← H(3,𝔽3)×#575 #13154708
Subgroups — H(3,𝔽3)×#575 #13154708
Order 1728.
1441 conjugacy classes of subgroups.
The subgroup lattice has 250 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#41×H(3,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | H(3,𝔽3)×#137 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | H(3,𝔽3)×AGL(1,ℤ/8) | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | H(3,𝔽3)×#135 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×M16×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C32×#575 | C3 | 1 | 4 | yes | maximal, complemented |
| 7 | #41×H(3,𝔽3) | — | 2 | 2 | no | — |
| 8 | #41×H(3,𝔽3) | — | 2 | 2 | no | — |
| 9 | #41×H(3,𝔽3) | C22 | 1 | 4 | yes | — |
| 10 | #41×H(3,𝔽3) | C22 | 1 | 4 | yes | complemented |
| 11 | C2×Q8×H(3,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C2×D4×H(3,𝔽3) | — | 2 | 2 | no | — |
| 13 | C2×D4×H(3,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 14 | C22×C4×H(3,𝔽3) | — | 2 | 2 | no | — |
| 15 | C22×C4×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 16 | Q16×H(3,𝔽3) | C22 | 1 | 4 | yes | complemented |
| 17 | SD16×H(3,𝔽3) | C22 | 1 | 8 | yes | complemented |
| 18 | D8×H(3,𝔽3) | C22 | 1 | 4 | yes | complemented |
| 19 | M16×H(3,𝔽3) | C22 | 1 | 4 | yes | complemented |
| 20 | C2×C8×H(3,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 21 | C2×C32×#41 | C2×C3 | 1 | 8 | yes | complemented |
| 22 | C32×#137 | C2×C3 | 1 | 16 | yes | complemented |
| 23 | C32×AGL(1,ℤ/8) | C2×C3 | 1 | 16 | yes | complemented |
| 24 | C32×#135 | C2×C3 | 1 | 16 | yes | complemented |
| 25 | C2×C32×M16 | C2×C3 | 1 | 4 | yes | complemented |
| 26 | C23×H(3,𝔽3) | — | 2 | 2 | no | — |
| 27 | C23×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 28 | Q8×H(3,𝔽3) | — | 2 | 2 | no | — |
| 29 | Q8×H(3,𝔽3) | C23 | 1 | 4 | yes | — |
| 30 | D4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 31 | D4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 32 | D4×H(3,𝔽3) | — | 2 | 2 | no | — |
| 33 | D4×H(3,𝔽3) | C23 | 1 | 4 | yes | — |
| 34 | C2×C4×H(3,𝔽3) | — | 2 | 2 | no | — |
| 35 | C2×C4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 36 | C2×C4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 37 | C2×C4×H(3,𝔽3) | D4 | 1 | 2 | yes | — |
| 38 | C2×C4×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×H(3,𝔽3) | C23 | 1 | 2 | yes | — |
| 40 | C2×C4×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 41 | C8×H(3,𝔽3) | C23 | 1 | 4 | yes | — |
| 42 | C3×#575 | — | 3 | 4 | no | — |
| 43 | C3×#575 | C32 | 1 | 1 | yes | characteristic |
| 44 | C32×#41 | — | 2 | 8 | no | — |
| 45 | C32×#41 | — | 2 | 8 | no | — |
| 46 | C32×#41 | C22×C3 | 1 | 16 | yes | — |
| 47 | C32×#41 | C22×C3 | 1 | 16 | yes | complemented |
