Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5305 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 5305 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 | #87497 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#5927 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#5926 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#5925 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#5924 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#5923 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C22×#985 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C2×#3584 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C2×D3×#41 | — | 4 | 1 | no | maximal |
| 11 | #5945 | C22 | 1 | 8 | yes | complemented |
| 12 | C2×2−1+4⋊C3 | C22 | 1 | 3 | yes | complemented |
| 13 | C2×#994 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×#41⋊C3 | C22 | 1 | 3 | yes | complemented |
| 15 | #5927 | C22 | 1 | 24 | yes | complemented |
| 16 | #5926 | C22 | 1 | 24 | yes | complemented |
| 17 | #5925 | C22 | 1 | 8 | yes | complemented |
| 18 | #5924 | C22 | 1 | 8 | yes | complemented |
| 19 | #5923 | C22 | 1 | 24 | yes | complemented |
| 20 | C2×#986 | C22 | 1 | 3 | yes | complemented |
| 21 | C2×#985 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×#985 | C22 | 1 | 3 | yes | complemented |
| 23 | C2×#985 | C22 | 1 | 3 | yes | complemented |
| 24 | C2×#985 | C22 | 1 | 3 | yes | complemented |
| 25 | C2×#985 | C22 | 1 | 24 | yes | complemented |
| 26 | C2×#984 | C22 | 1 | 3 | yes | complemented |
| 27 | C2×#983 | C22 | 1 | 6 | yes | complemented |
| 28 | C22×GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 29 | C22×#226 | C22 | 1 | 3 | yes | complemented |
| 30 | C2×#585 | — | 3 | 1 | no | — |
| 31 | C2×#585 | D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | #3584 | — | 3 | 16 | no | — |
| 33 | #3584 | — | 3 | 16 | no | — |
| 34 | C2×#578 | — | 3 | 1 | no | — |
| 35 | C2×#578 | — | 3 | 3 | no | — |
| 36 | C2×#577 | — | 3 | 3 | no | — |
| 37 | C2×#577 | — | 3 | 1 | no | — |
| 38 | C2×#577 | — | 3 | 3 | no | — |
| 39 | C2×#577 | — | 3 | 1 | no | — |
| 40 | C2×#576 | — | 3 | 3 | no | — |
| 41 | C2×#576 | — | 3 | 1 | no | — |
| 42 | C2×#575 | — | 3 | 3 | no | — |
| 43 | C2×#575 | — | 3 | 3 | no | — |
| 44 | C22×#135 | — | 3 | 3 | no | — |
| 45 | C22×#135 | — | 3 | 3 | no | — |
| 46 | C22×#131 | — | 3 | 1 | no | — |
| 47 | C2×C3×#41 | — | 4 | 1 | no | — |
| 48 | D3×#41 | — | 4 | 16 | no | — |
| 49 | C2×#239 | — | 4 | 1 | no | — |
| 50 | C2×D3×Q8 | — | 4 | 1 | no | — |
| 51 | C2×#237 | — | 4 | 3 | no | — |
| 52 | C2×D3×D4 | — | 4 | 3 | no | — |
| 53 | C2×#235 | — | 4 | 3 | no | — |
| 54 | C22×C4×D3 | — | 4 | 3 | no | — |
| 55 | 2−1+4⋊C3 | C23 | 1 | 12 | yes | complemented |
| 56 | #994 | C23 | 1 | 4 | yes | complemented |
| 57 | C2×#41⋊C3 | C23 | 1 | 3 | yes | — |
| 58 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×#41⋊C3 | C23 | 1 | 12 | yes | complemented |
| 60 | C22×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 61 | #986 | C23 | 1 | 12 | yes | complemented |
| 62 | #985 | C23 | 1 | 4 | yes | complemented |
| 63 | #985 | C23 | 1 | 12 | yes | complemented |
| 64 | #985 | C23 | 1 | 12 | yes | complemented |
| 65 | #985 | C23 | 1 | 12 | yes | complemented |
| 66 | #984 | C23 | 1 | 12 | yes | complemented |
| 67 | #983 | C23 | 1 | 24 | yes | complemented |
| 68 | C2×GL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 69 | C2×GL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 70 | C2×GL(2,𝔽3) | C23 | 1 | 12 | yes | complemented |
| 71 | C2×#226 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×#226 | C23 | 1 | 3 | yes | — |
| 73 | C2×#226 | C23 | 1 | 12 | yes | complemented |
| 74 | #585 | — | 3 | 8 | no | — |
| 75 | #585 | — | 3 | 16 | no | — |
| 76 | #585 | — | 6 | 8 | no | — |
| 77 | #585 | C2×D3 | 1 | 8 | yes | complemented |
| 78 | C2×2−1+4 | — | 3 | 1 | no | — |
| 79 | C2×2−1+4 | — | 3 | 3 | no | — |
| 80 | C2×2−1+4 | — | 3 | 1 | no | — |
| 81 | C2×2−1+4 | — | 6 | 1 | no | — |
| 82 | C2×2−1+4 | C2×D3 | 1 | 3 | yes | complemented |
| 83 | C2×H(5,𝔽2) | — | 3 | 3 | no | — |
| 84 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 85 | C2×H(5,𝔽2) | — | 3 | 3 | no | — |
| 86 | C2×H(5,𝔽2) | — | 6 | 3 | no | — |
| 87 | C2×H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 88 | C22×#41 | — | 3 | 1 | no | — |
| 89 | C22×#41 | — | 3 | 3 | no | — |
| 90 | C22×#41 | — | 3 | 3 | no | — |
| 91 | C22×#41 | — | 3 | 3 | no | — |
| 92 | C22×#41 | — | 6 | 3 | no | — |
| 93 | C22×#41 | — | 6 | 1 | no | — |
| 94 | C22×#41 | C2×D3 | 1 | 3 | yes | complemented |
| 95 | #578 | — | 3 | 16 | no | — |
| 96 | #578 | — | 3 | 24 | no | — |
| 97 | #578 | — | 3 | 24 | no | — |
| 98 | #577 | — | 3 | 48 | no | — |
| 99 | #577 | — | 3 | 8 | no | — |
| 100 | #577 | — | 3 | 16 | no | — |
| 101 | #577 | — | 3 | 24 | no | — |
| 102 | #577 | — | 3 | 24 | no | — |
| 103 | #577 | — | 3 | 8 | no | — |
| 104 | #576 | — | 3 | 48 | no | — |
| 105 | #576 | — | 3 | 8 | no | — |
| 106 | #576 | — | 3 | 8 | no | — |
| 107 | #575 | — | 3 | 24 | no | — |
| 108 | #575 | — | 3 | 48 | no | — |
| 109 | #575 | — | 3 | 24 | no | — |
| 110 | C2×#137 | — | 3 | 3 | no | — |
| 111 | C2×#137 | — | 3 | 3 | no | — |
| 112 | C2×#137 | — | 3 | 3 | no | — |
| 113 | C2×#137 | — | 3 | 6 | no | — |
| 114 | C2×#137 | — | 3 | 3 | no | — |
| 115 | C2×#137 | — | 3 | 6 | no | — |
| 116 | C2×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 117 | C2×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 118 | C2×AGL(1,ℤ/8) | — | 3 | 6 | no | — |
| 119 | C2×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 120 | C2×AGL(1,ℤ/8) | — | 3 | 6 | no | — |
| 121 | C2×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 122 | C2×#135 | — | 3 | 24 | no | — |
| 123 | C2×#135 | — | 3 | 48 | no | — |
| 124 | C2×#135 | — | 3 | 3 | no | — |
| 125 | C2×#135 | — | 3 | 6 | no | — |
| 126 | C2×#135 | — | 3 | 3 | no | — |
| 127 | C2×#135 | — | 3 | 3 | no | — |
| 128 | C2×#135 | — | 3 | 6 | no | — |
| 129 | C2×#135 | — | 3 | 3 | no | — |
| 130 | C2×#135 | — | 3 | 6 | no | — |
| 131 | C2×#135 | — | 3 | 1 | no | — |
| 132 | C2×#135 | — | 3 | 3 | no | — |
| 133 | C2×#135 | — | 3 | 3 | no | — |
| 134 | C2×#135 | — | 3 | 3 | no | — |
| 135 | C2×#135 | — | 3 | 24 | no | — |
| 136 | C22×Q16 | — | 3 | 3 | no | — |
| 137 | C22×Q16 | — | 3 | 3 | no | — |
| 138 | C22×SD16 | — | 3 | 3 | no | — |
| 139 | C22×SD16 | — | 3 | 3 | no | — |
| 140 | C22×SD16 | — | 3 | 3 | no | — |
| 141 | C22×SD16 | — | 3 | 3 | no | — |
| 142 | C22×D8 | — | 3 | 3 | no | — |
| 143 | C22×D8 | — | 3 | 3 | no | — |
| 144 | C2×#131 | — | 3 | 16 | no | — |
| 145 | C2×#131 | — | 3 | 1 | no | — |
| 146 | C2×#131 | — | 3 | 3 | no | — |
| 147 | C2×#131 | — | 3 | 1 | no | — |
| 148 | C2×#131 | — | 3 | 3 | no | — |
| 149 | C22×M16 | — | 3 | 3 | no | — |
| 150 | C23×C8 | — | 3 | 3 | no | — |
| 151 | C23×D3 | — | 4 | 3 | no | — |
| 152 | C3×#41 | — | 4 | 8 | no | — |
| 153 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 154 | C2×C3×D4 | — | 4 | 3 | no | — |
| 155 | C22×C3×C4 | — | 4 | 3 | no | — |
| 156 | C2×#67 | — | 4 | 6 | no | — |
| 157 | C22×Dic3 | — | 4 | 3 | no | — |
| 158 | #239 | — | 4 | 8 | no | — |
| 159 | D3×Q8 | — | 4 | 8 | no | — |
| 160 | #237 | — | 4 | 24 | no | — |
| 161 | D3×D4 | — | 4 | 24 | no | — |
| 162 | #235 | — | 4 | 24 | no | — |
| 163 | C2×D12 | — | 4 | 3 | no | — |
| 164 | C2×C4×D3 | — | 4 | 1 | no | — |
| 165 | C2×C4×D3 | — | 4 | 3 | no | — |
| 166 | C2×C4×D3 | — | 4 | 3 | no | — |
| 167 | C2×C4×D3 | — | 4 | 3 | no | — |
| 168 | C2×C4×D3 | — | 4 | 24 | no | — |
| 169 | C2×Dic6 | — | 4 | 3 | no | — |
| 170 | #41⋊C3 | C24 | 1 | 6 | yes | — |
| 171 | #41⋊C3 | C24 | 1 | 2 | yes | — |
| 172 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 173 | C2×SL(2,𝔽3) | C24 | 1 | 6 | yes | — |
| 174 | GL(2,𝔽3) | C24 | 1 | 2 | yes | — |
| 175 | GL(2,𝔽3) | C24 | 1 | 6 | yes | — |
| 176 | #226 | C24 | 1 | 2 | yes | — |
| 177 | #226 | C24 | 1 | 6 | yes | — |
| 178 | 2−1+4 | — | 3 | 8 | no | — |
| 179 | 2−1+4 | — | 3 | 12 | no | — |
| 180 | 2−1+4 | — | 3 | 24 | no | — |
| 181 | 2−1+4 | — | 3 | 8 | no | — |
| 182 | 2−1+4 | — | 6 | 4 | no | — |
| 183 | 2−1+4 | — | 6 | 16 | no | — |
| 184 | 2−1+4 | — | 6 | 12 | no | — |
| 185 | 2−1+4 | — | 6 | 4 | no | — |
| 186 | 2−1+4 | C22×D3 | 1 | 12 | yes | complemented |
| 187 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 188 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 189 | H(5,𝔽2) | — | 3 | 8 | no | — |
| 190 | H(5,𝔽2) | — | 3 | 24 | no | — |
| 191 | H(5,𝔽2) | — | 6 | 12 | no | — |
| 192 | H(5,𝔽2) | — | 6 | 48 | no | — |
| 193 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 194 | H(5,𝔽2) | — | 6 | 12 | no | — |
| 195 | H(5,𝔽2) | C22×D3 | 1 | 4 | yes | complemented |
| 196 | C2×#41 | — | 3 | 8 | no | — |
| 197 | C2×#41 | — | 3 | 8 | no | — |
| 198 | C2×#41 | — | 3 | 1 | no | — |
| 199 | C2×#41 | — | 3 | 3 | no | — |
| 200 | C2×#41 | — | 3 | 3 | no | — |
| 201 | C2×#41 | — | 3 | 3 | no | — |
| 202 | C2×#41 | — | 3 | 3 | no | — |
| 203 | C2×#41 | — | 3 | 12 | no | — |
| 204 | C2×#41 | — | 3 | 6 | no | — |
| 205 | C2×#41 | — | 3 | 24 | no | — |
| 206 | C2×#41 | — | 3 | 3 | no | — |
| 207 | C2×#41 | — | 3 | 1 | no | — |
| 208 | C2×#41 | — | 3 | 6 | no | — |
| 209 | C2×#41 | — | 3 | 3 | no | — |
| 210 | C2×#41 | — | 3 | 3 | no | — |
| 211 | C2×#41 | — | 3 | 24 | no | — |
| 212 | C2×#41 | — | 3 | 24 | no | — |
| 213 | C2×#41 | — | 6 | 3 | no | — |
| 214 | C2×#41 | — | 6 | 12 | no | — |
| 215 | C2×#41 | — | 6 | 3 | no | — |
| 216 | C2×#41 | — | 6 | 3 | no | — |
| 217 | C2×#41 | — | 6 | 3 | no | — |
| 218 | C2×#41 | — | 6 | 12 | no | — |
| 219 | C2×#41 | — | 6 | 12 | no | — |
| 220 | C2×#41 | — | 6 | 6 | no | — |
| 221 | C2×#41 | — | 6 | 3 | no | — |
| 222 | C2×#41 | — | 6 | 3 | no | — |
| 223 | C2×#41 | — | 6 | 48 | no | — |
| 224 | C2×#41 | — | 6 | 16 | no | — |
| 225 | C2×#41 | — | 6 | 1 | no | — |
| 226 | C2×#41 | — | 6 | 3 | no | — |
| 227 | C2×#41 | — | 6 | 3 | no | — |
| 228 | C2×#41 | S4 | 1 | 1 | yes | characteristic |
| 229 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 230 | C2×#41 | C22×D3 | 1 | 3 | yes | — |
| 231 | C2×#41 | C22×D3 | 1 | 12 | yes | complemented |
| 232 | C22×Q8 | — | 3 | 1 | no | — |
| 233 | C22×Q8 | — | 3 | 3 | no | — |
| 234 | C22×Q8 | — | 3 | 3 | no | — |
| 235 | C22×Q8 | — | 3 | 3 | no | — |
| 236 | C22×Q8 | — | 6 | 3 | no | — |
| 237 | C22×Q8 | — | 6 | 1 | no | — |
| 238 | C22×Q8 | C22×D3 | 1 | 3 | yes | — |
| 239 | C22×D4 | — | 3 | 3 | no | — |
| 240 | C22×D4 | — | 3 | 3 | no | — |
| 241 | C22×D4 | — | 3 | 3 | no | — |
| 242 | C22×D4 | — | 3 | 3 | no | — |
| 243 | C22×D4 | — | 3 | 3 | no | — |
| 244 | C22×D4 | — | 6 | 3 | no | — |
| 245 | C22×D4 | — | 6 | 6 | no | — |
| 246 | C22×D4 | — | 6 | 3 | no | — |
| 247 | C22×D4 | — | 6 | 3 | no | — |
| 248 | C22×D4 | — | 6 | 3 | no | — |
| 249 | C22×D4 | — | 6 | 3 | no | — |
| 250 | C23×C4 | — | 3 | 3 | no | — |
| 251 | C23×C4 | — | 6 | 1 | no | — |
| 252 | C23×C4 | — | 6 | 3 | no | — |
| 253 | C23×C4 | — | 6 | 3 | no | — |
| 254 | #137 | — | 3 | 24 | no | — |
| 255 | #137 | — | 3 | 24 | no | — |
| 256 | #137 | — | 3 | 24 | no | — |
| 257 | #137 | — | 3 | 12 | no | — |
| 258 | #137 | — | 3 | 24 | no | — |
| 259 | #137 | — | 3 | 48 | no | — |
| 260 | #137 | — | 3 | 12 | no | — |
| 261 | #137 | — | 3 | 24 | no | — |
| 262 | AGL(1,ℤ/8) | — | 3 | 24 | no | — |
| 263 | AGL(1,ℤ/8) | — | 3 | 24 | no | — |
| 264 | AGL(1,ℤ/8) | — | 3 | 48 | no | — |
| 265 | AGL(1,ℤ/8) | — | 3 | 12 | no | — |
| 266 | AGL(1,ℤ/8) | — | 3 | 24 | no | — |
| 267 | AGL(1,ℤ/8) | — | 3 | 48 | no | — |
| 268 | AGL(1,ℤ/8) | — | 3 | 12 | no | — |
| 269 | #135 | — | 3 | 4 | no | — |
| 270 | #135 | — | 3 | 24 | no | — |
| 271 | #135 | — | 3 | 12 | no | — |
| 272 | #135 | — | 3 | 48 | no | — |
| 273 | #135 | — | 3 | 24 | no | — |
| 274 | #135 | — | 3 | 12 | no | — |
| 275 | #135 | — | 3 | 24 | no | — |
| 276 | #135 | — | 3 | 48 | no | — |
| 277 | #135 | — | 3 | 12 | no | — |
| 278 | #135 | — | 3 | 24 | no | — |
| 279 | #135 | — | 3 | 48 | no | — |
| 280 | #135 | — | 3 | 4 | no | — |
| 281 | #135 | — | 3 | 12 | no | — |
| 282 | #135 | — | 3 | 12 | no | — |
| 283 | #135 | — | 3 | 12 | no | — |
| 284 | C2×Q16 | — | 3 | 3 | no | — |
| 285 | C2×Q16 | — | 3 | 6 | no | — |
| 286 | C2×Q16 | — | 3 | 3 | no | — |
| 287 | C2×Q16 | — | 3 | 12 | no | — |
| 288 | C2×Q16 | — | 3 | 24 | no | — |
| 289 | C2×Q16 | — | 3 | 1 | no | — |
| 290 | C2×Q16 | — | 3 | 3 | no | — |
| 291 | C2×Q16 | — | 3 | 12 | no | — |
| 292 | C2×SD16 | — | 3 | 3 | no | — |
| 293 | C2×SD16 | — | 3 | 1 | no | — |
| 294 | C2×SD16 | — | 3 | 3 | no | — |
| 295 | C2×SD16 | — | 3 | 6 | no | — |
| 296 | C2×SD16 | — | 3 | 3 | no | — |
| 297 | C2×SD16 | — | 3 | 3 | no | — |
| 298 | C2×SD16 | — | 3 | 6 | no | — |
| 299 | C2×SD16 | — | 3 | 3 | no | — |
| 300 | C2×SD16 | — | 3 | 24 | no | — |
| 301 | C2×SD16 | — | 3 | 24 | no | — |
| 302 | C2×SD16 | — | 3 | 12 | no | — |
| 303 | C2×SD16 | — | 3 | 24 | no | — |
| 304 | C2×SD16 | — | 3 | 1 | no | — |
| 305 | C2×SD16 | — | 3 | 3 | no | — |
| 306 | C2×SD16 | — | 3 | 12 | no | — |
| 307 | C2×D8 | — | 3 | 3 | no | — |
| 308 | C2×D8 | — | 3 | 1 | no | — |
| 309 | C2×D8 | — | 3 | 6 | no | — |
| 310 | C2×D8 | — | 3 | 3 | no | — |
| 311 | C2×D8 | — | 3 | 3 | no | — |
| 312 | C2×D8 | — | 3 | 24 | no | — |
| 313 | C2×D8 | — | 3 | 24 | no | — |
| 314 | #131 | — | 3 | 8 | no | — |
| 315 | #131 | — | 3 | 24 | no | — |
| 316 | #131 | — | 3 | 8 | no | — |
| 317 | #131 | — | 3 | 24 | no | — |
| 318 | C2×M16 | — | 3 | 6 | no | — |
| 319 | C2×M16 | — | 3 | 3 | no | — |
| 320 | C2×M16 | — | 3 | 3 | no | — |
| 321 | C2×M16 | — | 3 | 24 | no | — |
| 322 | C22×C8 | — | 3 | 1 | no | — |
| 323 | C22×C8 | — | 3 | 3 | no | — |
| 324 | C22×C8 | — | 3 | 3 | no | — |
| 325 | C22×C8 | — | 3 | 3 | no | — |
| 326 | C22×C8 | — | 3 | 3 | no | — |
| 327 | C22×C8 | — | 3 | 3 | no | — |
| 328 | C22×C8 | — | 3 | 24 | no | — |
| 329 | C23×C3 | — | 4 | 3 | no | — |
| 330 | C22×D3 | — | 4 | 1 | no | — |
| 331 | C22×D3 | — | 4 | 3 | no | — |
| 332 | C22×D3 | — | 4 | 12 | no | — |
| 333 | C22×D3 | — | 8 | 12 | no | — |
| 334 | C3×Q8 | — | 4 | 4 | no | — |
| 335 | C3×D4 | — | 4 | 12 | no | — |
| 336 | C2×C3×C4 | — | 4 | 3 | no | — |
| 337 | C2×C3×C4 | — | 4 | 12 | no | — |
| 338 | C2×C3×C4 | — | 4 | 1 | no | — |
| 339 | #67 | — | 4 | 24 | no | — |
| 340 | C2×Dic3 | — | 4 | 3 | no | — |
| 341 | C2×Dic3 | — | 4 | 1 | no | — |
| 342 | C2×Dic3 | — | 4 | 12 | no | — |
| 343 | D12 | — | 4 | 12 | no | — |
| 344 | C4×D3 | — | 4 | 4 | no | — |
| 345 | C4×D3 | — | 4 | 12 | no | — |
| 346 | C4×D3 | — | 4 | 12 | no | — |
| 347 | C4×D3 | — | 4 | 12 | no | — |
| 348 | Dic6 | — | 4 | 12 | no | — |
| 349 | SL(2,𝔽3) | C25 | 1 | 1 | yes | characteristic |
| 350 | C24 | — | 3 | 3 | no | — |
| 351 | C24 | — | 6 | 1 | no | — |
| 352 | C24 | — | 6 | 3 | no | — |
| 353 | C24 | — | 6 | 3 | no | — |
| 354 | #41 | — | 3 | 4 | no | — |
| 355 | #41 | — | 3 | 4 | no | — |
| 356 | #41 | — | 3 | 12 | no | — |
| 357 | #41 | — | 3 | 12 | no | — |
| 358 | #41 | — | 3 | 24 | no | — |
| 359 | #41 | — | 3 | 2 | no | — |
| 360 | #41 | — | 3 | 12 | no | — |
| 361 | #41 | — | 3 | 6 | no | — |
| 362 | #41 | — | 3 | 12 | no | — |
| 363 | #41 | — | 3 | 24 | no | — |
| 364 | #41 | — | 3 | 12 | no | — |
| 365 | #41 | — | 3 | 4 | no | — |
| 366 | #41 | — | 3 | 24 | no | — |
| 367 | #41 | — | 3 | 12 | no | — |
| 368 | #41 | — | 3 | 12 | no | — |
| 369 | #41 | — | 6 | 6 | no | — |
| 370 | #41 | — | 6 | 6 | no | — |
| 371 | #41 | — | 6 | 12 | no | — |
| 372 | #41 | — | 6 | 24 | no | — |
| 373 | #41 | — | 6 | 6 | no | — |
| 374 | #41 | — | 6 | 2 | no | — |
| 375 | #41 | — | 6 | 24 | no | — |
| 376 | #41 | — | 6 | 12 | no | — |
| 377 | #41 | — | 6 | 24 | no | — |
| 378 | #41 | — | 6 | 24 | no | — |
| 379 | #41 | — | 6 | 6 | no | — |
| 380 | #41 | — | 6 | 6 | no | — |
| 381 | #41 | — | 6 | 48 | no | — |
| 382 | #41 | — | 6 | 24 | no | — |
| 383 | #41 | — | 6 | 24 | no | — |
| 384 | #41 | — | 6 | 8 | no | — |
| 385 | #41 | — | 6 | 24 | no | — |
| 386 | #41 | — | 6 | 24 | no | — |
| 387 | #41 | C2×S4 | 1 | 8 | yes | — |
| 388 | #41 | C23×D3 | 1 | 2 | yes | — |
| 389 | #41 | C23×D3 | 1 | 6 | yes | — |
| 390 | C2×Q8 | — | 3 | 8 | no | — |
| 391 | C2×Q8 | — | 3 | 3 | no | — |
| 392 | C2×Q8 | — | 3 | 3 | no | — |
| 393 | C2×Q8 | — | 3 | 6 | no | — |
| 394 | C2×Q8 | — | 3 | 12 | no | — |
| 395 | C2×Q8 | — | 3 | 3 | no | — |
| 396 | C2×Q8 | — | 3 | 1 | no | — |
| 397 | C2×Q8 | — | 3 | 12 | no | — |
| 398 | C2×Q8 | — | 3 | 12 | no | — |
| 399 | C2×Q8 | — | 6 | 6 | no | — |
| 400 | C2×Q8 | — | 6 | 3 | no | — |
| 401 | C2×Q8 | — | 6 | 6 | no | — |
| 402 | C2×Q8 | — | 6 | 6 | no | — |
| 403 | C2×Q8 | — | 6 | 24 | no | — |
| 404 | C2×Q8 | — | 6 | 8 | no | — |
| 405 | C2×Q8 | — | 6 | 3 | no | — |
| 406 | C2×Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 407 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 408 | C2×Q8 | C23×D3 | 1 | 6 | yes | — |
| 409 | C2×D4 | — | 3 | 6 | no | — |
| 410 | C2×D4 | — | 3 | 24 | no | — |
| 411 | C2×D4 | — | 3 | 12 | no | — |
| 412 | C2×D4 | — | 3 | 1 | no | — |
| 413 | C2×D4 | — | 3 | 3 | no | — |
| 414 | C2×D4 | — | 3 | 12 | no | — |
| 415 | C2×D4 | — | 3 | 3 | no | — |
| 416 | C2×D4 | — | 3 | 1 | no | — |
| 417 | C2×D4 | — | 3 | 12 | no | — |
| 418 | C2×D4 | — | 3 | 12 | no | — |
| 419 | C2×D4 | — | 3 | 3 | no | — |
| 420 | C2×D4 | — | 6 | 6 | no | — |
| 421 | C2×D4 | — | 6 | 3 | no | — |
| 422 | C2×D4 | — | 6 | 6 | no | — |
| 423 | C2×D4 | — | 6 | 3 | no | — |
| 424 | C2×D4 | — | 6 | 6 | no | — |
| 425 | C2×D4 | — | 6 | 6 | no | — |
| 426 | C2×D4 | — | 6 | 24 | no | — |
| 427 | C2×D4 | — | 6 | 48 | no | — |
| 428 | C2×D4 | — | 6 | 24 | no | — |
| 429 | C2×D4 | — | 6 | 3 | no | — |
| 430 | C2×D4 | — | 6 | 1 | no | — |
| 431 | C2×D4 | — | 6 | 3 | no | — |
| 432 | C2×D4 | — | 6 | 3 | no | — |
| 433 | C2×D4 | — | 6 | 24 | no | — |
| 434 | C2×D4 | — | 6 | 24 | no | — |
| 435 | C2×D4 | — | 6 | 6 | no | — |
| 436 | C2×D4 | — | 6 | 24 | no | — |
| 437 | C2×D4 | — | 6 | 3 | no | — |
| 438 | C2×D4 | C2×S4 | 1 | 3 | yes | — |
| 439 | C22×C4 | — | 3 | 3 | no | — |
| 440 | C22×C4 | — | 3 | 3 | no | — |
| 441 | C22×C4 | — | 3 | 12 | no | — |
| 442 | C22×C4 | — | 3 | 3 | no | — |
| 443 | C22×C4 | — | 3 | 12 | no | — |
| 444 | C22×C4 | — | 3 | 1 | no | — |
| 445 | C22×C4 | — | 3 | 3 | no | — |
| 446 | C22×C4 | — | 6 | 8 | no | — |
| 447 | C22×C4 | — | 6 | 3 | no | — |
| 448 | C22×C4 | — | 6 | 6 | no | — |
| 449 | C22×C4 | — | 6 | 24 | no | — |
| 450 | C22×C4 | — | 6 | 3 | no | — |
| 451 | C22×C4 | — | 6 | 3 | no | — |
| 452 | C22×C4 | — | 6 | 3 | no | — |
| 453 | C22×C4 | — | 6 | 3 | no | — |
| 454 | C22×C4 | — | 6 | 24 | no | — |
| 455 | C22×C4 | — | 6 | 3 | no | — |
| 456 | C22×C4 | — | 6 | 1 | no | — |
| 457 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 458 | Q16 | — | 3 | 12 | no | — |
| 459 | Q16 | — | 3 | 2 | no | — |
| 460 | Q16 | — | 3 | 24 | no | — |
| 461 | Q16 | — | 3 | 12 | no | — |
| 462 | Q16 | — | 3 | 6 | no | — |
| 463 | Q16 | — | 3 | 2 | no | — |
| 464 | Q16 | — | 3 | 6 | no | — |
| 465 | SD16 | — | 3 | 12 | no | — |
| 466 | SD16 | — | 3 | 4 | no | — |
| 467 | SD16 | — | 3 | 12 | no | — |
| 468 | SD16 | — | 3 | 2 | no | — |
| 469 | SD16 | — | 3 | 24 | no | — |
| 470 | SD16 | — | 3 | 12 | no | — |
| 471 | SD16 | — | 3 | 12 | no | — |
| 472 | SD16 | — | 3 | 24 | no | — |
| 473 | SD16 | — | 3 | 12 | no | — |
| 474 | SD16 | — | 3 | 6 | no | — |
| 475 | SD16 | — | 3 | 2 | no | — |
| 476 | SD16 | — | 3 | 6 | no | — |
| 477 | D8 | — | 3 | 12 | no | — |
| 478 | D8 | — | 3 | 4 | no | — |
| 479 | D8 | — | 3 | 24 | no | — |
| 480 | D8 | — | 3 | 12 | no | — |
| 481 | D8 | — | 3 | 12 | no | — |
| 482 | M16 | — | 3 | 24 | no | — |
| 483 | M16 | — | 3 | 12 | no | — |
| 484 | M16 | — | 3 | 12 | no | — |
| 485 | C2×C8 | — | 3 | 1 | no | — |
| 486 | C2×C8 | — | 3 | 1 | no | — |
| 487 | C2×C8 | — | 3 | 4 | no | — |
| 488 | C2×C8 | — | 3 | 3 | no | — |
| 489 | C2×C8 | — | 3 | 3 | no | — |
| 490 | C2×C8 | — | 3 | 12 | no | — |
| 491 | C2×C8 | — | 3 | 12 | no | — |
| 492 | C2×C8 | — | 3 | 12 | no | — |
| 493 | C2×C8 | — | 3 | 12 | no | — |
| 494 | C2×C8 | — | 3 | 12 | no | — |
| 495 | C22×C3 | — | 4 | 1 | no | — |
| 496 | C22×C3 | — | 4 | 6 | no | — |
| 497 | C22×C3 | — | 8 | 6 | no | — |
| 498 | C2×D3 | — | 4 | 2 | no | — |
| 499 | C2×D3 | — | 4 | 6 | no | — |
| 500 | C2×D3 | — | 4 | 4 | no | — |
| 501 | C2×D3 | — | 8 | 6 | no | — |
| 502 | C2×D3 | — | 8 | 24 | no | — |
| 503 | C3×C4 | — | 4 | 6 | no | — |
| 504 | C3×C4 | — | 4 | 2 | no | — |
| 505 | Dic3 | — | 4 | 6 | no | — |
| 506 | Dic3 | — | 4 | 2 | no | — |
| 507 | C23 | — | 3 | 1 | no | — |
| 508 | C23 | — | 3 | 3 | no | — |
| 509 | C23 | — | 3 | 12 | no | — |
| 510 | C23 | — | 6 | 4 | no | — |
| 511 | C23 | — | 6 | 12 | no | — |
| 512 | C23 | — | 6 | 1 | no | — |
| 513 | C23 | — | 6 | 3 | no | — |
| 514 | C23 | — | 6 | 12 | no | — |
| 515 | C23 | — | 12 | 6 | no | — |
| 516 | C23 | — | 24 | 6 | no | — |
| 517 | C23 | — | 24 | 2 | no | — |
| 518 | C23 | — | 24 | 6 | no | — |
| 519 | C23 | C22×S4 | 1 | 3 | yes | — |
| 520 | Q8 | — | 3 | 1 | no | — |
| 521 | Q8 | — | 3 | 6 | no | — |
| 522 | Q8 | — | 3 | 12 | no | — |
| 523 | Q8 | — | 3 | 6 | no | — |
| 524 | Q8 | — | 3 | 2 | no | — |
| 525 | Q8 | — | 6 | 3 | no | — |
| 526 | Q8 | — | 6 | 3 | no | — |
| 527 | Q8 | — | 6 | 1 | no | — |
| 528 | Q8 | — | 6 | 12 | no | — |
| 529 | Q8 | — | 6 | 12 | no | — |
| 530 | Q8 | C22×S4 | 1 | 4 | yes | — |
| 531 | Q8 | C24×D3 | 1 | 1 | yes | characteristic |
| 532 | D4 | — | 3 | 2 | no | — |
| 533 | D4 | — | 3 | 6 | no | — |
| 534 | D4 | — | 3 | 24 | no | — |
| 535 | D4 | — | 3 | 6 | no | — |
| 536 | D4 | — | 3 | 2 | no | — |
| 537 | D4 | — | 3 | 12 | no | — |
| 538 | D4 | — | 6 | 1 | no | — |
| 539 | D4 | — | 6 | 3 | no | — |
| 540 | D4 | — | 6 | 12 | no | — |
| 541 | D4 | — | 6 | 1 | no | — |
| 542 | D4 | — | 6 | 3 | no | — |
| 543 | D4 | — | 6 | 12 | no | — |
| 544 | D4 | — | 6 | 12 | no | — |
| 545 | D4 | — | 6 | 4 | no | — |
| 546 | D4 | — | 6 | 12 | no | — |
| 547 | D4 | — | 6 | 12 | no | — |
| 548 | D4 | — | 6 | 24 | no | — |
| 549 | D4 | — | 6 | 12 | no | — |
| 550 | D4 | C22×S4 | 1 | 12 | yes | — |
| 551 | C2×C4 | — | 3 | 1 | no | — |
| 552 | C2×C4 | — | 3 | 6 | no | — |
| 553 | C2×C4 | — | 3 | 3 | no | — |
| 554 | C2×C4 | — | 3 | 6 | no | — |
| 555 | C2×C4 | — | 3 | 2 | no | — |
| 556 | C2×C4 | — | 3 | 6 | no | — |
| 557 | C2×C4 | — | 3 | 2 | no | — |
| 558 | C2×C4 | — | 3 | 12 | no | — |
| 559 | C2×C4 | — | 3 | 12 | no | — |
| 560 | C2×C4 | — | 3 | 6 | no | — |
| 561 | C2×C4 | — | 6 | 12 | no | — |
| 562 | C2×C4 | — | 6 | 24 | no | — |
| 563 | C2×C4 | — | 6 | 12 | no | — |
| 564 | C2×C4 | — | 6 | 3 | no | — |
| 565 | C2×C4 | — | 6 | 1 | no | — |
| 566 | C2×C4 | — | 6 | 12 | no | — |
| 567 | C2×C4 | — | 6 | 12 | no | — |
| 568 | C2×C4 | — | 6 | 4 | no | — |
| 569 | C2×C4 | — | 6 | 12 | no | — |
| 570 | C2×C4 | — | 6 | 12 | no | — |
| 571 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 572 | C2×C4 | C22×S4 | 1 | 12 | yes | — |
| 573 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 574 | C8 | — | 3 | 2 | no | — |
| 575 | C8 | — | 3 | 2 | no | — |
| 576 | C8 | — | 3 | 6 | no | — |
| 577 | C8 | — | 3 | 6 | no | — |
| 578 | C2×C3 | — | 4 | 1 | no | — |
| 579 | C2×C3 | — | 4 | 2 | no | — |
| 580 | C2×C3 | — | 8 | 6 | no | — |
| 581 | D3 | — | 4 | 4 | no | — |
| 582 | D3 | — | 8 | 6 | no | — |
| 583 | C22 | — | 2 | 6 | no | — |
| 584 | C22 | — | 3 | 2 | no | — |
| 585 | C22 | — | 3 | 6 | no | — |
| 586 | C22 | — | 6 | 2 | no | — |
| 587 | C22 | — | 6 | 6 | no | — |
| 588 | C22 | — | 6 | 2 | no | — |
| 589 | C22 | — | 6 | 6 | no | — |
| 590 | C22 | — | 12 | 12 | no | — |
| 591 | C22 | — | 12 | 2 | no | — |
| 592 | C22 | — | 12 | 6 | no | — |
| 593 | C22 | — | 24 | 12 | no | — |
| 594 | C22 | — | 24 | 4 | no | — |
| 595 | C22 | — | 24 | 12 | no | — |
| 596 | C22 | C23×S4 | 1 | 6 | yes | — |
| 597 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 598 | C4 | — | 3 | 1 | no | — |
| 599 | C4 | — | 3 | 1 | no | — |
| 600 | C4 | — | 3 | 6 | no | — |
| 601 | C4 | — | 6 | 6 | no | — |
| 602 | C4 | — | 6 | 2 | no | — |
| 603 | C4 | C23×S4 | 1 | 2 | yes | — |
| 604 | C4 | C23×S4 | 1 | 6 | yes | — |
| 605 | C3 | — | 4 | 1 | no | Sylow |
| 606 | C2 | — | 2 | 6 | no | — |
| 607 | C2 | — | 6 | 2 | no | — |
| 608 | C2 | — | 6 | 6 | no | — |
| 609 | C2 | — | 12 | 2 | no | — |
| 610 | C2 | — | 12 | 6 | no | — |
| 611 | C2 | #87497 | 1 | 2 | yes | direct product factor |
| 612 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C2×#585 | D3 |
| socle | C22 | C23×S4 |
| hypercenter | C2×#41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5945 | C2 | semidirect |
| 2. #87497 | C2 | direct (N × H) |
| 3. C2×#5927 | C2 | semidirect |
| 4. C2×#5926 | C2 | semidirect |
| 5. C2×#5925 | C2 | semidirect |
| 6. C2×#5924 | C2 | semidirect |
| 7. C2×#5923 | C2 | semidirect |
| 8. C22×#985 | C2 | semidirect |
| 11. #5945 | C22 | semidirect |
| 12. C2×2−1+4⋊C3 | C22 | semidirect |
| 13. C2×#994 | C22 | semidirect |
| 14. C22×#41⋊C3 | C22 | semidirect |
| 15. #5927 | C22 | semidirect |
| 16. #5926 | C22 | semidirect |
| 17. #5925 | C22 | semidirect |
| 18. #5924 | C22 | semidirect |
| 19. #5923 | C22 | semidirect |
| 20. C2×#986 | C22 | semidirect |
| 21. C2×#985 | C22 | semidirect |
| 22. C2×#985 | C22 | semidirect |
| 23. C2×#985 | C22 | semidirect |
| 24. C2×#985 | C22 | semidirect |
| 25. C2×#985 | C22 | semidirect |
| 26. C2×#984 | C22 | semidirect |
| 27. C2×#983 | C22 | semidirect |
| 28. C22×GL(2,𝔽3) | C22 | semidirect |
| 29. C22×#226 | C22 | semidirect |
| 31. C2×#585 | D3 | semidirect |
| 55. 2−1+4⋊C3 | C23 | semidirect |
| 56. #994 | C23 | semidirect |
| 59. C2×#41⋊C3 | C23 | semidirect |
| 61. #986 | C23 | semidirect |
| 62. #985 | C23 | semidirect |
| 63. #985 | C23 | semidirect |
| 64. #985 | C23 | semidirect |
| 65. #985 | C23 | semidirect |
| 66. #984 | C23 | semidirect |
| 67. #983 | C23 | semidirect |
| 70. C2×GL(2,𝔽3) | C23 | semidirect |
| 73. C2×#226 | C23 | semidirect |
| 77. #585 | C2×D3 | semidirect |
| 82. C2×2−1+4 | C2×D3 | semidirect |
| 87. C2×H(5,𝔽2) | C2×D3 | semidirect |
| 94. C22×#41 | C2×D3 | semidirect |
| 186. 2−1+4 | C22×D3 | semidirect |
| 195. H(5,𝔽2) | C22×D3 | semidirect |
| 231. C2×#41 | C22×D3 | semidirect |
| 611. C2 | #87497 | 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 |
| #87497 | 2 |
| C2×#5927 | 2 |
| C2×#5926 | 2 |
| C2×#5925 | 2 |
| C2×#5924 | 2 |
| C2×#5923 | 2 |
| C22×#985 | 2 |
| C2×#3584 | 3 |
| C2×D3×#41 | 4 |