
| 222 | C2×SD16 | — | 3 | 6 | no | — |
| 223 | C2×SD16 | — | 3 | 6 | no | — |
| 224 | C2×SD16 | — | 3 | 48 | no | — |
| 225 | C2×SD16 | — | 3 | 48 | no | — |
| 226 | C2×SD16 | — | 3 | 6 | no | — |
| 227 | C2×SD16 | — | 3 | 2 | no | — |
| 228 | C2×D8 | — | 3 | 2 | no | — |
| 229 | C2×D8 | — | 3 | 6 | no | — |
| 230 | #131 | — | 3 | 16 | no | — |
| 231 | #131 | — | 3 | 48 | no | — |
| 232 | C2×M16 | — | 3 | 3 | no | — |
| 233 | C2×M16 | — | 3 | 1 | no | — |
| 234 | C2×M16 | — | 3 | 6 | no | — |
| 235 | C2×M16 | — | 3 | 6 | no | — |
| 236 | C2×M16 | — | 3 | 24 | no | — |
| 237 | C2×M16 | — | 3 | 24 | no | — |
| 238 | C22×C8 | — | 3 | 6 | no | — |
| 239 | C22×C8 | — | 3 | 6 | no | — |
| 240 | C23×C3 | — | 4 | 3 | no | — |
| 241 | C22×D3 | — | 4 | 2 | no | — |
| 242 | C3×Q8 | — | 4 | 4 | no | — |
| 243 | C3×D4 | — | 4 | 12 | no | — |
| 244 | C2×C3×C4 | — | 4 | 1 | no | — |
| 245 | C2×C3×C4 | — | 4 | 3 | no | — |
| 246 | C2×C3×C4 | — | 4 | 12 | no | — |
| 247 | #67 | — | 4 | 24 | no | — |
| 248 | C2×Dic3 | — | 4 | 6 | no | — |
| 249 | C2×Dic3 | — | 4 | 24 | no | — |
| 250 | D12 | — | 4 | 4 | no | — |
| 251 | C4×D3 | — | 4 | 24 | no | — |
| 252 | Dic6 | — | 4 | 12 | no | — |
| 253 | Dic6 | — | 4 | 24 | no | — |
| 254 | SL(2,𝔽3) | C25 | 1 | 1 | yes | characteristic |
| 255 | C24 | — | 3 | 3 | no | — |
| 256 | C24 | — | 6 | 1 | no | — |
| 257 | C24 | — | 6 | 1 | no | — |
| 258 | #41 | — | 3 | 4 | no | — |
| 259 | #41 | — | 3 | 4 | no | — |
| 260 | #41 | — | 3 | 12 | no | — |
| 261 | #41 | — | 3 | 12 | no | — |
| 262 | #41 | — | 3 | 24 | no | — |
| 263 | #41 | — | 3 | 12 | no | — |
| 264 | #41 | — | 3 | 6 | no | — |
| 265 | #41 | — | 3 | 2 | no | — |
| 266 | #41 | — | 3 | 12 | no | — |
| 267 | #41 | — | 3 | 24 | no | — |
| 268 | #41 | — | 3 | 24 | no | — |
| 269 | #41 | — | 3 | 24 | no | — |
| 270 | #41 | — | 6 | 6 | no | — |
| 271 | #41 | — | 6 | 6 | no | — |
| 272 | #41 | — | 6 | 12 | no | — |
| 273 | #41 | — | 6 | 24 | no | — |
| 274 | #41 | — | 6 | 12 | no | — |
| 275 | #41 | — | 6 | 24 | no | — |
| 276 | #41 | — | 6 | 12 | no | — |
| 277 | #41 | — | 6 | 96 | no | — |
| 278 | #41 | — | 6 | 48 | no | — |
| 279 | #41 | — | 6 | 48 | no | — |
| 280 | #41 | — | 6 | 24 | no | — |
| 281 | #41 | C2×S4 | 1 | 8 | yes | — |
| 282 | #41 | C23×D3 | 1 | 6 | yes | — |
| 283 | #41 | C23×D3 | 1 | 2 | yes | — |
| 284 | C2×Q8 | — | 3 | 8 | no | — |
| 285 | C2×Q8 | — | 3 | 3 | no | — |
| 286 | C2×Q8 | — | 3 | 3 | no | — |
| 287 | C2×Q8 | — | 3 | 6 | no | — |
| 288 | C2×Q8 | — | 3 | 12 | no | — |
| 289 | C2×Q8 | — | 3 | 6 | no | — |
| 290 | C2×Q8 | — | 3 | 24 | no | — |
| 291 | C2×Q8 | — | 3 | 24 | no | — |
| 292 | C2×Q8 | — | 3 | 12 | no | — |
| 293 | C2×Q8 | — | 6 | 6 | no | — |
| 294 | C2×Q8 | — | 6 | 6 | no | — |
| 295 | C2×Q8 | — | 6 | 6 | no | — |
| 296 | C2×Q8 | — | 6 | 24 | no | — |
| 297 | C2×Q8 | — | 6 | 12 | no | — |
| 298 | C2×Q8 | — | 6 | 24 | no | — |
| 299 | C2×Q8 | — | 6 | 12 | no | — |
| 300 | C2×Q8 | — | 6 | 48 | no | — |
| 301 | C2×Q8 | — | 6 | 16 | no | — |
| 302 | C2×Q8 | — | 6 | 3 | no | — |
| 303 | C2×Q8 | — | 6 | 6 | no | — |
| 304 | C2×Q8 | — | 6 | 24 | no | — |
| 305 | C2×Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 306 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 307 | C2×Q8 | C23×D3 | 1 | 6 | yes | — |
| 308 | C2×D4 | — | 3 | 6 | no | — |
| 309 | C2×D4 | — | 3 | 24 | no | — |
| 310 | C2×D4 | — | 3 | 12 | no | — |
| 311 | C2×D4 | — | 3 | 1 | no | — |
| 312 | C2×D4 | — | 3 | 3 | no | — |
| 313 | C2×D4 | — | 3 | 12 | no | — |
| 314 | C2×D4 | — | 3 | 2 | no | — |
| 315 | C2×D4 | — | 3 | 4 | no | — |
| 316 | C2×D4 | — | 3 | 3 | no | — |
| 317 | C2×D4 | — | 6 | 6 | no | — |
| 318 | C2×D4 | — | 6 | 3 | no | — |
| 319 | C2×D4 | — | 6 | 6 | no | — |
| 320 | C2×D4 | — | 6 | 2 | no | — |
| 321 | C2×D4 | — | 6 | 24 | no | — |
| 322 | C2×D4 | — | 6 | 24 | no | — |
| 323 | C2×D4 | — | 6 | 6 | no | — |
| 324 | C2×D4 | — | 6 | 6 | no | — |
| 325 | C2×D4 | — | 6 | 1 | no | — |
| 326 | C2×D4 | C2×S4 | 1 | 3 | yes | — |
| 327 | C22×C4 | — | 3 | 3 | no | — |
| 328 | C22×C4 | — | 3 | 3 | no | — |
| 329 | C22×C4 | — | 3 | 1 | no | — |
| 330 | C22×C4 | — | 3 | 12 | no | — |
| 331 | C22×C4 | — | 3 | 3 | no | — |
| 332 | C22×C4 | — | 3 | 3 | no | — |
| 333 | C22×C4 | — | 3 | 12 | no | — |
| 334 | C22×C4 | — | 6 | 8 | no | — |
| 335 | C22×C4 | — | 6 | 3 | no | — |
| 336 | C22×C4 | — | 6 | 6 | no | — |
| 337 | C22×C4 | — | 6 | 24 | no | — |
| 338 | C22×C4 | — | 6 | 3 | no | — |
| 339 | C22×C4 | — | 6 | 24 | no | — |
| 340 | C22×C4 | — | 6 | 6 | no | — |
| 341 | C22×C4 | — | 6 | 6 | no | — |
| 342 | C22×C4 | — | 6 | 6 | no | — |
| 343 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 344 | Q16 | — | 3 | 24 | no | — |
| 345 | Q16 | — | 3 | 48 | no | — |
| 346 | Q16 | — | 3 | 12 | no | — |
| 347 | Q16 | — | 3 | 12 | no | — |
| 348 | SD16 | — | 3 | 48 | no | — |
| 349 | SD16 | — | 3 | 24 | no | — |
| 350 | SD16 | — | 3 | 24 | no | — |
| 351 | SD16 | — | 3 | 24 | no | — |
| 352 | SD16 | — | 3 | 4 | no | — |
| 353 | SD16 | — | 3 | 4 | no | — |
| 354 | D8 | — | 3 | 8 | no | — |
| 355 | D8 | — | 3 | 24 | no | — |
| 356 | M16 | — | 3 | 12 | no | — |
| 357 | M16 | — | 3 | 4 | no | — |
| 358 | M16 | — | 3 | 24 | no | — |
| 359 | M16 | — | 3 | 24 | no | — |
| 360 | C2×C8 | — | 3 | 6 | no | — |
| 361 | C2×C8 | — | 3 | 2 | no | — |
| 362 | C2×C8 | — | 3 | 24 | no | — |
| 363 | C2×C8 | — | 3 | 24 | no | — |
| 364 | C22×C3 | — | 4 | 6 | no | — |
| 365 | C22×C3 | — | 4 | 1 | no | — |
| 366 | C22×C3 | — | 8 | 6 | no | — |
| 367 | C2×D3 | — | 4 | 4 | no | — |
| 368 | C2×D3 | — | 8 | 4 | no | — |
| 369 | C3×C4 | — | 4 | 2 | no | — |
| 370 | C3×C4 | — | 4 | 6 | no | — |
| 371 | Dic3 | — | 4 | 12 | no | — |
| 372 | C23 | — | 3 | 1 | no | — |
| 373 | C23 | — | 3 | 3 | no | — |
| 374 | C23 | — | 3 | 12 | no | — |
| 375 | C23 | — | 6 | 4 | no | — |
| 376 | C23 | — | 6 | 4 | no | — |
| 377 | C23 | — | 6 | 2 | no | — |
| 378 | C23 | — | 12 | 6 | no | — |
| 379 | C23 | — | 12 | 4 | no | — |
| 380 | C23 | — | 24 | 2 | no | — |
| 381 | C23 | C22×S4 | 1 | 3 | yes | — |
| 382 | Q8 | — | 3 | 1 | no | — |
| 383 | Q8 | — | 3 | 6 | no | — |
| 384 | Q8 | — | 3 | 12 | no | — |
| 385 | Q8 | — | 3 | 12 | no | — |
| 386 | Q8 | — | 6 | 3 | no | — |
| 387 | Q8 | — | 6 | 6 | no | — |
| 388 | Q8 | — | 6 | 24 | no | — |
| 389 | Q8 | — | 6 | 12 | no | — |
| 390 | Q8 | — | 6 | 24 | no | — |
| 391 | Q8 | C22×S4 | 1 | 4 | yes | — |
| 392 | Q8 | C24×D3 | 1 | 1 | yes | characteristic |
| 393 | D4 | — | 3 | 2 | no | — |
| 394 | D4 | — | 3 | 6 | no | — |
| 395 | D4 | — | 3 | 24 | no | — |
| 396 | D4 | — | 3 | 4 | no | — |
| 397 | D4 | — | 3 | 12 | no | — |
| 398 | D4 | — | 6 | 1 | no | — |
| 399 | D4 | — | 6 | 3 | no | — |
| 400 | D4 | — | 6 | 12 | no | — |
| 401 | D4 | — | 6 | 2 | no | — |
| 402 | D4 | — | 6 | 24 | no | — |
| 403 | D4 | — | 6 | 24 | no | — |
| 404 | D4 | — | 6 | 4 | no | — |
| 405 | D4 | C22×S4 | 1 | 12 | yes | — |
| 406 | C2×C4 | — | 3 | 6 | no | — |
| 407 | C2×C4 | — | 3 | 6 | no | — |
| 408 | C2×C4 | — | 3 | 1 | no | — |
| 409 | C2×C4 | — | 3 | 3 | no | — |
| 410 | C2×C4 | — | 3 | 6 | no | — |
| 411 | C2×C4 | — | 3 | 2 | no | — |
| 412 | C2×C4 | — | 3 | 2 | no | — |
| 413 | C2×C4 | — | 3 | 12 | no | — |
| 414 | C2×C4 | — | 3 | 12 | no | — |
| 415 | C2×C4 | — | 3 | 6 | no | — |
| 416 | C2×C4 | — | 6 | 12 | no | — |
| 417 | C2×C4 | — | 6 | 24 | no | — |
| 418 | C2×C4 | — | 6 | 12 | no | — |
| 419 | C2×C4 | — | 6 | 24 | no | — |
| 420 | C2×C4 | — | 6 | 24 | no | — |
| 421 | C2×C4 | — | 6 | 6 | no | — |
| 422 | C2×C4 | — | 6 | 24 | no | — |
| 423 | C2×C4 | C22×S4 | 1 | 12 | yes | — |
| 424 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 425 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 426 | C8 | — | 3 | 12 | no | — |
| 427 | C8 | — | 3 | 4 | no | — |
| 428 | C2×C3 | — | 4 | 1 | no | — |
| 429 | C2×C3 | — | 4 | 2 | no | — |
| 430 | C2×C3 | — | 8 | 6 | no | — |
| 431 | D3 | — | 8 | 4 | no | — |
| 432 | C22 | — | 2 | 6 | no | — |
| 433 | C22 | — | 3 | 2 | no | — |
| 434 | C22 | — | 3 | 6 | no | — |
| 435 | C22 | — | 6 | 2 | no | — |
| 436 | C22 | — | 6 | 6 | no | — |
| 437 | C22 | — | 6 | 4 | no | — |
| 438 | C22 | — | 12 | 12 | no | — |
| 439 | C22 | — | 12 | 8 | no | — |
| 440 | C22 | — | 12 | 4 | no | — |
| 441 | C22 | — | 24 | 4 | no | — |
| 442 | C22 | C23×S4 | 1 | 6 | yes | — |
| 443 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 444 | C4 | — | 3 | 1 | no | — |
| 445 | C4 | — | 3 | 1 | no | — |
| 446 | C4 | — | 3 | 6 | no | — |
| 447 | C4 | — | 6 | 12 | no | — |
| 448 | C4 | C23×S4 | 1 | 2 | yes | — |
| 449 | C4 | C23×S4 | 1 | 6 | yes | — |
| 450 | C3 | — | 4 | 1 | no | Sylow |
| 451 | C2 | — | 2 | 6 | no | — |
| 452 | C2 | — | 6 | 2 | no | — |
| 453 | C2 | — | 6 | 6 | no | — |
| 454 | C2 | — | 12 | 4 | no | — |
| 455 | C2 | #87499 | 1 | 2 | yes | direct product factor |
| 456 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C2×#585 | D3 |
| hypercenter | C2×#41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5945 | C2 | semidirect |
| 2. #87499 | C2 | direct (N × H) |
| 3. C2×#5927 | C2 | semidirect |
| 4. C2×#5924 | C2 | semidirect |
| 5. C2×#5923 | C2 | semidirect |
| 6. C22×#984 | C2 | semidirect |
| 9. #5945 | C22 | semidirect |
| 10. C2×2−1+4⋊C3 | C22 | semidirect |
| 11. C2×#994 | C22 | semidirect |
| 13. #5927 | C22 | semidirect |
| 14. #5924 | C22 | semidirect |
| 15. #5923 | C22 | semidirect |
| 16. C2×#986 | C22 | semidirect |
| 17. C2×#985 | C22 | semidirect |
| 18. C2×#984 | C22 | semidirect |
| 19. C2×#984 | C22 | semidirect |
| 20. C2×#984 | C22 | semidirect |
| 21. C2×#983 | C22 | semidirect |
| 22. C22×#226 | C22 | semidirect |
| 23. C2×#585 | D3 | semidirect |
| 42. 2−1+4⋊C3 | C23 | semidirect |
| 43. #994 | C23 | semidirect |
| 48. #986 | C23 | semidirect |
| 49. #985 | C23 | semidirect |
| 50. #984 | C23 | semidirect |
| 51. #984 | C23 | semidirect |
| 52. #983 | C23 | semidirect |
| 55. C2×#226 | C23 | semidirect |
| 57. #585 | C2×D3 | semidirect |
| 455. C2 | #87499 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5945 | 2 |
| #87499 | 2 |
| C2×#5927 | 2 |
| C2×#5924 | 2 |
| C2×#5923 | 2 |
| C22×#984 | 2 |
| C2×#3586 | 3 |
| C2×#1010 | 4 |