| 48 | C2×C32×Q8 | C22×C3 | 1 | 8 | yes | complemented |
| 49 | C2×C32×D4 | — | 2 | 8 | no | — |
| 50 | C2×C32×D4 | C22×C3 | 1 | 8 | yes | complemented |
| 51 | C22×C32×C4 | — | 2 | 8 | no | — |
| 52 | C22×C32×C4 | C22×C3 | 1 | 4 | yes | — |
| 53 | C32×Q16 | C22×C3 | 1 | 16 | yes | complemented |
| 54 | C32×SD16 | C22×C3 | 1 | 32 | yes | complemented |
| 55 | C32×D8 | C22×C3 | 1 | 16 | yes | complemented |
| 56 | C32×M16 | C22×C3 | 1 | 16 | yes | complemented |
| 57 | C2×C32×C8 | C22×C3 | 1 | 8 | yes | complemented |
| 58 | C22×H(3,𝔽3) | — | 2 | 4 | no | — |
| 59 | C22×H(3,𝔽3) | — | 4 | 1 | no | — |
| 60 | C22×H(3,𝔽3) | — | 4 | 4 | no | — |
| 61 | C22×H(3,𝔽3) | C2×D4 | 1 | 2 | yes | — |
| 62 | C22×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 64 | C4×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C4×H(3,𝔽3) | C2×D4 | 1 | 2 | yes | — |
| 66 | C4×H(3,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 67 | C2×C3×#41 | — | 3 | 8 | no | — |
| 68 | C2×C3×#41 | C2×C32 | 1 | 2 | yes | — |
| 69 | C3×#137 | — | 3 | 16 | no | — |
| 70 | C3×#137 | C2×C32 | 1 | 4 | yes | — |
| 71 | C3×AGL(1,ℤ/8) | — | 3 | 16 | no | — |
| 72 | C3×AGL(1,ℤ/8) | C2×C32 | 1 | 4 | yes | — |
| 73 | C3×#135 | — | 3 | 16 | no | — |
| 74 | C3×#135 | C2×C32 | 1 | 4 | yes | — |
| 75 | C2×C3×M16 | — | 3 | 4 | no | — |
| 76 | C2×C3×M16 | C2×C32 | 1 | 1 | yes | characteristic |
| 77 | C23×C32 | — | 2 | 8 | no | — |
| 78 | C23×C32 | C3×D4 | 1 | 4 | yes | — |
| 79 | C32×Q8 | — | 2 | 8 | no | — |
| 80 | C32×Q8 | C23×C3 | 1 | 16 | yes | — |
| 81 | C32×D4 | — | 2 | 16 | no | — |
| 82 | C32×D4 | — | 2 | 16 | no | — |
| 83 | C32×D4 | — | 2 | 8 | no | — |
| 84 | C32×D4 | C23×C3 | 1 | 16 | yes | — |
| 85 | C2×C32×C4 | — | 2 | 8 | no | — |
| 86 | C2×C32×C4 | — | 2 | 16 | no | — |
| 87 | C2×C32×C4 | — | 2 | 16 | no | — |
| 88 | C2×C32×C4 | C3×D4 | 1 | 8 | yes | — |
| 89 | C2×C32×C4 | C3×D4 | 1 | 4 | yes | — |
| 90 | C2×C32×C4 | C23×C3 | 1 | 4 | yes | — |
| 91 | C2×C32×C4 | C23×C3 | 1 | 8 | yes | — |
| 92 | C32×C8 | C23×C3 | 1 | 16 | yes | — |
| 93 | #575 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 94 | C2×H(3,𝔽3) | — | 2 | 2 | no | — |
| 95 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 96 | C2×H(3,𝔽3) | — | 4 | 4 | no | — |
| 97 | C2×H(3,𝔽3) | C22×D4 | 1 | 1 | yes | characteristic |
| 98 | C3×#41 | — | 2 | 2 | no | — |
| 99 | C3×#41 | — | 2 | 2 | no | — |
| 100 | C3×#41 | — | 3 | 16 | no | — |
| 101 | C3×#41 | — | 3 | 16 | no | — |
| 102 | C3×#41 | — | 6 | 8 | no | — |
| 103 | C3×#41 | — | 6 | 8 | no | — |
| 104 | C3×#41 | C22×C32 | 1 | 4 | yes | — |
| 105 | C3×#41 | C22×C32 | 1 | 4 | yes | — |
| 106 | C2×C3×Q8 | — | 3 | 8 | no | — |
| 107 | C2×C3×Q8 | C22×C32 | 1 | 2 | yes | — |
| 108 | C2×C3×D4 | — | 2 | 2 | no | — |
| 109 | C2×C3×D4 | — | 3 | 8 | no | — |
| 110 | C2×C3×D4 | — | 6 | 8 | no | — |
| 111 | C2×C3×D4 | C22×C32 | 1 | 2 | yes | — |
| 112 | C22×C3×C4 | — | 2 | 2 | no | — |
| 113 | C22×C3×C4 | — | 3 | 4 | no | — |
| 114 | C22×C3×C4 | — | 6 | 8 | no | — |
| 115 | C22×C3×C4 | C22×C32 | 1 | 1 | yes | characteristic |
| 116 | C3×Q16 | — | 3 | 16 | no | — |
| 117 | C3×Q16 | C22×C32 | 1 | 4 | yes | — |
| 118 | C3×SD16 | — | 3 | 32 | no | — |
| 119 | C3×SD16 | C22×C32 | 1 | 8 | yes | — |
| 120 | C3×D8 | — | 3 | 16 | no | — |
| 121 | C3×D8 | C22×C32 | 1 | 4 | yes | — |
| 122 | C3×M16 | — | 3 | 16 | no | — |
| 123 | C3×M16 | C22×C32 | 1 | 4 | yes | — |
| 124 | C2×C3×C8 | — | 3 | 8 | no | — |
| 125 | C2×C3×C8 | C22×C32 | 1 | 2 | yes | — |
| 126 | C22×C32 | — | 2 | 16 | no | — |
| 127 | C22×C32 | — | 4 | 4 | no | — |
| 128 | C22×C32 | — | 4 | 16 | no | — |
| 129 | C22×C32 | C2×C3×D4 | 1 | 8 | yes | — |
| 130 | C22×C32 | C2×C3×D4 | 1 | 4 | yes | — |
| 131 | C32×C4 | — | 2 | 16 | no | — |
| 132 | C32×C4 | C2×C3×D4 | 1 | 8 | yes | — |
| 133 | C32×C4 | C2×C3×D4 | 1 | 4 | yes | — |
| 134 | C32×C4 | C24×C3 | 1 | 4 | yes | — |
| 135 | C2×#41 | C2×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 136 | #137 | C2×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 137 | AGL(1,ℤ/8) | C2×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 138 | #135 | C2×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 139 | C2×M16 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 140 | H(3,𝔽3) | #575 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 141 | C23×C3 | — | 2 | 2 | no | — |
| 142 | C23×C3 | — | 3 | 4 | no | — |
| 143 | C23×C3 | — | 6 | 8 | no | — |
| 144 | C23×C3 | C32×D4 | 1 | 1 | yes | characteristic |
| 145 | C3×Q8 | — | 2 | 2 | no | — |
| 146 | C3×Q8 | — | 3 | 16 | no | — |
| 147 | C3×Q8 | — | 6 | 8 | no | — |
| 148 | C3×Q8 | C23×C32 | 1 | 4 | yes | — |
| 149 | C3×D4 | — | 2 | 4 | no | — |
| 150 | C3×D4 | — | 2 | 4 | no | — |
| 151 | C3×D4 | — | 2 | 2 | no | — |
| 152 | C3×D4 | — | 3 | 16 | no | — |
| 153 | C3×D4 | — | 6 | 8 | no | — |
| 154 | C3×D4 | — | 6 | 16 | no | — |
| 155 | C3×D4 | — | 6 | 16 | no | — |
| 156 | C3×D4 | C23×C32 | 1 | 4 | yes | — |
| 157 | C2×C3×C4 | — | 2 | 2 | no | — |
| 158 | C2×C3×C4 | — | 2 | 4 | no | — |
| 159 | C2×C3×C4 | — | 2 | 4 | no | — |
| 160 | C2×C3×C4 | — | 3 | 8 | no | — |
| 161 | C2×C3×C4 | — | 3 | 4 | no | — |
| 162 | C2×C3×C4 | — | 3 | 4 | no | — |
| 163 | C2×C3×C4 | — | 3 | 8 | no | — |
| 164 | C2×C3×C4 | — | 6 | 16 | no | — |
| 165 | C2×C3×C4 | — | 6 | 16 | no | — |
| 166 | C2×C3×C4 | — | 6 | 8 | no | — |
| 167 | C2×C3×C4 | C32×D4 | 1 | 1 | yes | characteristic |
| 168 | C2×C3×C4 | C32×D4 | 1 | 2 | yes | — |
| 169 | C2×C3×C4 | C23×C32 | 1 | 2 | yes | — |
| 170 | C2×C3×C4 | C23×C32 | 1 | 1 | yes | characteristic |
| 171 | C3×C8 | — | 3 | 16 | no | — |
| 172 | C3×C8 | C23×C32 | 1 | 4 | yes | — |
| 173 | C2×C32 | — | 2 | 8 | no | — |
| 174 | C2×C32 | — | 2 | 4 | no | — |
| 175 | C2×C32 | — | 4 | 16 | no | — |
| 176 | C2×C32 | C22×C3×D4 | 1 | 4 | yes | — |
| 177 | #41 | — | 2 | 2 | no | — |
| 178 | #41 | — | 2 | 2 | no | — |
| 179 | #41 | C22×H(3,𝔽3) | 1 | 4 | yes | — |
| 180 | #41 | C22×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 181 | C2×Q8 | C22×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 182 | C2×D4 | — | 2 | 2 | no | — |
| 183 | C2×D4 | C22×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 184 | C22×C4 | — | 2 | 2 | no | — |
| 185 | C22×C4 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 186 | Q16 | C22×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 187 | SD16 | C22×H(3,𝔽3) | 1 | 8 | yes | complemented |
| 188 | D8 | C22×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 189 | M16 | C22×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 190 | C2×C8 | C22×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 191 | C22×C3 | — | 2 | 4 | no | — |
| 192 | C22×C3 | — | 3 | 8 | no | — |
| 193 | C22×C3 | — | 3 | 4 | no | — |
| 194 | C22×C3 | — | 4 | 1 | no | — |
| 195 | C22×C3 | — | 4 | 4 | no | — |
| 196 | C22×C3 | — | 6 | 16 | no | — |
| 197 | C22×C3 | — | 12 | 4 | no | — |
| 198 | C22×C3 | — | 12 | 16 | no | — |
| 199 | C22×C3 | C2×C32×D4 | 1 | 2 | yes | — |
| 200 | C22×C3 | C2×C32×D4 | 1 | 1 | yes | characteristic |
| 201 | C3×C4 | — | 2 | 4 | no | — |
| 202 | C3×C4 | — | 3 | 4 | no | — |
| 203 | C3×C4 | — | 3 | 4 | no | — |
| 204 | C3×C4 | — | 3 | 8 | no | — |
| 205 | C3×C4 | — | 6 | 16 | no | — |
| 206 | C3×C4 | C2×C32×D4 | 1 | 2 | yes | — |
| 207 | C3×C4 | C2×C32×D4 | 1 | 1 | yes | characteristic |
| 208 | C3×C4 | C24×C32 | 1 | 1 | yes | characteristic |
| 209 | C32 | C3×#575 | 1 | 4 | yes | complemented |
| 210 | C23 | — | 2 | 2 | no | — |
| 211 | C23 | D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 212 | Q8 | — | 2 | 2 | no | — |
| 213 | Q8 | C23×H(3,𝔽3) | 1 | 4 | yes | — |
| 214 | D4 | — | 2 | 4 | no | — |
| 215 | D4 | — | 2 | 4 | no | — |
| 216 | D4 | — | 2 | 2 | no | — |
| 217 | D4 | C23×H(3,𝔽3) | 1 | 4 | yes | — |
| 218 | C2×C4 | — | 2 | 2 | no | — |
| 219 | C2×C4 | — | 2 | 4 | no | — |
| 220 | C2×C4 | — | 2 | 4 | no | — |
| 221 | C2×C4 | D4×H(3,𝔽3) | 1 | 2 | yes | — |
| 222 | C2×C4 | D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | C23×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | C23×H(3,𝔽3) | 1 | 2 | yes | — |
| 225 | C8 | C23×H(3,𝔽3) | 1 | 4 | yes | — |
| 226 | C2×C3 | — | 2 | 2 | no | — |
| 227 | C2×C3 | — | 2 | 1 | no | — |
| 228 | C2×C3 | — | 3 | 4 | no | — |
| 229 | C2×C3 | — | 4 | 4 | no | — |
| 230 | C2×C3 | — | 6 | 8 | no | — |
| 231 | C2×C3 | — | 6 | 4 | no | — |
| 232 | C2×C3 | — | 12 | 16 | no | — |
| 233 | C2×C3 | C22×C32×D4 | 1 | 1 | yes | characteristic |
| 234 | C22 | — | 2 | 4 | no | — |
| 235 | C22 | — | 4 | 1 | no | — |
| 236 | C22 | — | 4 | 4 | no | — |
| 237 | C22 | C2×D4×H(3,𝔽3) | 1 | 2 | yes | — |
| 238 | C22 | C2×D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 2 | 4 | no | — |
| 240 | C4 | C2×D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 241 | C4 | C2×D4×H(3,𝔽3) | 1 | 2 | yes | — |
| 242 | C4 | C24×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 243 | C3 | — | 3 | 4 | no | — |
| 244 | C3 | C32×#575 | 1 | 1 | yes | characteristic |
| 245 | C2 | — | 2 | 2 | no | — |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 4 | 4 | no | — |
| 248 | C2 | C22×D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C2×#41×H(3,𝔽3) | C2 | semidirect |
| 2. H(3,𝔽3)×#137 | C2 | semidirect |
| 3. H(3,𝔽3)×AGL(1,ℤ/8) | C2 | semidirect |
| 4. H(3,𝔽3)×#135 | C2 | semidirect |
| 5. C2×M16×H(3,𝔽3) | C2 | semidirect |
| 6. C32×#575 | C3 | semidirect |
| 10. #41×H(3,𝔽3) | C22 | semidirect |
| 11. C2×Q8×H(3,𝔽3) | C22 | semidirect |
| 13. C2×D4×H(3,𝔽3) | C22 | semidirect |
| 16. Q16×H(3,𝔽3) | C22 | semidirect |
| 17. SD16×H(3,𝔽3) | C22 | semidirect |
| 18. D8×H(3,𝔽3) | C22 | semidirect |
| 19. M16×H(3,𝔽3) | C22 | semidirect |
| 20. C2×C8×H(3,𝔽3) | C22 | semidirect |
| 21. C2×C32×#41 | C2×C3 | semidirect |
| 22. C32×#137 | C2×C3 | semidirect |
| 23. C32×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 24. C32×#135 | C2×C3 | semidirect |
| 25. C2×C32×M16 | C2×C3 | semidirect |
| 47. C32×#41 | C22×C3 | semidirect |
| 48. C2×C32×Q8 | C22×C3 | semidirect |
| 50. C2×C32×D4 | C22×C3 | semidirect |
| 53. C32×Q16 | C22×C3 | semidirect |
| 54. C32×SD16 | C22×C3 | semidirect |
| 55. C32×D8 | C22×C3 | semidirect |
| 56. C32×M16 | C22×C3 | semidirect |
| 57. C2×C32×C8 | C22×C3 | semidirect |
| 93. #575 | H(3,𝔽3) | direct (N × H) |
| 135. C2×#41 | C2×H(3,𝔽3) | semidirect |
| 136. #137 | C2×H(3,𝔽3) | semidirect |
| 137. AGL(1,ℤ/8) | C2×H(3,𝔽3) | semidirect |
| 138. #135 | C2×H(3,𝔽3) | semidirect |
| 139. C2×M16 | C2×H(3,𝔽3) | semidirect |
| 140. H(3,𝔽3) | #575 | direct (N × H) |
| 180. #41 | C22×H(3,𝔽3) | semidirect |
| 181. C2×Q8 | C22×H(3,𝔽3) | semidirect |
| 183. C2×D4 | C22×H(3,𝔽3) | semidirect |
| 186. Q16 | C22×H(3,𝔽3) | semidirect |
| 187. SD16 | C22×H(3,𝔽3) | semidirect |
| 188. D8 | C22×H(3,𝔽3) | semidirect |
| 189. M16 | C22×H(3,𝔽3) | semidirect |
| 190. C2×C8 | C22×H(3,𝔽3) | semidirect |
| 209. C32 | C3×#575 | semidirect |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta