← C2×#72210024 #72240569
Subgroups — C2×#72210024 #72240569
Order 2048.
Only the 4512 normal subgroups are shown; non-normal subgroups are omitted.
The subgroup lattice has 224 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#8093108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#391709 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#417896 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C2×#417969 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | C2×#6628896 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#449475 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #72210024 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 8 | C22×H(7,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C23×B(4,𝔽2) | C22 | 1 | 12 | yes | — |
| 10 | #8093108 | C22 | 1 | 16 | yes | complemented |
| 11 | C22×#2846 | C22 | 1 | 3 | yes | complemented |
| 12 | C22×#2846 | C22 | 1 | 4 | yes | complemented |
| 13 | C2×#35367 | C22 | 1 | 6 | yes | complemented |
| 14 | C2×#35367 | C22 | 1 | 3 | yes | complemented |
| 15 | C2×#35346 | C22 | 1 | 3 | yes | — |
| 16 | C2×#35346 | C22 | 1 | 6 | yes | — |
| 17 | #6628896 | C22 | 1 | 48 | yes | complemented |
| 18 | C22×#2032 | C22 | 1 | 8 | yes | complemented |
| 19 | C2×#23468 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×#20609 | C22 | 1 | 12 | yes | complemented |
| 21 | C2×#20585 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×#20585 | C22 | 1 | 3 | yes | complemented |
| 23 | #449475 | C22 | 1 | 96 | yes | complemented |
| 24 | #417969 | C22 | 1 | 192 | yes | complemented |
| 25 | #417896 | C22 | 1 | 96 | yes | complemented |
| 26 | #391709 | C22 | 1 | 48 | yes | complemented |
| 27 | C2×#11172 | C22 | 1 | 12 | yes | complemented |
| 28 | C2×#11169 | C22 | 1 | 6 | yes | complemented |
| 29 | C2×#10991 | C22 | 1 | 24 | yes | complemented |
| 30 | C2×#10991 | C22 | 1 | 12 | yes | complemented |
| 31 | C2×#10965 | C22 | 1 | 12 | yes | complemented |
| 32 | C2×#10965 | C22 | 1 | 6 | yes | complemented |
| 33 | C2×#10938 | C22 | 1 | 12 | yes | complemented |
| 34 | C28 | C23 | 1 | 1 | yes | characteristic, complemented |
| 35 | C2×H(7,𝔽2) | C23 | 1 | 8 | yes | complemented |
| 36 | C23×H(5,𝔽2) | C23 | 1 | 1 | yes | characteristic |
| 37 | C23×H(5,𝔽2) | C23 | 1 | 6 | yes | — |
| 38 | C22×#543 | C23 | 1 | 1 | yes | characteristic, complemented |
| 39 | C22×#543 | C23 | 1 | 2 | yes | complemented |
| 40 | C23×#120 | C23 | 1 | 3 | yes | — |
| 41 | C23×#120 | C23 | 1 | 6 | yes | — |
| 42 | C23×#120 | C23 | 1 | 12 | yes | — |
| 43 | C22×B(4,𝔽2) | C23 | 1 | 96 | yes | — |
| 44 | C23×#99 | C23 | 1 | 12 | yes | — |
| 45 | C23×#99 | C23 | 1 | 24 | yes | — |
| 46 | C2×#2846 | C23 | 1 | 24 | yes | complemented |
| 47 | C2×#2846 | C23 | 1 | 32 | yes | complemented |
| 48 | #35367 | C23 | 1 | 24 | yes | complemented |
| 49 | #35367 | C23 | 1 | 48 | yes | complemented |
| 50 | #35346 | D4 | 1 | 24 | yes | complemented |
| 51 | #35346 | C23 | 1 | 24 | yes | — |
| 52 | #35346 | C23 | 1 | 48 | yes | — |
| 53 | C2×#2404 | D4 | 1 | 12 | yes | complemented |
| 54 | C2×#2403 | D4 | 1 | 12 | yes | complemented |
| 55 | C2×#2308 | C23 | 1 | 3 | yes | — |
| 56 | C2×#2308 | C23 | 1 | 6 | yes | — |
| 57 | C2×#2308 | C23 | 1 | 6 | yes | — |
| 58 | C2×#2308 | C23 | 1 | 12 | yes | — |
| 59 | C22×#379 | C23 | 1 | 3 | yes | complemented |
| 60 | C22×#379 | C23 | 1 | 12 | yes | complemented |
| 61 | C22×#379 | C23 | 1 | 6 | yes | complemented |
| 62 | C2×#2273 | C23 | 1 | 3 | yes | complemented |
| 63 | C2×#2032 | C23 | 1 | 64 | yes | complemented |
| 64 | #23468 | C23 | 1 | 32 | yes | complemented |
| 65 | #20609 | C23 | 1 | 96 | yes | complemented |
| 66 | #20585 | C23 | 1 | 24 | yes | complemented |
| 67 | #20585 | C23 | 1 | 48 | yes | complemented |
| 68 | C22×#342 | C23 | 1 | 12 | yes | complemented |
| 69 | C22×#342 | C23 | 1 | 24 | yes | complemented |
| 70 | #11172 | C23 | 1 | 96 | yes | complemented |
| 71 | #11169 | C23 | 1 | 48 | yes | complemented |
| 72 | #10991 | C23 | 1 | 96 | yes | complemented |
| 73 | #10991 | C23 | 1 | 192 | yes | complemented |
| 74 | #10965 | C23 | 1 | 48 | yes | complemented |
| 75 | #10965 | C23 | 1 | 96 | yes | complemented |
| 76 | #10938 | C23 | 1 | 96 | yes | complemented |
| 77 | C27 | C24 | 1 | 3 | yes | — |
| 78 | C27 | C24 | 1 | 4 | yes | complemented |
| 79 | C22×H(5,𝔽2) | C2×D4 | 1 | 12 | yes | — |
| 80 | C22×H(5,𝔽2) | C24 | 1 | 4 | yes | — |
| 81 | C22×H(5,𝔽2) | C24 | 1 | 24 | yes | — |
| 82 | C23×#41 | C2×D4 | 1 | 6 | yes | — |
| 83 | C24×D4 | C2×D4 | 1 | 6 | yes | — |
| 84 | C24×D4 | C2×D4 | 1 | 6 | yes | — |
| 85 | C24×D4 | C24 | 1 | 3 | yes | — |
| 86 | C24×D4 | C24 | 1 | 4 | yes | — |
| 87 | C2×#543 | C24 | 1 | 4 | yes | complemented |
| 88 | C2×#543 | C24 | 1 | 8 | yes | complemented |
| 89 | C22×#120 | C2×D4 | 1 | 12 | yes | — |
| 90 | C22×#120 | C2×D4 | 1 | 24 | yes | — |
| 91 | C22×#120 | C2×D4 | 1 | 24 | yes | — |
| 92 | C22×#120 | C24 | 1 | 12 | yes | — |
| 93 | C22×#120 | C24 | 1 | 24 | yes | — |
| 94 | C22×#120 | C24 | 1 | 48 | yes | — |
| 95 | C23×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 96 | C23×C22⋊C4 | C24 | 1 | 6 | yes | — |
| 97 | C23×C22⋊C4 | C24 | 1 | 3 | yes | — |
| 98 | C23×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 99 | C22×#99 | C24 | 1 | 48 | yes | — |
| 100 | C22×#99 | C24 | 1 | 96 | yes | — |
| 101 | #2404 | C2×D4 | 1 | 96 | yes | complemented |
| 102 | #2403 | C2×D4 | 1 | 96 | yes | complemented |
| 103 | #2308 | C2×D4 | 1 | 12 | yes | — |
| 104 | #2308 | C2×D4 | 1 | 24 | yes | — |
| 105 | #2308 | C24 | 1 | 24 | yes | — |
| 106 | #2308 | C24 | 1 | 48 | yes | — |
| 107 | #2308 | C24 | 1 | 12 | yes | — |
| 108 | #2308 | C24 | 1 | 24 | yes | — |
| 109 | C2×#386 | C2×D4 | 1 | 24 | yes | — |
| 110 | C2×#381 | C2×D4 | 1 | 6 | yes | — |
| 111 | C2×#379 | C2×D4 | 1 | 6 | yes | — |
| 112 | C2×#379 | C24 | 1 | 12 | yes | complemented |
| 113 | C2×#379 | C24 | 1 | 24 | yes | complemented |
| 114 | C2×#379 | C24 | 1 | 48 | yes | complemented |
| 115 | #2273 | C24 | 1 | 12 | yes | complemented |
| 116 | C2×#342 | C24 | 1 | 96 | yes | complemented |
| 117 | C2×#342 | C24 | 1 | 48 | yes | complemented |
| 118 | C26 | #120 | 1 | 2 | yes | complemented |
| 119 | C26 | C22×D4 | 1 | 3 | yes | — |
| 120 | C26 | C22×D4 | 1 | 6 | yes | — |
| 121 | C26 | C22×D4 | 1 | 6 | yes | — |
| 122 | C26 | H(5,𝔽2) | 1 | 4 | yes | — |
| 123 | C26 | C25 | 1 | 6 | yes | — |
| 124 | C26 | C25 | 1 | 1 | yes | characteristic |
| 125 | C22×#41 | #120 | 1 | 24 | yes | complemented |
| 126 | C22×#41 | C22×D4 | 1 | 24 | yes | — |
| 127 | C23×Q8 | #120 | 1 | 2 | yes | complemented |
| 128 | C23×D4 | #120 | 1 | 6 | yes | complemented |
| 129 | C23×D4 | #120 | 1 | 24 | yes | complemented |
| 130 | C23×D4 | C22×D4 | 1 | 12 | yes | — |
| 131 | C23×D4 | C22×D4 | 1 | 12 | yes | — |
| 132 | C23×D4 | C22×D4 | 1 | 24 | yes | — |
| 133 | C23×D4 | C22×D4 | 1 | 6 | yes | — |
| 134 | C23×D4 | C22×D4 | 1 | 24 | yes | — |
| 135 | C23×D4 | H(5,𝔽2) | 1 | 32 | yes | — |
| 136 | C23×D4 | H(5,𝔽2) | 1 | 8 | yes | — |
| 137 | C23×D4 | C25 | 1 | 6 | yes | — |
| 138 | C23×D4 | C25 | 1 | 8 | yes | — |
| 139 | C24×C4 | #120 | 1 | 6 | yes | complemented |
| 140 | C24×C4 | C22×D4 | 1 | 3 | yes | — |
| 141 | C24×C4 | C22×D4 | 1 | 6 | yes | — |
| 142 | C24×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 143 | C2×#120 | C22×D4 | 1 | 96 | yes | — |
| 144 | C22×C22⋊C4 | C22×D4 | 1 | 12 | yes | — |
| 145 | C22×C22⋊C4 | C22×D4 | 1 | 12 | yes | — |
| 146 | C22×C22⋊C4 | C22×D4 | 1 | 6 | yes | — |
| 147 | C22×C22⋊C4 | C22×D4 | 1 | 24 | yes | — |
| 148 | C22×C22⋊C4 | C25 | 1 | 12 | yes | — |
| 149 | C22×C22⋊C4 | C25 | 1 | 6 | yes | — |
| 150 | C22×C22⋊C4 | C25 | 1 | 24 | yes | — |
| 151 | #386 | C22×D4 | 1 | 96 | yes | — |
| 152 | #381 | C22×D4 | 1 | 24 | yes | — |
| 153 | #379 | C22×D4 | 1 | 24 | yes | — |
| 154 | C25 | C2×#120 | 1 | 6 | yes | — |
| 155 | C25 | C2×#120 | 1 | 6 | yes | — |
| 156 | C25 | C2×#120 | 1 | 8 | yes | complemented |
| 157 | C25 | H(3,𝔽4) | 1 | 16 | yes | complemented |
| 158 | C25 | C23×D4 | 1 | 12 | yes | — |
| 159 | C25 | C23×D4 | 1 | 6 | yes | — |
| 160 | C25 | C23×D4 | 1 | 3 | yes | — |
| 161 | C25 | C23×D4 | 1 | 3 | yes | — |
| 162 | C25 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 163 | C25 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 164 | C25 | C2×H(5,𝔽2) | 1 | 16 | yes | — |
| 165 | C25 | C2×H(5,𝔽2) | 1 | 8 | yes | — |
| 166 | C25 | C26 | 1 | 1 | yes | characteristic |
| 167 | C22×Q8 | C2×#120 | 1 | 8 | yes | complemented |
| 168 | C22×D4 | C2×#120 | 1 | 24 | yes | — |
| 169 | C22×D4 | C2×#120 | 1 | 24 | yes | complemented |
| 170 | C22×D4 | H(3,𝔽4) | 1 | 32 | yes | complemented |
| 171 | C22×D4 | C23×D4 | 1 | 24 | yes | — |
| 172 | C22×D4 | C2×H(5,𝔽2) | 1 | 32 | yes | — |
| 173 | C23×C4 | C2×#120 | 1 | 6 | yes | — |
| 174 | C23×C4 | C2×#120 | 1 | 6 | yes | — |
| 175 | C23×C4 | C2×#120 | 1 | 24 | yes | complemented |
| 176 | C23×C4 | H(3,𝔽4) | 1 | 16 | yes | complemented |
| 177 | C23×C4 | C23×D4 | 1 | 6 | yes | — |
| 178 | C23×C4 | C23×D4 | 1 | 12 | yes | — |
| 179 | C23×C4 | C2×H(5,𝔽2) | 1 | 8 | yes | — |
| 180 | C2×C22⋊C4 | C23×D4 | 1 | 24 | yes | — |
| 181 | C24 | C22×#120 | 1 | 3 | yes | — |
| 182 | C24 | C22×#120 | 1 | 1 | yes | characteristic |
| 183 | C24 | C22×#120 | 1 | 12 | yes | — |
| 184 | C24 | #3484 | 1 | 12 | yes | — |
| 185 | C24 | #3484 | 1 | 24 | yes | — |
| 186 | C24 | #3484 | 1 | 6 | yes | — |
| 187 | C24 | #3484 | 1 | 24 | yes | — |
| 188 | C24 | #3484 | 1 | 6 | yes | — |
| 189 | C24 | C2×H(3,𝔽4) | 1 | 8 | yes | — |
| 190 | C24 | C2×H(3,𝔽4) | 1 | 8 | yes | — |
| 191 | C24 | C2×H(3,𝔽4) | 1 | 16 | yes | complemented |
| 192 | C24 | #3532 | 1 | 24 | yes | — |
| 193 | C24 | #3532 | 1 | 24 | yes | — |
| 194 | C24 | C24×D4 | 1 | 3 | yes | — |
| 195 | C24 | C22×H(5,𝔽2) | 1 | 4 | yes | — |
| 196 | C24 | C22×H(5,𝔽2) | 1 | 8 | yes | — |
| 197 | C22×C4 | C22×#120 | 1 | 12 | yes | — |
| 198 | C22×C4 | C2×H(3,𝔽4) | 1 | 16 | yes | complemented |
| 199 | C23 | #35346 | 1 | 6 | yes | — |
| 200 | C23 | #35346 | 1 | 1 | yes | characteristic |
| 201 | C23 | #35346 | 1 | 4 | yes | — |
| 202 | C23 | #35346 | 1 | 12 | yes | — |
| 203 | C23 | #35346 | 1 | 1 | yes | characteristic |
| 204 | C23 | #45439 | 1 | 24 | yes | — |
| 205 | C23 | #45439 | 1 | 24 | yes | — |
| 206 | C23 | #45439 | 1 | 24 | yes | — |
| 207 | C23 | #45439 | 1 | 24 | yes | — |
| 208 | C23 | C23×#120 | 1 | 1 | yes | characteristic |
| 209 | C23 | C22×H(3,𝔽4) | 1 | 4 | yes | — |
| 210 | C23 | C2×#3532 | 1 | 12 | yes | — |
| 211 | C23 | C2×#3484 | 1 | 6 | yes | — |
| 212 | C23 | C2×#3484 | 1 | 12 | yes | — |
| 213 | C22 | #5021138 | 1 | 2 | yes | — |
| 214 | C22 | #5021138 | 1 | 12 | yes | — |
| 215 | C22 | #5021138 | 1 | 2 | yes | — |
| 216 | C22 | #6499874 | 1 | 4 | yes | — |
| 217 | C22 | C2×#35346 | 1 | 1 | yes | characteristic |
| 218 | C22 | C2×#35346 | 1 | 2 | yes | — |
| 219 | C22 | C2×#45439 | 1 | 12 | yes | — |
| 220 | C2 | #72210024 | 1 | 4 | yes | direct product factor |
| 221 | C2 | C2×#5021138 | 1 | 1 | yes | characteristic |
| 222 | C2 | C2×#6499874 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta