← C3×D4×H(5,𝔽2) #10681588
Subgroups — C3×D4×H(5,𝔽2) #10681588
Order 768.
12414 conjugacy classes of subgroups.
The subgroup lattice has 410 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3×H(5,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#3498 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C3×#3496 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | C3×#3488 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | C3×#3484 | C2 | 1 | 18 | yes | maximal, complemented |
| 6 | C3×D4×#41 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×C3×D42 | C2 | 1 | 9 | yes | maximal, complemented |
| 8 | C3×C4×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | D4×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 10 | C2×C3×H(5,𝔽2) | — | 2 | 32 | no | — |
| 11 | C2×C3×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×H(5,𝔽2) | C22 | 1 | 18 | yes | complemented |
| 13 | C2×C3×H(5,𝔽2) | C22 | 1 | 12 | yes | complemented |
| 14 | C22×C3×#41 | C22 | 1 | 12 | yes | complemented |
| 15 | C23×C3×D4 | C22 | 1 | 18 | yes | complemented |
| 16 | C23×C3×D4 | C22 | 1 | 6 | yes | complemented |
| 17 | C3×#550 | C22 | 1 | 6 | yes | complemented |
| 18 | C3×#550 | C22 | 1 | 12 | yes | complemented |
| 19 | C3×D4×Q8 | C22 | 1 | 2 | yes | complemented |
| 20 | C3×#548 | C22 | 1 | 12 | yes | complemented |
| 21 | C3×#547 | C22 | 1 | 18 | yes | complemented |
| 22 | C3×#546 | C22 | 1 | 36 | yes | complemented |
| 23 | C3×#546 | C22 | 1 | 72 | yes | complemented |
| 24 | C3×D42 | C22 | 1 | 18 | yes | complemented |
| 25 | C3×D42 | C22 | 1 | 72 | yes | complemented |
| 26 | C3×D42 | C22 | 1 | 36 | yes | complemented |
| 27 | C3×D42 | C22 | 1 | 36 | yes | complemented |
| 28 | C3×#537 | C22 | 1 | 18 | yes | complemented |
| 29 | C3×#535 | C22 | 1 | 36 | yes | complemented |
| 30 | C3×#534 | C22 | 1 | 36 | yes | complemented |
| 31 | C3×#532 | C22 | 1 | 18 | yes | complemented |
| 32 | C2×C3×#127 | C22 | 1 | 9 | yes | complemented |
| 33 | C3×#525 | C22 | 1 | 36 | yes | complemented |
| 34 | C2×C3×#121 | C22 | 1 | 18 | yes | complemented |
| 35 | C2×C3×#121 | C22 | 1 | 18 | yes | complemented |
| 36 | C2×C3×#120 | C22 | 1 | 36 | yes | complemented |
| 37 | C3×#520 | C22 | 1 | 6 | yes | complemented |
| 38 | C3×#518 | C22 | 1 | 9 | yes | complemented |
| 39 | C3×C4×#41 | C22 | 1 | 6 | yes | complemented |
| 40 | C2×C3×C4×D4 | C22 | 1 | 9 | yes | complemented |
| 41 | C2×C3×C4×D4 | C22 | 1 | 9 | yes | complemented |
| 42 | C22×H(5,𝔽2) | C2×C3 | 1 | 2 | yes | complemented |
| 43 | #3498 | C2×C3 | 1 | 6 | yes | complemented |
| 44 | #3496 | C2×C3 | 1 | 9 | yes | complemented |
| 45 | #3488 | C2×C3 | 1 | 12 | yes | complemented |
| 46 | #3484 | C2×C3 | 1 | 18 | yes | complemented |
| 47 | D4×#41 | C2×C3 | 1 | 6 | yes | complemented |
| 48 | C2×D42 | C2×C3 | 1 | 9 | yes | complemented |
| 49 | C4×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 50 | C25×C3 | C23 | 1 | 12 | yes | complemented |
| 51 | C3×H(5,𝔽2) | — | 2 | 96 | no | — |
| 52 | C3×H(5,𝔽2) | — | 2 | 144 | no | — |
| 53 | C3×H(5,𝔽2) | D4 | 1 | 16 | yes | direct product factor |
| 54 | C2×C3×#41 | — | 2 | 96 | no | — |
| 55 | C2×C3×#41 | C23 | 1 | 6 | yes | — |
| 56 | C2×C3×#41 | C23 | 1 | 36 | yes | complemented |
| 57 | C2×C3×#41 | C23 | 1 | 36 | yes | complemented |
| 58 | C2×C3×#41 | C23 | 1 | 12 | yes | complemented |
| 59 | C22×C3×Q8 | C23 | 1 | 4 | yes | complemented |
| 60 | C22×C3×D4 | — | 2 | 144 | no | — |
| 61 | C22×C3×D4 | — | 4 | 24 | no | — |
| 62 | C22×C3×D4 | C23 | 1 | 9 | yes | — |
| 63 | C22×C3×D4 | C23 | 1 | 9 | yes | — |
| 64 | C22×C3×D4 | C23 | 1 | 36 | yes | complemented |
| 65 | C22×C3×D4 | C23 | 1 | 36 | yes | complemented |
| 66 | C22×C3×D4 | C23 | 1 | 18 | yes | complemented |
| 67 | C22×C3×D4 | C23 | 1 | 72 | yes | complemented |
| 68 | C22×C3×D4 | C23 | 1 | 36 | yes | complemented |
| 69 | C22×C3×D4 | C23 | 1 | 18 | yes | complemented |
| 70 | C22×C3×D4 | C23 | 1 | 18 | yes | complemented |
| 71 | C23×C3×C4 | C23 | 1 | 18 | yes | complemented |
| 72 | C23×C3×C4 | C23 | 1 | 6 | yes | complemented |
| 73 | C3×C4⋊Q8 | C23 | 1 | 6 | yes | complemented |
| 74 | C3×#127 | C23 | 1 | 18 | yes | complemented |
| 75 | C3×#127 | C23 | 1 | 36 | yes | complemented |
| 76 | C3×#124 | C23 | 1 | 36 | yes | complemented |
| 77 | C3×#123 | C23 | 1 | 72 | yes | complemented |
| 78 | C3×C22⋊Q8 | C23 | 1 | 12 | yes | complemented |
| 79 | C3×C22⋊Q8 | C23 | 1 | 36 | yes | complemented |
| 80 | C3×#121 | C23 | 1 | 36 | yes | complemented |
| 81 | C3×#121 | C23 | 1 | 36 | yes | complemented |
| 82 | C3×#121 | C23 | 1 | 72 | yes | complemented |
| 83 | C3×#121 | C23 | 1 | 72 | yes | complemented |
| 84 | C3×#121 | C23 | 1 | 36 | yes | complemented |
| 85 | C3×#121 | C23 | 1 | 36 | yes | complemented |
| 86 | C3×#120 | C23 | 1 | 72 | yes | complemented |
| 87 | C3×#120 | C23 | 1 | 144 | yes | complemented |
| 88 | C3×C4×Q8 | C23 | 1 | 2 | yes | complemented |
| 89 | C3×C4×Q8 | C23 | 1 | 6 | yes | complemented |
| 90 | C3×C4×D4 | C23 | 1 | 6 | yes | — |
| 91 | C3×C4×D4 | C23 | 1 | 18 | yes | complemented |
| 92 | C3×C4×D4 | C23 | 1 | 36 | yes | complemented |
| 93 | C3×C4×D4 | C23 | 1 | 36 | yes | complemented |
| 94 | C3×C4×D4 | C23 | 1 | 18 | yes | complemented |
| 95 | C3×C4×D4 | C23 | 1 | 18 | yes | complemented |
| 96 | C3×C4×D4 | C23 | 1 | 36 | yes | complemented |
| 97 | C3×#117 | C23 | 1 | 18 | yes | complemented |
| 98 | C2×C3×C4⋊C4 | C23 | 1 | 9 | yes | complemented |
| 99 | C2×C3×C4⋊C4 | C23 | 1 | 9 | yes | complemented |
| 100 | C2×C3×C22⋊C4 | C23 | 1 | 18 | yes | complemented |
| 101 | C2×C3×C22⋊C4 | C23 | 1 | 18 | yes | complemented |
| 102 | C2×C3×C22⋊C4 | C23 | 1 | 36 | yes | complemented |
| 103 | C2×C3×C42 | C23 | 1 | 9 | yes | complemented |
| 104 | C2×H(5,𝔽2) | — | 2 | 32 | no | — |
| 105 | C2×H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic |
| 106 | C2×H(5,𝔽2) | C22×C3 | 1 | 18 | yes | complemented |
| 107 | C2×H(5,𝔽2) | C22×C3 | 1 | 12 | yes | complemented |
| 108 | C22×#41 | C22×C3 | 1 | 12 | yes | complemented |
| 109 | C23×D4 | C22×C3 | 1 | 18 | yes | complemented |
| 110 | C23×D4 | C22×C3 | 1 | 6 | yes | complemented |
| 111 | #550 | C22×C3 | 1 | 6 | yes | complemented |
| 112 | #550 | C22×C3 | 1 | 12 | yes | complemented |
| 113 | D4×Q8 | C22×C3 | 1 | 2 | yes | complemented |
| 114 | #548 | C22×C3 | 1 | 12 | yes | complemented |
| 115 | #547 | C22×C3 | 1 | 18 | yes | complemented |
| 116 | #546 | C22×C3 | 1 | 36 | yes | complemented |
| 117 | #546 | C22×C3 | 1 | 72 | yes | complemented |
| 118 | D42 | C22×C3 | 1 | 18 | yes | complemented |
| 119 | D42 | C22×C3 | 1 | 72 | yes | complemented |
| 120 | D42 | C22×C3 | 1 | 36 | yes | complemented |
| 121 | D42 | C22×C3 | 1 | 36 | yes | complemented |
| 122 | #537 | C22×C3 | 1 | 18 | yes | complemented |
| 123 | #535 | C22×C3 | 1 | 36 | yes | complemented |
| 124 | #534 | C22×C3 | 1 | 36 | yes | complemented |
| 125 | #532 | C22×C3 | 1 | 18 | yes | complemented |
| 126 | C2×#127 | C22×C3 | 1 | 9 | yes | complemented |
| 127 | #525 | C22×C3 | 1 | 36 | yes | complemented |
| 128 | C2×#121 | C22×C3 | 1 | 18 | yes | complemented |
| 129 | C2×#121 | C22×C3 | 1 | 18 | yes | complemented |
| 130 | C2×#120 | C22×C3 | 1 | 36 | yes | complemented |
| 131 | #520 | C22×C3 | 1 | 6 | yes | complemented |
| 132 | #518 | C22×C3 | 1 | 9 | yes | complemented |
| 133 | C4×#41 | C22×C3 | 1 | 6 | yes | complemented |
| 134 | C2×C4×D4 | C22×C3 | 1 | 9 | yes | complemented |
| 135 | C2×C4×D4 | C22×C3 | 1 | 9 | yes | complemented |
| 136 | C24×C3 | — | 2 | 48 | no | — |
| 137 | C24×C3 | — | 4 | 24 | no | — |
| 138 | C24×C3 | — | 8 | 12 | no | — |
| 139 | C24×C3 | C24 | 1 | 18 | yes | — |
| 140 | C24×C3 | C24 | 1 | 36 | yes | — |
| 141 | C24×C3 | C24 | 1 | 6 | yes | — |
| 142 | C3×#41 | — | 2 | 48 | no | — |
| 143 | C3×#41 | — | 2 | 144 | no | — |
| 144 | C3×#41 | — | 2 | 144 | no | — |
| 145 | C3×#41 | C2×D4 | 1 | 48 | yes | complemented |
| 146 | C2×C3×Q8 | — | 2 | 16 | no | — |
| 147 | C2×C3×Q8 | C24 | 1 | 12 | yes | — |
| 148 | C2×C3×Q8 | C24 | 1 | 2 | yes | — |
| 149 | C2×C3×D4 | — | 2 | 144 | no | — |
| 150 | C2×C3×D4 | — | 2 | 144 | no | — |
| 151 | C2×C3×D4 | — | 2 | 288 | no | — |
| 152 | C2×C3×D4 | — | 2 | 36 | no | — |
| 153 | C2×C3×D4 | — | 2 | 72 | no | — |
| 154 | C2×C3×D4 | — | 4 | 36 | no | — |
| 155 | C2×C3×D4 | — | 4 | 72 | no | — |
| 156 | C2×C3×D4 | — | 8 | 18 | no | — |
| 157 | C2×C3×D4 | C2×D4 | 1 | 72 | yes | complemented |
| 158 | C2×C3×D4 | C24 | 1 | 18 | yes | — |
| 159 | C2×C3×D4 | C24 | 1 | 9 | yes | — |
| 160 | C2×C3×D4 | C24 | 1 | 72 | yes | — |
| 161 | C2×C3×D4 | C24 | 1 | 36 | yes | — |
| 162 | C2×C3×D4 | C24 | 1 | 36 | yes | — |
| 163 | C2×C3×D4 | C24 | 1 | 36 | yes | — |
| 164 | C2×C3×D4 | C24 | 1 | 18 | yes | — |
| 165 | C2×C3×D4 | C24 | 1 | 1 | yes | characteristic |
| 166 | C22×C3×C4 | — | 2 | 72 | no | — |
| 167 | C22×C3×C4 | — | 4 | 12 | no | — |
| 168 | C22×C3×C4 | C24 | 1 | 9 | yes | — |
| 169 | C22×C3×C4 | C24 | 1 | 18 | yes | — |
| 170 | C22×C3×C4 | C24 | 1 | 12 | yes | — |
| 171 | C22×C3×C4 | C24 | 1 | 36 | yes | — |
| 172 | C22×C3×C4 | C24 | 1 | 9 | yes | — |
| 173 | C22×C3×C4 | C24 | 1 | 18 | yes | — |
| 174 | C22×C3×C4 | C24 | 1 | 9 | yes | — |
| 175 | C3×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 176 | C3×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 177 | C3×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 178 | C3×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 179 | C3×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 180 | C3×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 181 | C3×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 182 | C3×C22⋊C4 | C24 | 1 | 72 | yes | — |
| 183 | C3×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 184 | C3×C42 | C24 | 1 | 6 | yes | — |
| 185 | C3×C42 | C24 | 1 | 18 | yes | — |
| 186 | C25 | C23×C3 | 1 | 12 | yes | complemented |
| 187 | H(5,𝔽2) | — | 2 | 96 | no | — |
| 188 | H(5,𝔽2) | — | 2 | 144 | no | — |
| 189 | H(5,𝔽2) | C3×D4 | 1 | 16 | yes | direct product factor |
| 190 | C2×#41 | — | 2 | 96 | no | — |
| 191 | C2×#41 | C23×C3 | 1 | 6 | yes | — |
| 192 | C2×#41 | C23×C3 | 1 | 36 | yes | complemented |
| 193 | C2×#41 | C23×C3 | 1 | 36 | yes | complemented |
| 194 | C2×#41 | C23×C3 | 1 | 12 | yes | complemented |
| 195 | C22×Q8 | C23×C3 | 1 | 4 | yes | complemented |
| 196 | C22×D4 | — | 2 | 144 | no | — |
| 197 | C22×D4 | — | 4 | 24 | no | — |
| 198 | C22×D4 | C23×C3 | 1 | 9 | yes | — |
| 199 | C22×D4 | C23×C3 | 1 | 9 | yes | — |
| 200 | C22×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 201 | C22×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 202 | C22×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 203 | C22×D4 | C23×C3 | 1 | 72 | yes | complemented |
| 204 | C22×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 205 | C22×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 206 | C22×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 207 | C23×C4 | C23×C3 | 1 | 18 | yes | complemented |
| 208 | C23×C4 | C23×C3 | 1 | 6 | yes | complemented |
| 209 | C4⋊Q8 | C23×C3 | 1 | 6 | yes | complemented |
| 210 | #127 | C23×C3 | 1 | 18 | yes | complemented |
| 211 | #127 | C23×C3 | 1 | 36 | yes | complemented |
| 212 | #124 | C23×C3 | 1 | 36 | yes | complemented |
| 213 | #123 | C23×C3 | 1 | 72 | yes | complemented |
| 214 | C22⋊Q8 | C23×C3 | 1 | 12 | yes | complemented |
| 215 | C22⋊Q8 | C23×C3 | 1 | 36 | yes | complemented |
| 216 | #121 | C23×C3 | 1 | 36 | yes | complemented |
| 217 | #121 | C23×C3 | 1 | 36 | yes | complemented |
| 218 | #121 | C23×C3 | 1 | 72 | yes | complemented |
| 219 | #121 | C23×C3 | 1 | 72 | yes | complemented |
| 220 | #121 | C23×C3 | 1 | 36 | yes | complemented |
| 221 | #121 | C23×C3 | 1 | 36 | yes | complemented |
| 222 | #120 | C23×C3 | 1 | 72 | yes | complemented |
| 223 | #120 | C23×C3 | 1 | 144 | yes | complemented |
| 224 | C4×Q8 | C23×C3 | 1 | 2 | yes | complemented |
| 225 | C4×Q8 | C23×C3 | 1 | 6 | yes | complemented |
| 226 | C4×D4 | C23×C3 | 1 | 6 | yes | — |
| 227 | C4×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 228 | C4×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 229 | C4×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 230 | C4×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 231 | C4×D4 | C23×C3 | 1 | 18 | yes | complemented |
| 232 | C4×D4 | C23×C3 | 1 | 36 | yes | complemented |
| 233 | #117 | C23×C3 | 1 | 18 | yes | complemented |
| 234 | C2×C4⋊C4 | C23×C3 | 1 | 9 | yes | complemented |
| 235 | C2×C4⋊C4 | C23×C3 | 1 | 9 | yes | complemented |
| 236 | C2×C22⋊C4 | C23×C3 | 1 | 18 | yes | complemented |
| 237 | C2×C22⋊C4 | C23×C3 | 1 | 18 | yes | complemented |
| 238 | C2×C22⋊C4 | C23×C3 | 1 | 36 | yes | complemented |
| 239 | C2×C42 | C23×C3 | 1 | 9 | yes | complemented |
| 240 | C23×C3 | — | 2 | 36 | no | — |
| 241 | C23×C3 | — | 2 | 36 | no | — |
| 242 | C23×C3 | — | 2 | 72 | no | — |
| 243 | C23×C3 | — | 4 | 6 | no | — |
| 244 | C23×C3 | — | 4 | 6 | no | — |
| 245 | C23×C3 | — | 4 | 18 | no | — |
| 246 | C23×C3 | — | 4 | 36 | no | — |
| 247 | C23×C3 | — | 4 | 36 | no | — |
| 248 | C23×C3 | — | 8 | 96 | no | — |
| 249 | C23×C3 | C22×D4 | 1 | 24 | yes | complemented |
| 250 | C23×C3 | C25 | 1 | 18 | yes | — |
| 251 | C23×C3 | C25 | 1 | 9 | yes | — |
| 252 | C23×C3 | C25 | 1 | 2 | yes | — |
| 253 | C3×Q8 | — | 2 | 24 | no | — |
| 254 | C3×Q8 | C22×D4 | 1 | 8 | yes | complemented |
| 255 | C3×D4 | — | 2 | 144 | no | — |
| 256 | C3×D4 | — | 2 | 36 | no | — |
| 257 | C3×D4 | — | 2 | 72 | no | — |
| 258 | C3×D4 | — | 2 | 18 | no | — |
| 259 | C3×D4 | — | 4 | 36 | no | — |
| 260 | C3×D4 | — | 4 | 36 | no | — |
| 261 | C3×D4 | C22×D4 | 1 | 72 | yes | complemented |
| 262 | C3×D4 | H(5,𝔽2) | 1 | 4 | yes | direct product factor |
| 263 | C2×C3×C4 | — | 2 | 36 | no | — |
| 264 | C2×C3×C4 | — | 2 | 24 | no | — |
| 265 | C2×C3×C4 | — | 2 | 72 | no | — |
| 266 | C2×C3×C4 | — | 2 | 18 | no | — |
| 267 | C2×C3×C4 | — | 4 | 18 | no | — |
| 268 | C2×C3×C4 | — | 4 | 9 | no | — |
| 269 | C2×C3×C4 | C22×D4 | 1 | 36 | yes | complemented |
| 270 | C2×C3×C4 | C25 | 1 | 6 | yes | — |
| 271 | C2×C3×C4 | C25 | 1 | 9 | yes | — |
| 272 | C2×C3×C4 | C25 | 1 | 12 | yes | — |
| 273 | C2×C3×C4 | C25 | 1 | 6 | yes | — |
| 274 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 275 | C24 | — | 2 | 48 | no | — |
| 276 | C24 | — | 4 | 24 | no | — |
| 277 | C24 | — | 8 | 12 | no | — |
| 278 | C24 | C24×C3 | 1 | 18 | yes | — |
| 279 | C24 | C24×C3 | 1 | 36 | yes | — |
| 280 | C24 | C24×C3 | 1 | 6 | yes | — |
| 281 | #41 | — | 2 | 48 | no | — |
| 282 | #41 | — | 2 | 144 | no | — |
| 283 | #41 | — | 2 | 144 | no | — |
| 284 | #41 | C2×C3×D4 | 1 | 48 | yes | complemented |
| 285 | C2×Q8 | — | 2 | 16 | no | — |
| 286 | C2×Q8 | C24×C3 | 1 | 12 | yes | — |
| 287 | C2×Q8 | C24×C3 | 1 | 2 | yes | — |
| 288 | C2×D4 | — | 2 | 144 | no | — |
| 289 | C2×D4 | — | 2 | 144 | no | — |
| 290 | C2×D4 | — | 2 | 288 | no | — |
| 291 | C2×D4 | — | 2 | 36 | no | — |
| 292 | C2×D4 | — | 2 | 72 | no | — |
| 293 | C2×D4 | — | 4 | 36 | no | — |
| 294 | C2×D4 | — | 4 | 72 | no | — |
| 295 | C2×D4 | — | 8 | 18 | no | — |
| 296 | C2×D4 | C2×C3×D4 | 1 | 72 | yes | complemented |
| 297 | C2×D4 | C24×C3 | 1 | 18 | yes | — |
| 298 | C2×D4 | C24×C3 | 1 | 9 | yes | — |
| 299 | C2×D4 | C24×C3 | 1 | 72 | yes | — |
| 300 | C2×D4 | C24×C3 | 1 | 36 | yes | — |
| 301 | C2×D4 | C24×C3 | 1 | 36 | yes | — |
| 302 | C2×D4 | C24×C3 | 1 | 36 | yes | — |
| 303 | C2×D4 | C24×C3 | 1 | 18 | yes | — |
| 304 | C2×D4 | C24×C3 | 1 | 1 | yes | characteristic |
| 305 | C22×C4 | — | 2 | 72 | no | — |
| 306 | C22×C4 | — | 4 | 12 | no | — |
| 307 | C22×C4 | C24×C3 | 1 | 9 | yes | — |
| 308 | C22×C4 | C24×C3 | 1 | 18 | yes | — |
| 309 | C22×C4 | C24×C3 | 1 | 12 | yes | — |
| 310 | C22×C4 | C24×C3 | 1 | 36 | yes | — |
| 311 | C22×C4 | C24×C3 | 1 | 9 | yes | — |
| 312 | C22×C4 | C24×C3 | 1 | 18 | yes | — |
| 313 | C22×C4 | C24×C3 | 1 | 9 | yes | — |
| 314 | C4⋊C4 | C24×C3 | 1 | 6 | yes | — |
| 315 | C4⋊C4 | C24×C3 | 1 | 18 | yes | — |
| 316 | C4⋊C4 | C24×C3 | 1 | 18 | yes | — |
| 317 | C4⋊C4 | C24×C3 | 1 | 6 | yes | — |
| 318 | C4⋊C4 | C24×C3 | 1 | 12 | yes | — |
| 319 | C22⋊C4 | C24×C3 | 1 | 12 | yes | — |
| 320 | C22⋊C4 | C24×C3 | 1 | 36 | yes | — |
| 321 | C22⋊C4 | C24×C3 | 1 | 72 | yes | — |
| 322 | C22⋊C4 | C24×C3 | 1 | 36 | yes | — |
| 323 | C42 | C24×C3 | 1 | 6 | yes | — |
| 324 | C42 | C24×C3 | 1 | 18 | yes | — |
| 325 | C22×C3 | — | 2 | 2 | no | — |
| 326 | C22×C3 | — | 2 | 9 | no | — |
| 327 | C22×C3 | — | 2 | 9 | no | — |
| 328 | C22×C3 | — | 2 | 2 | no | — |
| 329 | C22×C3 | — | 2 | 18 | no | — |
| 330 | C22×C3 | — | 2 | 18 | no | — |
| 331 | C22×C3 | — | 2 | 18 | no | — |
| 332 | C22×C3 | — | 4 | 24 | no | — |
| 333 | C22×C3 | — | 4 | 72 | no | — |
| 334 | C22×C3 | — | 8 | 72 | no | — |
| 335 | C22×C3 | C23×D4 | 1 | 18 | yes | — |
| 336 | C22×C3 | C2×H(5,𝔽2) | 1 | 4 | yes | complemented |
| 337 | C22×C3 | C26 | 1 | 1 | yes | characteristic |
| 338 | C3×C4 | — | 2 | 12 | no | — |
| 339 | C3×C4 | — | 2 | 9 | no | — |
| 340 | C3×C4 | — | 2 | 6 | no | — |
| 341 | C3×C4 | C23×D4 | 1 | 12 | yes | — |
| 342 | C3×C4 | C2×H(5,𝔽2) | 1 | 2 | yes | complemented |
| 343 | C23 | — | 2 | 36 | no | — |
| 344 | C23 | — | 2 | 36 | no | — |
| 345 | C23 | — | 2 | 72 | no | — |
| 346 | C23 | — | 4 | 6 | no | — |
| 347 | C23 | — | 4 | 6 | no | — |
| 348 | C23 | — | 4 | 18 | no | — |
| 349 | C23 | — | 4 | 36 | no | — |
| 350 | C23 | — | 4 | 36 | no | — |
| 351 | C23 | — | 8 | 96 | no | — |
| 352 | C23 | C22×C3×D4 | 1 | 24 | yes | complemented |
| 353 | C23 | C25×C3 | 1 | 18 | yes | — |
| 354 | C23 | C25×C3 | 1 | 9 | yes | — |
| 355 | C23 | C25×C3 | 1 | 2 | yes | — |
| 356 | Q8 | — | 2 | 24 | no | — |
| 357 | Q8 | C22×C3×D4 | 1 | 8 | yes | complemented |
| 358 | D4 | — | 2 | 144 | no | — |
| 359 | D4 | — | 2 | 36 | no | — |
| 360 | D4 | — | 2 | 72 | no | — |
| 361 | D4 | — | 2 | 18 | no | — |
| 362 | D4 | — | 4 | 36 | no | — |
| 363 | D4 | — | 4 | 36 | no | — |
| 364 | D4 | C22×C3×D4 | 1 | 72 | yes | complemented |
| 365 | D4 | C3×H(5,𝔽2) | 1 | 4 | yes | direct product factor |
| 366 | C2×C4 | — | 2 | 36 | no | — |
| 367 | C2×C4 | — | 2 | 24 | no | — |
| 368 | C2×C4 | — | 2 | 72 | no | — |
| 369 | C2×C4 | — | 2 | 18 | no | — |
| 370 | C2×C4 | — | 4 | 18 | no | — |
| 371 | C2×C4 | — | 4 | 9 | no | — |
| 372 | C2×C4 | C22×C3×D4 | 1 | 36 | yes | complemented |
| 373 | C2×C4 | C25×C3 | 1 | 6 | yes | — |
| 374 | C2×C4 | C25×C3 | 1 | 9 | yes | — |
| 375 | C2×C4 | C25×C3 | 1 | 12 | yes | — |
| 376 | C2×C4 | C25×C3 | 1 | 6 | yes | — |
| 377 | C2×C4 | C25×C3 | 1 | 1 | yes | characteristic |
| 378 | C2×C3 | — | 2 | 4 | no | — |
| 379 | C2×C3 | — | 2 | 18 | no | — |
| 380 | C2×C3 | — | 4 | 18 | no | — |
| 381 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 382 | C2×C3 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 383 | C2×C3 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 384 | C22 | — | 2 | 2 | no | — |
| 385 | C22 | — | 2 | 9 | no | — |
| 386 | C22 | — | 2 | 9 | no | — |
| 387 | C22 | — | 2 | 2 | no | — |
| 388 | C22 | — | 2 | 18 | no | — |
| 389 | C22 | — | 2 | 18 | no | — |
| 390 | C22 | — | 2 | 18 | no | — |
| 391 | C22 | — | 4 | 24 | no | — |
| 392 | C22 | — | 4 | 72 | no | — |
| 393 | C22 | — | 8 | 72 | no | — |
| 394 | C22 | C23×C3×D4 | 1 | 18 | yes | — |
| 395 | C22 | C2×C3×H(5,𝔽2) | 1 | 4 | yes | complemented |
| 396 | C22 | C26×C3 | 1 | 1 | yes | characteristic |
| 397 | C4 | — | 2 | 12 | no | — |
| 398 | C4 | — | 2 | 9 | no | — |
| 399 | C4 | — | 2 | 6 | no | — |
| 400 | C4 | C23×C3×D4 | 1 | 12 | yes | — |
| 401 | C4 | C2×C3×H(5,𝔽2) | 1 | 2 | yes | complemented |
| 402 | C3 | D4×H(5,𝔽2) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 403 | C2 | — | 2 | 4 | no | — |
| 404 | C2 | — | 2 | 18 | no | — |
| 405 | C2 | — | 4 | 18 | no | — |
| 406 | C2 | C24×C3×D4 | 1 | 1 | yes | characteristic |
| 407 | C2 | C22×C3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 408 | C2 | C3×H(7,𝔽2) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C22×C3×H(5,𝔽2) | C2 | semidirect |
| 2. C3×#3498 | C2 | semidirect |
| 3. C3×#3496 | C2 | semidirect |
| 4. C3×#3488 | C2 | semidirect |
| 5. C3×#3484 | C2 | semidirect |
| 6. C3×D4×#41 | C2 | semidirect |
| 7. C2×C3×D42 | C2 | semidirect |
| 8. C3×C4×H(5,𝔽2) | C2 | semidirect |
| 9. D4×H(5,𝔽2) | C3 | direct (N × H) |
| 12. C2×C3×H(5,𝔽2) | C22 | semidirect |
| 13. C2×C3×H(5,𝔽2) | C22 | semidirect |
| 14. C22×C3×#41 | C22 | semidirect |
| 15. C23×C3×D4 | C22 | semidirect |
| 16. C23×C3×D4 | C22 | semidirect |
| 17. C3×#550 | C22 | semidirect |
| 18. C3×#550 | C22 | semidirect |
| 19. C3×D4×Q8 | C22 | semidirect |
| 20. C3×#548 | C22 | semidirect |
| 21. C3×#547 | C22 | semidirect |
| 22. C3×#546 | C22 | semidirect |
| 23. C3×#546 | C22 | semidirect |
| 24. C3×D42 | C22 | semidirect |
| 25. C3×D42 | C22 | semidirect |
| 26. C3×D42 | C22 | semidirect |
| 27. C3×D42 | C22 | semidirect |
| 28. C3×#537 | C22 | semidirect |
| 29. C3×#535 | C22 | semidirect |
| 30. C3×#534 | C22 | semidirect |
| 31. C3×#532 | C22 | semidirect |
| 32. C2×C3×#127 | C22 | semidirect |
| 33. C3×#525 | C22 | semidirect |
| 34. C2×C3×#121 | C22 | semidirect |
| 35. C2×C3×#121 | C22 | semidirect |
| 36. C2×C3×#120 | C22 | semidirect |
| 37. C3×#520 | C22 | semidirect |
| 38. C3×#518 | C22 | semidirect |
| 39. C3×C4×#41 | C22 | semidirect |
| 40. C2×C3×C4×D4 | C22 | semidirect |
| 41. C2×C3×C4×D4 | C22 | semidirect |
| 42. C22×H(5,𝔽2) | C2×C3 | semidirect |
| 43. #3498 | C2×C3 | semidirect |
| 44. #3496 | C2×C3 | semidirect |
| 45. #3488 | C2×C3 | semidirect |
| 46. #3484 | C2×C3 | semidirect |
| 47. D4×#41 | C2×C3 | semidirect |
| 48. C2×D42 | C2×C3 | semidirect |
| 49. C4×H(5,𝔽2) | C2×C3 | semidirect |
| 50. C25×C3 | C23 | semidirect |
| 53. C3×H(5,𝔽2) | D4 | direct (N × H) |
| 56. C2×C3×#41 | C23 | semidirect |
| 57. C2×C3×#41 | C23 | semidirect |
| 58. C2×C3×#41 | C23 | semidirect |
| 59. C22×C3×Q8 | C23 | semidirect |
| 64. C22×C3×D4 | C23 | semidirect |
| 65. C22×C3×D4 | C23 | semidirect |
| 66. C22×C3×D4 | C23 | semidirect |
| 67. C22×C3×D4 | C23 | semidirect |
| 68. C22×C3×D4 | C23 | semidirect |
| 69. C22×C3×D4 | C23 | semidirect |
| 70. C22×C3×D4 | C23 | semidirect |
| 71. C23×C3×C4 | C23 | semidirect |
| 72. C23×C3×C4 | C23 | semidirect |
| 73. C3×C4⋊Q8 | C23 | semidirect |
| 74. C3×#127 | C23 | semidirect |
| 75. C3×#127 | C23 | semidirect |
| 76. C3×#124 | C23 | semidirect |
| 77. C3×#123 | C23 | semidirect |
| 78. C3×C22⋊Q8 | C23 | semidirect |
| 79. C3×C22⋊Q8 | C23 | semidirect |
| 80. C3×#121 | C23 | semidirect |
| 81. C3×#121 | C23 | semidirect |
| 82. C3×#121 | C23 | semidirect |
| 83. C3×#121 | C23 | semidirect |
| 84. C3×#121 | C23 | semidirect |
| 85. C3×#121 | C23 | semidirect |
| 86. C3×#120 | C23 | semidirect |
| 87. C3×#120 | C23 | semidirect |
| 88. C3×C4×Q8 | C23 | semidirect |
| 89. C3×C4×Q8 | C23 | semidirect |
| 91. C3×C4×D4 | C23 | semidirect |
| 92. C3×C4×D4 | C23 | semidirect |
| 93. C3×C4×D4 | C23 | semidirect |
| 94. C3×C4×D4 | C23 | semidirect |
| 95. C3×C4×D4 | C23 | semidirect |
| 96. C3×C4×D4 | C23 | semidirect |
| 97. C3×#117 | C23 | semidirect |
| 98. C2×C3×C4⋊C4 | C23 | semidirect |
| 99. C2×C3×C4⋊C4 | C23 | semidirect |
| 100. C2×C3×C22⋊C4 | C23 | semidirect |
| 101. C2×C3×C22⋊C4 | C23 | semidirect |
| 102. C2×C3×C22⋊C4 | C23 | semidirect |
| 103. C2×C3×C42 | C23 | semidirect |
| 106. C2×H(5,𝔽2) | C22×C3 | semidirect |
| 107. C2×H(5,𝔽2) | C22×C3 | semidirect |
| 108. C22×#41 | C22×C3 | semidirect |
| 109. C23×D4 | C22×C3 | semidirect |
| 110. C23×D4 | C22×C3 | semidirect |
| 111. #550 | C22×C3 | semidirect |
| 112. #550 | C22×C3 | semidirect |
| 113. D4×Q8 | C22×C3 | semidirect |
| 114. #548 | C22×C3 | semidirect |
| 115. #547 | C22×C3 | semidirect |
| 116. #546 | C22×C3 | semidirect |
| 117. #546 | C22×C3 | semidirect |
| 118. D42 | C22×C3 | semidirect |
| 119. D42 | C22×C3 | semidirect |
| 120. D42 | C22×C3 | semidirect |
| 121. D42 | C22×C3 | semidirect |
| 122. #537 | C22×C3 | semidirect |
| 123. #535 | C22×C3 | semidirect |
| 124. #534 | C22×C3 | semidirect |
| 125. #532 | C22×C3 | semidirect |
| 126. C2×#127 | C22×C3 | semidirect |
| 127. #525 | C22×C3 | semidirect |
| 128. C2×#121 | C22×C3 | semidirect |
| 129. C2×#121 | C22×C3 | semidirect |
| 130. C2×#120 | C22×C3 | semidirect |
| 131. #520 | C22×C3 | semidirect |
| 132. #518 | C22×C3 | semidirect |
| 133. C4×#41 | C22×C3 | semidirect |
| 134. C2×C4×D4 | C22×C3 | semidirect |
| 135. C2×C4×D4 | C22×C3 | semidirect |
| 145. C3×#41 | C2×D4 | semidirect |
| 157. C2×C3×D4 | C2×D4 | semidirect |
| 186. C25 | C23×C3 | semidirect |
| 189. H(5,𝔽2) | C3×D4 | direct (N × H) |
| 192. C2×#41 | C23×C3 | semidirect |
| 193. C2×#41 | C23×C3 | semidirect |
| 194. C2×#41 | C23×C3 | semidirect |
| 195. C22×Q8 | C23×C3 | semidirect |
| 200. C22×D4 | C23×C3 | semidirect |
| 201. C22×D4 | C23×C3 | semidirect |
| 202. C22×D4 | C23×C3 | semidirect |
| 203. C22×D4 | C23×C3 | semidirect |
| 204. C22×D4 | C23×C3 | semidirect |
| 205. C22×D4 | C23×C3 | semidirect |
| 206. C22×D4 | C23×C3 | semidirect |
| 207. C23×C4 | C23×C3 | semidirect |
| 208. C23×C4 | C23×C3 | semidirect |
| 209. C4⋊Q8 | C23×C3 | semidirect |
| 210. #127 | C23×C3 | semidirect |
| 211. #127 | C23×C3 | semidirect |
| 212. #124 | C23×C3 | semidirect |
| 213. #123 | C23×C3 | semidirect |
| 214. C22⋊Q8 | C23×C3 | semidirect |
| 215. C22⋊Q8 | C23×C3 | semidirect |
| 216. #121 | C23×C3 | semidirect |
| 217. #121 | C23×C3 | semidirect |
| 218. #121 | C23×C3 | semidirect |
| 219. #121 | C23×C3 | semidirect |
| 220. #121 | C23×C3 | semidirect |
| 221. #121 | C23×C3 | semidirect |
| 222. #120 | C23×C3 | semidirect |
| 223. #120 | C23×C3 | semidirect |
| 224. C4×Q8 | C23×C3 | semidirect |
| 225. C4×Q8 | C23×C3 | semidirect |
| 227. C4×D4 | C23×C3 | semidirect |
| 228. C4×D4 | C23×C3 | semidirect |
| 229. C4×D4 | C23×C3 | semidirect |
| 230. C4×D4 | C23×C3 | semidirect |
| 231. C4×D4 | C23×C3 | semidirect |
| 232. C4×D4 | C23×C3 | semidirect |
| 233. #117 | C23×C3 | semidirect |
| 234. C2×C4⋊C4 | C23×C3 | semidirect |
| 235. C2×C4⋊C4 | C23×C3 | semidirect |
| 236. C2×C22⋊C4 | C23×C3 | semidirect |
| 237. C2×C22⋊C4 | C23×C3 | semidirect |
| 238. C2×C22⋊C4 | C23×C3 | semidirect |
| 239. C2×C42 | C23×C3 | semidirect |
| 249. C23×C3 | C22×D4 | semidirect |
| 254. C3×Q8 | C22×D4 | semidirect |
| 261. C3×D4 | C22×D4 | semidirect |
| 262. C3×D4 | H(5,𝔽2) | direct (N × H) |
| 269. C2×C3×C4 | C22×D4 | semidirect |
| 284. #41 | C2×C3×D4 | semidirect |
| 296. C2×D4 | C2×C3×D4 | semidirect |
| 336. C22×C3 | C2×H(5,𝔽2) | semidirect |
| 342. C3×C4 | C2×H(5,𝔽2) | semidirect |
| 352. C23 | C22×C3×D4 | semidirect |
| 357. Q8 | C22×C3×D4 | semidirect |
| 364. D4 | C22×C3×D4 | semidirect |
| 365. D4 | C3×H(5,𝔽2) | direct (N × H) |
| 372. C2×C4 | C22×C3×D4 | semidirect |
| 395. C22 | C2×C3×H(5,𝔽2) | semidirect |
| 401. C4 | C2×C3×H(5,𝔽2) | semidirect |
| 402. C3 | D4×H(5,𝔽2) | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta