← C2×D3×H(3,𝔽4) #11702058
Subgroups — C2×D3×H(3,𝔽4) #11702058
Order 768.
10246 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D3×H(3,𝔽4) | C2 | 1 | 32 | yes | maximal, direct product factor |
| 3 | C2×#5683 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | C2×D3×#124 | C2 | 1 | 9 | yes | maximal, complemented |
| 5 | C2×D3×#120 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×#5594 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C22×H(3,𝔽4) | — | 3 | 1 | no | maximal, Sylow |
| 8 | C25×D3 | C22 | 1 | 2 | yes | complemented |
| 9 | C22×D3×Q8 | C22 | 1 | 3 | yes | complemented |
| 10 | C22×D3×D4 | C22 | 1 | 9 | yes | — |
| 11 | C3×H(3,𝔽4) | C22 | 1 | 16 | yes | complemented |
| 12 | C2×C3×#124 | C22 | 1 | 9 | yes | complemented |
| 13 | C2×C3×#120 | C22 | 1 | 6 | yes | complemented |
| 14 | C2×#953 | C22 | 1 | 6 | yes | complemented |
| 15 | C2×#947 | C22 | 1 | 9 | yes | complemented |
| 16 | C2×#937 | C22 | 1 | 18 | yes | complemented |
| 17 | C2×#936 | C22 | 1 | 9 | yes | complemented |
| 18 | #5683 | C22 | 1 | 144 | yes | complemented |
| 19 | D3×#124 | C22 | 1 | 144 | yes | complemented |
| 20 | #5594 | C22 | 1 | 96 | yes | complemented |
| 21 | D3×#120 | C22 | 1 | 96 | yes | complemented |
| 22 | C2×#885 | C22 | 1 | 36 | yes | complemented |
| 23 | C2×#882 | C22 | 1 | 18 | yes | complemented |
| 24 | C2×D3×C22⋊C4 | C22 | 1 | 18 | yes | complemented |
| 25 | C2×#875 | C22 | 1 | 9 | yes | complemented |
| 26 | C2×C42×D3 | C22 | 1 | 3 | yes | complemented |
| 27 | C2×H(3,𝔽4) | — | 3 | 32 | no | — |
| 28 | C2×H(3,𝔽4) | — | 3 | 9 | no | — |
| 29 | C2×H(3,𝔽4) | — | 3 | 6 | no | — |
| 30 | C2×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 31 | C22×#124 | — | 3 | 9 | no | — |
| 32 | C22×#120 | — | 3 | 6 | no | — |
| 33 | C25×C3 | C23 | 1 | 2 | yes | complemented |
| 34 | C24×D3 | — | 4 | 24 | no | — |
| 35 | C24×D3 | C23 | 1 | 6 | yes | — |
| 36 | C24×D3 | C23 | 1 | 16 | yes | complemented |
| 37 | C24×D3 | C23 | 1 | 6 | yes | complemented |
| 38 | C22×C3×Q8 | C23 | 1 | 3 | yes | complemented |
| 39 | C22×C3×D4 | C23 | 1 | 9 | yes | — |
| 40 | C22×#67 | C23 | 1 | 18 | yes | — |
| 41 | C2×D3×Q8 | — | 4 | 12 | no | — |
| 42 | C2×D3×Q8 | C23 | 1 | 24 | yes | complemented |
| 43 | C2×D3×D4 | — | 4 | 36 | no | — |
| 44 | C2×D3×D4 | C23 | 1 | 72 | yes | — |
| 45 | C22×D12 | C23 | 1 | 9 | yes | — |
| 46 | C22×C4×D3 | C23 | 1 | 9 | yes | — |
| 47 | C22×Dic6 | C23 | 1 | 9 | yes | complemented |
| 48 | C3×#124 | C23 | 1 | 72 | yes | complemented |
| 49 | C3×#120 | C23 | 1 | 48 | yes | complemented |
| 50 | C2×C3×C22⋊C4 | C23 | 1 | 18 | yes | complemented |
| 51 | C2×C3×C42 | C23 | 1 | 3 | yes | complemented |
| 52 | #953 | C23 | 1 | 48 | yes | complemented |
| 53 | C2×#217 | C23 | 1 | 18 | yes | complemented |
| 54 | #947 | C23 | 1 | 72 | yes | complemented |
| 55 | #937 | C23 | 1 | 144 | yes | complemented |
| 56 | #936 | C23 | 1 | 72 | yes | complemented |
| 57 | C2×(C2×D3)⋊C4 | C23 | 1 | 36 | yes | complemented |
| 58 | C2×C4×Dic3 | C23 | 1 | 9 | yes | complemented |
| 59 | #885 | C23 | 1 | 288 | yes | complemented |
| 60 | #882 | C23 | 1 | 144 | yes | complemented |
| 61 | D3×C22⋊C4 | C23 | 1 | 144 | yes | complemented |
| 62 | #875 | C23 | 1 | 72 | yes | complemented |
| 63 | C42×D3 | C23 | 1 | 24 | yes | complemented |
| 64 | C26 | — | 3 | 2 | no | — |
| 65 | C23×Q8 | — | 3 | 3 | no | — |
| 66 | C23×D4 | — | 3 | 9 | no | — |
| 67 | H(3,𝔽4) | — | 3 | 144 | no | — |
| 68 | H(3,𝔽4) | — | 3 | 96 | no | — |
| 69 | H(3,𝔽4) | C2×D3 | 1 | 16 | yes | direct product factor |
| 70 | C2×#124 | — | 3 | 9 | no | — |
| 71 | C2×#124 | — | 3 | 9 | no | — |
| 72 | C2×#124 | — | 3 | 144 | no | — |
| 73 | C2×#124 | — | 3 | 36 | no | — |
| 74 | C2×#124 | — | 3 | 9 | no | — |
| 75 | C2×#124 | C2×D3 | 1 | 9 | yes | complemented |
| 76 | C2×#120 | — | 3 | 6 | no | — |
| 77 | C2×#120 | — | 3 | 18 | no | — |
| 78 | C2×#120 | — | 3 | 96 | no | — |
| 79 | C2×#120 | — | 3 | 18 | no | — |
| 80 | C2×#120 | C2×D3 | 1 | 6 | yes | complemented |
| 81 | C22×C22⋊C4 | — | 3 | 18 | no | — |
| 82 | C22×C42 | — | 3 | 3 | no | — |
| 83 | C24×C3 | — | 4 | 12 | no | — |
| 84 | C24×C3 | C24 | 1 | 6 | yes | — |
| 85 | C24×C3 | C24 | 1 | 8 | yes | complemented |
| 86 | C23×D3 | — | 2 | 72 | no | — |
| 87 | C23×D3 | — | 4 | 96 | no | — |
| 88 | C23×D3 | — | 4 | 36 | no | — |
| 89 | C23×D3 | — | 4 | 128 | no | — |
| 90 | C23×D3 | C24 | 1 | 24 | yes | — |
| 91 | C23×D3 | C24 | 1 | 6 | yes | — |
| 92 | C23×D3 | C24 | 1 | 1 | yes | characteristic |
| 93 | C23×D3 | C24 | 1 | 24 | yes | complemented |
| 94 | C2×C3×Q8 | — | 4 | 6 | no | — |
| 95 | C2×C3×Q8 | C24 | 1 | 12 | yes | complemented |
| 96 | C2×C3×D4 | — | 4 | 18 | no | — |
| 97 | C2×C3×D4 | C24 | 1 | 36 | yes | — |
| 98 | C22×C3×C4 | C24 | 1 | 9 | yes | — |
| 99 | C2×#67 | — | 4 | 36 | no | — |
| 100 | C2×#67 | C24 | 1 | 72 | yes | — |
| 101 | C22×Dic3 | C24 | 1 | 9 | yes | — |
| 102 | D3×Q8 | — | 4 | 48 | no | — |
| 103 | D3×D4 | — | 4 | 144 | no | — |
| 104 | C2×D12 | — | 4 | 18 | no | — |
| 105 | C2×D12 | C24 | 1 | 36 | yes | — |
| 106 | C2×C4×D3 | — | 2 | 36 | no | — |
| 107 | C2×C4×D3 | C24 | 1 | 36 | yes | — |
| 108 | C2×Dic6 | — | 4 | 18 | no | — |
| 109 | C2×Dic6 | C24 | 1 | 36 | yes | complemented |
| 110 | C3×C22⋊C4 | C24 | 1 | 72 | yes | complemented |
| 111 | C3×C42 | C24 | 1 | 12 | yes | complemented |
| 112 | #217 | C24 | 1 | 72 | yes | complemented |
| 113 | (C2×D3)⋊C4 | C24 | 1 | 144 | yes | complemented |
| 114 | C4×Dic3 | C24 | 1 | 36 | yes | complemented |
| 115 | C25 | — | 3 | 6 | no | — |
| 116 | C25 | — | 3 | 6 | no | — |
| 117 | C25 | — | 3 | 16 | no | — |
| 118 | C25 | — | 12 | 24 | no | — |
| 119 | C25 | C22×D3 | 1 | 2 | yes | complemented |
| 120 | C22×Q8 | — | 3 | 24 | no | — |
| 121 | C22×Q8 | — | 3 | 9 | no | — |
| 122 | C22×Q8 | — | 12 | 12 | no | — |
| 123 | C22×Q8 | C22×D3 | 1 | 3 | yes | complemented |
| 124 | C22×D4 | — | 3 | 18 | no | — |
| 125 | C22×D4 | — | 3 | 72 | no | — |
| 126 | C22×D4 | — | 3 | 9 | no | — |
| 127 | C22×D4 | — | 12 | 36 | no | — |
| 128 | C22×D4 | C22×D3 | 1 | 9 | yes | — |
| 129 | C23×C4 | — | 3 | 9 | no | — |
| 130 | #124 | — | 3 | 72 | no | — |
| 131 | #124 | — | 3 | 72 | no | — |
| 132 | #124 | — | 3 | 288 | no | — |
| 133 | #124 | — | 3 | 72 | no | — |
| 134 | #124 | C22×D3 | 1 | 72 | yes | complemented |
| 135 | #120 | — | 3 | 48 | no | — |
| 136 | #120 | — | 3 | 144 | no | — |
| 137 | #120 | — | 3 | 144 | no | — |
| 138 | #120 | C22×D3 | 1 | 48 | yes | complemented |
| 139 | C2×C22⋊C4 | — | 3 | 18 | no | — |
| 140 | C2×C22⋊C4 | — | 3 | 36 | no | — |
| 141 | C2×C22⋊C4 | — | 3 | 144 | no | — |
| 142 | C2×C22⋊C4 | C22×D3 | 1 | 18 | yes | complemented |
| 143 | C2×C42 | — | 3 | 9 | no | — |
| 144 | C2×C42 | — | 3 | 24 | no | — |
| 145 | C2×C42 | C22×D3 | 1 | 3 | yes | complemented |
| 146 | C23×C3 | — | 2 | 36 | no | — |
| 147 | C23×C3 | — | 4 | 24 | no | — |
| 148 | C23×C3 | — | 4 | 32 | no | — |
| 149 | C23×C3 | C25 | 1 | 12 | yes | — |
| 150 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 151 | C22×D3 | — | 2 | 36 | no | — |
| 152 | C22×D3 | — | 2 | 144 | no | — |
| 153 | C22×D3 | — | 4 | 72 | no | — |
| 154 | C22×D3 | — | 4 | 96 | no | — |
| 155 | C22×D3 | — | 4 | 256 | no | — |
| 156 | C22×D3 | — | 4 | 96 | no | — |
| 157 | C22×D3 | H(5,𝔽2) | 1 | 12 | yes | — |
| 158 | C22×D3 | C25 | 1 | 12 | yes | — |
| 159 | C22×D3 | C25 | 1 | 2 | yes | — |
| 160 | C3×Q8 | — | 4 | 12 | no | — |
| 161 | C3×D4 | — | 4 | 36 | no | — |
| 162 | C2×C3×C4 | — | 2 | 18 | no | — |
| 163 | C2×C3×C4 | C25 | 1 | 18 | yes | — |
| 164 | #67 | — | 4 | 72 | no | — |
| 165 | C2×Dic3 | — | 2 | 18 | no | — |
| 166 | C2×Dic3 | C25 | 1 | 18 | yes | — |
| 167 | D12 | — | 4 | 36 | no | — |
| 168 | C4×D3 | — | 2 | 72 | no | — |
| 169 | Dic6 | — | 4 | 36 | no | — |
| 170 | C24 | — | 3 | 1 | no | — |
| 171 | C24 | — | 3 | 24 | no | — |
| 172 | C24 | — | 3 | 6 | no | — |
| 173 | C24 | — | 3 | 24 | no | — |
| 174 | C24 | — | 4 | 12 | no | — |
| 175 | C24 | — | 6 | 72 | no | — |
| 176 | C24 | — | 12 | 36 | no | — |
| 177 | C24 | — | 12 | 96 | no | — |
| 178 | C24 | — | 12 | 128 | no | — |
| 179 | C24 | C23×D3 | 1 | 6 | yes | — |
| 180 | C24 | C23×D3 | 1 | 8 | yes | complemented |
| 181 | C2×Q8 | — | 3 | 36 | no | — |
| 182 | C2×Q8 | — | 4 | 6 | no | — |
| 183 | C2×Q8 | — | 12 | 18 | no | — |
| 184 | C2×Q8 | — | 12 | 48 | no | — |
| 185 | C2×Q8 | C23×D3 | 1 | 12 | yes | complemented |
| 186 | C2×D4 | — | 3 | 72 | no | — |
| 187 | C2×D4 | — | 3 | 36 | no | — |
| 188 | C2×D4 | — | 4 | 18 | no | — |
| 189 | C2×D4 | — | 12 | 18 | no | — |
| 190 | C2×D4 | — | 12 | 144 | no | — |
| 191 | C2×D4 | — | 12 | 36 | no | — |
| 192 | C2×D4 | C23×D3 | 1 | 36 | yes | — |
| 193 | C22×C4 | — | 3 | 9 | no | — |
| 194 | C22×C4 | — | 3 | 36 | no | — |
| 195 | C22×C4 | — | 6 | 36 | no | — |
| 196 | C22×C4 | C23×D3 | 1 | 9 | yes | — |
| 197 | C22⋊C4 | — | 3 | 72 | no | — |
| 198 | C22⋊C4 | — | 3 | 144 | no | — |
| 199 | C22⋊C4 | C23×D3 | 1 | 72 | yes | complemented |
| 200 | C42 | — | 3 | 36 | no | — |
| 201 | C42 | C23×D3 | 1 | 12 | yes | complemented |
| 202 | C22×C3 | — | 2 | 36 | no | — |
| 203 | C22×C3 | — | 4 | 32 | no | — |
| 204 | C22×C3 | — | 4 | 24 | no | — |
| 205 | C22×C3 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 206 | C22×C3 | C26 | 1 | 1 | yes | characteristic |
| 207 | C2×D3 | — | 2 | 36 | no | — |
| 208 | C2×D3 | — | 4 | 96 | no | — |
| 209 | C2×D3 | — | 4 | 24 | no | — |
| 210 | C2×D3 | — | 4 | 96 | no | — |
| 211 | C2×D3 | H(3,𝔽4) | 1 | 16 | yes | direct product factor |
| 212 | C2×D3 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 213 | C3×C4 | — | 2 | 18 | no | — |
| 214 | Dic3 | — | 2 | 18 | no | — |
| 215 | C23 | — | 2 | 36 | no | — |
| 216 | C23 | — | 3 | 12 | no | — |
| 217 | C23 | — | 3 | 2 | no | — |
| 218 | C23 | — | 3 | 12 | no | — |
| 219 | C23 | — | 4 | 24 | no | — |
| 220 | C23 | — | 4 | 32 | no | — |
| 221 | C23 | — | 6 | 144 | no | — |
| 222 | C23 | — | 6 | 36 | no | — |
| 223 | C23 | — | 12 | 72 | no | — |
| 224 | C23 | — | 12 | 96 | no | — |
| 225 | C23 | — | 12 | 256 | no | — |
| 226 | C23 | — | 12 | 96 | no | — |
| 227 | C23 | C24×D3 | 1 | 12 | yes | — |
| 228 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 229 | Q8 | — | 4 | 12 | no | — |
| 230 | Q8 | — | 12 | 36 | no | — |
| 231 | D4 | — | 4 | 36 | no | — |
| 232 | D4 | — | 12 | 36 | no | — |
| 233 | D4 | — | 12 | 72 | no | — |
| 234 | C2×C4 | — | 2 | 18 | no | — |
| 235 | C2×C4 | — | 3 | 18 | no | — |
| 236 | C2×C4 | — | 6 | 72 | no | — |
| 237 | C2×C4 | — | 6 | 18 | no | — |
| 238 | C2×C4 | C24×D3 | 1 | 18 | yes | — |
| 239 | C2×C3 | — | 4 | 12 | no | — |
| 240 | C2×C3 | C2×H(3,𝔽4) | 1 | 4 | yes | complemented |
| 241 | C2×C3 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 242 | D3 | — | 4 | 12 | no | — |
| 243 | D3 | C2×H(3,𝔽4) | 1 | 8 | yes | direct product factor |
| 244 | C22 | — | 2 | 36 | no | — |
| 245 | C22 | — | 3 | 12 | no | — |
| 246 | C22 | — | 3 | 16 | no | — |
| 247 | C22 | — | 4 | 32 | no | — |
| 248 | C22 | — | 4 | 24 | no | — |
| 249 | C22 | — | 6 | 36 | no | — |
| 250 | C22 | — | 12 | 96 | no | — |
| 251 | C22 | — | 12 | 96 | no | — |
| 252 | C22 | — | 12 | 24 | no | — |
| 253 | C22 | D3×H(5,𝔽2) | 1 | 6 | yes | — |
| 254 | C22 | C25×D3 | 1 | 1 | yes | characteristic |
| 255 | C4 | — | 2 | 18 | no | — |
| 256 | C4 | — | 6 | 18 | no | — |
| 257 | C3 | C22×H(3,𝔽4) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 258 | C2 | — | 3 | 8 | no | — |
| 259 | C2 | — | 4 | 12 | no | — |
| 260 | C2 | — | 12 | 12 | no | — |
| 261 | C2 | D3×H(3,𝔽4) | 1 | 4 | yes | direct product factor |
| 262 | C2 | C2×D3×H(5,𝔽2) | 1 | 3 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C2×C3×H(3,𝔽4) | C2 | semidirect |
| 2. D3×H(3,𝔽4) | C2 | direct (N × H) |
| 3. C2×#5683 | C2 | semidirect |
| 4. C2×D3×#124 | C2 | semidirect |
| 5. C2×D3×#120 | C2 | semidirect |
| 6. C2×#5594 | C2 | semidirect |
| 8. C25×D3 | C22 | semidirect |
| 9. C22×D3×Q8 | C22 | semidirect |
| 11. C3×H(3,𝔽4) | C22 | semidirect |
| 12. C2×C3×#124 | C22 | semidirect |
| 13. C2×C3×#120 | C22 | semidirect |
| 14. C2×#953 | C22 | semidirect |
| 15. C2×#947 | C22 | semidirect |
| 16. C2×#937 | C22 | semidirect |
| 17. C2×#936 | C22 | semidirect |
| 18. #5683 | C22 | semidirect |
| 19. D3×#124 | C22 | semidirect |
| 20. #5594 | C22 | semidirect |
| 21. D3×#120 | C22 | semidirect |
| 22. C2×#885 | C22 | semidirect |
| 23. C2×#882 | C22 | semidirect |
| 24. C2×D3×C22⋊C4 | C22 | semidirect |
| 25. C2×#875 | C22 | semidirect |
| 26. C2×C42×D3 | C22 | semidirect |
| 30. C2×H(3,𝔽4) | D3 | direct (N × H) |
| 33. C25×C3 | C23 | semidirect |
| 36. C24×D3 | C23 | semidirect |
| 37. C24×D3 | C23 | semidirect |
| 38. C22×C3×Q8 | C23 | semidirect |
| 42. C2×D3×Q8 | C23 | semidirect |
| 47. C22×Dic6 | C23 | semidirect |
| 48. C3×#124 | C23 | semidirect |
| 49. C3×#120 | C23 | semidirect |
| 50. C2×C3×C22⋊C4 | C23 | semidirect |
| 51. C2×C3×C42 | C23 | semidirect |
| 52. #953 | C23 | semidirect |
| 53. C2×#217 | C23 | semidirect |
| 54. #947 | C23 | semidirect |
| 55. #937 | C23 | semidirect |
| 56. #936 | C23 | semidirect |
| 57. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 58. C2×C4×Dic3 | C23 | semidirect |
| 59. #885 | C23 | semidirect |
| 60. #882 | C23 | semidirect |
| 61. D3×C22⋊C4 | C23 | semidirect |
| 62. #875 | C23 | semidirect |
| 63. C42×D3 | C23 | semidirect |
| 69. H(3,𝔽4) | C2×D3 | direct (N × H) |
| 75. C2×#124 | C2×D3 | semidirect |
| 80. C2×#120 | C2×D3 | semidirect |
| 85. C24×C3 | C24 | semidirect |
| 93. C23×D3 | C24 | semidirect |
| 95. C2×C3×Q8 | C24 | semidirect |
| 109. C2×Dic6 | C24 | semidirect |
| 110. C3×C22⋊C4 | C24 | semidirect |
| 111. C3×C42 | C24 | semidirect |
| 112. #217 | C24 | semidirect |
| 113. (C2×D3)⋊C4 | C24 | semidirect |
| 114. C4×Dic3 | C24 | semidirect |
| 119. C25 | C22×D3 | semidirect |
| 123. C22×Q8 | C22×D3 | semidirect |
| 134. #124 | C22×D3 | semidirect |
| 138. #120 | C22×D3 | semidirect |
| 142. C2×C22⋊C4 | C22×D3 | semidirect |
| 145. C2×C42 | C22×D3 | semidirect |
| 180. C24 | C23×D3 | semidirect |
| 185. C2×Q8 | C23×D3 | semidirect |
| 199. C22⋊C4 | C23×D3 | semidirect |
| 201. C42 | C23×D3 | semidirect |
| 211. C2×D3 | H(3,𝔽4) | direct (N × H) |
| 240. C2×C3 | C2×H(3,𝔽4) | semidirect |
| 243. D3 | C2×H(3,𝔽4) | direct (N × H) |
| 257. C3 | C22×H(3,𝔽4) | semidirect |
| 261. C2 | D3×H(3,𝔽4) | 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