Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
370 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 36 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 | #11722577 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722438 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720767 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10596307 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5836⋊C3 | — | 3 | 1 | no | maximal |
| 6 | #41⋊H(3,𝔽3) | — | 2 | 1 | no | — |
| 7 | #41⋊H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | #88623 | — | 2 | 1 | no | — |
| 9 | #3755⋊C3 | — | 4 | 1 | no | maximal |
| 10 | C32×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #64367 | C2×C3 | 1 | 1 | yes | characteristic |
| 12 | C3×#994 | — | 2 | 1 | no | — |
| 13 | C3×#994 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #1010⋊C3 | — | 3 | 1 | no | — |
| 15 | #64160 | — | 3 | 1 | no | — |
| 16 | #64159 | — | 3 | 1 | no | — |
| 17 | #64157 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×#3754 | — | 3 | 1 | no | — |
| 19 | #64141 | — | 3 | 1 | no | — |
| 20 | C3⋊(#131⋊C3) | — | 3 | 1 | no | — |
| 21 | Q8×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | C4⋊((C3⋊D3)⋊C3) | — | 4 | 1 | no | — |
| 23 | Q8⋊H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 24 | (C3⋊(C3⋊C8))⋊C3 | — | 4 | 1 | no | — |
| 25 | #5836 | — | 3 | 1 | no | — |
| 26 | #5836 | — | 9 | 1 | no | — |
| 27 | #577⋊C3 | — | 9 | 1 | no | — |
| 28 | C32×#41 | — | 2 | 1 | no | — |
| 29 | C32×#41 | — | 3 | 1 | no | — |
| 30 | C32×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 31 | C2×C32×D4 | — | 3 | 1 | no | — |
| 32 | C2×C32×D4 | — | 6 | 1 | no | — |
| 33 | Q8×C3⋊D3 | — | 2 | 1 | no | — |
| 34 | #3830 | — | 6 | 1 | no | — |
| 35 | #3828 | — | 3 | 1 | no | — |
| 36 | C2×#634 | — | 3 | 1 | no | — |
| 37 | C3×#41⋊C3 | — | 2 | 2 | no | — |
| 38 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 39 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 40 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 42 | #3756 | — | 3 | 1 | no | — |
| 43 | #3755 | A4 | 1 | 1 | yes | characteristic |
| 44 | #3754 | — | 3 | 2 | no | — |
| 45 | #3754 | — | 3 | 1 | no | — |
| 46 | #3753 | — | 3 | 1 | no | — |
| 47 | #3748 | — | 3 | 1 | no | — |
| 48 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 49 | C3×#215 | — | 12 | 1 | no | — |
| 50 | C2×(C3⋊D3)⋊C3 | — | 8 | 1 | no | — |
| 51 | C4×H(3,𝔽3) | — | 4 | 1 | no | — |
| 52 | C4×H(3,𝔽3) | — | 8 | 1 | no | — |
| 53 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 54 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 55 | #1010 | — | 3 | 1 | no | — |
| 56 | #1010 | — | 9 | 1 | no | — |
| 57 | 2−1+4⋊C3 | — | 9 | 1 | no | — |
| 58 | #994 | — | 3 | 1 | no | — |
| 59 | #994 | — | 6 | 1 | no | — |
| 60 | #951 | — | 9 | 3 | no | — |
| 61 | #951 | — | 9 | 1 | no | — |
| 62 | #950 | — | 9 | 1 | no | — |
| 63 | #950 | — | 9 | 3 | no | — |
| 64 | #948 | — | 3 | 1 | no | — |
| 65 | #948 | — | 9 | 1 | no | — |
| 66 | C2×#214 | — | 9 | 3 | no | — |
| 67 | C2×#214 | — | 9 | 1 | no | — |
| 68 | #932 | — | 9 | 3 | no | — |
| 69 | #932 | — | 9 | 1 | no | — |
| 70 | #131⋊C3 | — | 9 | 1 | no | — |
| 71 | C23×C32 | — | 6 | 1 | no | — |
| 72 | C32×Q8 | — | 4 | 1 | no | — |
| 73 | C32×Q8 | — | 8 | 1 | no | — |
| 74 | C32×Q8 | C3×D4 | 1 | 1 | yes | characteristic |
| 75 | C32×Q8 | C2×A4 | 1 | 1 | yes | characteristic |
| 76 | C32×D4 | — | 3 | 1 | no | — |
| 77 | C32×D4 | — | 3 | 1 | no | — |
| 78 | C32×D4 | — | 6 | 1 | no | — |
| 79 | C32×D4 | — | 6 | 1 | no | — |
| 80 | C2×C32×C4 | — | 3 | 1 | no | — |
| 81 | C2×C32×C4 | — | 6 | 1 | no | — |
| 82 | #638 | — | 6 | 1 | no | — |
| 83 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 84 | C4⋊(C3⋊D3) | C2×A4 | 1 | 1 | yes | characteristic |
| 85 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 86 | #634 | — | 3 | 1 | no | — |
| 87 | #634 | — | 6 | 1 | no | — |
| 88 | C3×D12 | — | 12 | 1 | no | — |
| 89 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 90 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 91 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 92 | C3⋊(C3⋊C8) | C2×A4 | 1 | 1 | yes | characteristic |
| 93 | C3×C3⋊C8 | — | 12 | 1 | no | — |
| 94 | #577 | — | 9 | 1 | no | Sylow |
| 95 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 96 | (C3⋊D3)⋊C3 | — | 16 | 1 | no | — |
| 97 | C3×#41 | — | 2 | 1 | no | — |
| 98 | C3×#41 | — | 3 | 3 | no | — |
| 99 | C3×#41 | — | 3 | 1 | no | — |
| 100 | C3×#41 | — | 3 | 1 | no | — |
| 101 | C3×#41 | — | 6 | 1 | no | — |
| 102 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 103 | C2×C3×D4 | — | 3 | 3 | no | — |
| 104 | C2×C3×D4 | — | 3 | 1 | no | — |
| 105 | C2×C3×D4 | — | 6 | 3 | no | — |
| 106 | C2×C3×D4 | — | 6 | 1 | no | — |
| 107 | D3×Q8 | — | 6 | 1 | no | — |
| 108 | D3×Q8 | — | 18 | 1 | no | — |
| 109 | #237 | — | 18 | 3 | no | — |
| 110 | #237 | — | 18 | 1 | no | — |
| 111 | #235 | — | 9 | 1 | no | — |
| 112 | #235 | — | 9 | 3 | no | — |
| 113 | C2×Dic6 | — | 9 | 3 | no | — |
| 114 | C2×Dic6 | — | 9 | 1 | no | — |
| 115 | #41⋊C3 | — | 3 | 1 | no | — |
| 116 | #41⋊C3 | — | 6 | 1 | no | — |
| 117 | #41⋊C3 | — | 6 | 2 | no | — |
| 118 | #41⋊C3 | — | 6 | 1 | no | — |
| 119 | C2×SL(2,𝔽3) | — | 18 | 1 | no | — |
| 120 | C3×SD16 | — | 36 | 1 | no | — |
| 121 | #216 | — | 9 | 3 | no | — |
| 122 | #216 | — | 9 | 1 | no | — |
| 123 | #215 | — | 3 | 1 | no | — |
| 124 | #215 | — | 9 | 1 | no | — |
| 125 | #214 | — | 9 | 1 | no | — |
| 126 | #214 | — | 9 | 6 | no | — |
| 127 | #214 | — | 9 | 2 | no | — |
| 128 | #214 | — | 9 | 3 | no | — |
| 129 | #213 | — | 9 | 1 | no | — |
| 130 | #213 | — | 9 | 3 | no | — |
| 131 | #208 | — | 9 | 3 | no | — |
| 132 | #208 | — | 9 | 1 | no | — |
| 133 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 134 | C2×C3⋊C8 | — | 9 | 3 | no | — |
| 135 | C22×C32 | — | 3 | 1 | no | — |
| 136 | C22×C32 | — | 6 | 1 | no | — |
| 137 | C22×C32 | — | 24 | 1 | no | — |
| 138 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 139 | C2×C3×D3 | — | 24 | 1 | no | — |
| 140 | C32×C4 | — | 2 | 1 | no | — |
| 141 | C32×C4 | — | 3 | 1 | no | — |
| 142 | C32×C4 | — | 4 | 1 | no | — |
| 143 | C32×C4 | — | 8 | 1 | no | — |
| 144 | C32×C4 | — | 8 | 2 | no | — |
| 145 | C32×C4 | — | 8 | 1 | no | — |
| 146 | C32×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 147 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 148 | 2−1+4 | — | 9 | 1 | no | — |
| 149 | H(5,𝔽2) | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 150 | #137 | — | 27 | 1 | no | — |
| 151 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 152 | #135 | — | 27 | 1 | no | — |
| 153 | C2×SD16 | — | 27 | 1 | no | — |
| 154 | #131 | — | 9 | 1 | no | — |
| 155 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 156 | C23×C3 | — | 6 | 3 | no | — |
| 157 | C23×C3 | — | 6 | 1 | no | — |
| 158 | C3×Q8 | — | 3 | 1 | no | — |
| 159 | C3×Q8 | — | 3 | 1 | no | — |
| 160 | C3×Q8 | — | 12 | 1 | no | — |
| 161 | C3×Q8 | — | 24 | 1 | no | — |
| 162 | C3×Q8 | C3×#67 | 1 | 1 | yes | characteristic |
| 163 | C3×Q8 | D3×A4 | 1 | 1 | yes | characteristic |
| 164 | C3×D4 | — | 3 | 1 | no | — |
| 165 | C3×D4 | — | 3 | 3 | no | — |
| 166 | C3×D4 | — | 3 | 3 | no | — |
| 167 | C3×D4 | — | 3 | 1 | no | — |
| 168 | C3×D4 | — | 6 | 1 | no | — |
| 169 | C3×D4 | — | 6 | 1 | no | — |
| 170 | C3×D4 | — | 6 | 3 | no | — |
| 171 | C3×D4 | — | 6 | 3 | no | — |
| 172 | C3×D4 | — | 36 | 1 | no | — |
| 173 | C2×C3×C4 | — | 3 | 3 | no | — |
| 174 | C2×C3×C4 | — | 3 | 1 | no | — |
| 175 | C2×C3×C4 | — | 6 | 1 | no | — |
| 176 | C2×C3×C4 | — | 6 | 3 | no | — |
| 177 | #67 | — | 18 | 1 | no | — |
| 178 | #67 | — | 18 | 3 | no | — |
| 179 | C2×Dic3 | — | 18 | 3 | no | — |
| 180 | C2×Dic3 | — | 18 | 1 | no | — |
| 181 | D12 | — | 3 | 1 | no | — |
| 182 | D12 | — | 9 | 1 | no | — |
| 183 | C4×D3 | — | 18 | 1 | no | — |
| 184 | C4×D3 | — | 18 | 3 | no | — |
| 185 | Dic6 | — | 9 | 3 | no | — |
| 186 | Dic6 | — | 9 | 1 | no | — |
| 187 | Dic6 | — | 18 | 3 | no | — |
| 188 | Dic6 | — | 18 | 1 | no | — |
| 189 | SL(2,𝔽3) | — | 3 | 1 | no | — |
| 190 | SL(2,𝔽3) | — | 6 | 1 | no | — |
| 191 | C3×C8 | — | 36 | 1 | no | — |
| 192 | C3⋊C8 | — | 3 | 1 | no | — |
| 193 | C3⋊C8 | — | 9 | 1 | no | — |
| 194 | C3⋊C8 | — | 9 | 3 | no | — |
| 195 | C3⋊C8 | — | 9 | 1 | no | — |
| 196 | C2×C32 | — | 4 | 1 | no | — |
| 197 | C2×C32 | — | 6 | 1 | no | — |
| 198 | C2×C32 | — | 8 | 1 | no | — |
| 199 | C2×C32 | — | 12 | 1 | no | — |
| 200 | C2×C32 | D4×A4 | 1 | 1 | yes | characteristic |
| 201 | C3⋊D3 | — | 4 | 1 | no | — |
| 202 | C3×D3 | — | 48 | 1 | no | — |
| 203 | #41 | — | 2 | 1 | no | — |
| 204 | #41 | — | 3 | 1 | no | — |
| 205 | #41 | — | 27 | 1 | no | — |
| 206 | #41 | — | 54 | 1 | no | — |
| 207 | #41 | C2×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 208 | C2×Q8 | — | 18 | 1 | no | — |
| 209 | C2×Q8 | — | 27 | 1 | no | — |
| 210 | C2×D4 | — | 3 | 1 | no | — |
| 211 | C2×D4 | — | 6 | 1 | no | — |
| 212 | Q16 | — | 27 | 1 | no | — |
| 213 | SD16 | — | 9 | 1 | no | — |
| 214 | SD16 | — | 27 | 1 | no | — |
| 215 | SD16 | — | 27 | 2 | no | — |
| 216 | D8 | — | 27 | 1 | no | — |
| 217 | M16 | — | 27 | 1 | no | — |
| 218 | C2×C8 | — | 27 | 1 | no | — |
| 219 | C22×C3 | — | 3 | 3 | no | — |
| 220 | C22×C3 | — | 3 | 1 | no | — |
| 221 | C22×C3 | — | 6 | 3 | no | — |
| 222 | C22×C3 | — | 6 | 1 | no | — |
| 223 | C22×C3 | — | 24 | 1 | no | — |
| 224 | C22×C3 | — | 24 | 3 | no | — |
| 225 | C22×C3 | — | 72 | 1 | no | — |
| 226 | C2×D3 | — | 6 | 1 | no | — |
| 227 | C2×D3 | — | 18 | 1 | no | — |
| 228 | C3×C4 | — | 2 | 1 | no | — |
| 229 | C3×C4 | — | 3 | 3 | no | — |
| 230 | C3×C4 | — | 3 | 1 | no | — |
| 231 | C3×C4 | — | 3 | 1 | no | — |
| 232 | C3×C4 | — | 6 | 1 | no | — |
| 233 | C3×C4 | — | 12 | 1 | no | — |
| 234 | C3×C4 | — | 24 | 1 | no | — |
| 235 | C3×C4 | — | 24 | 2 | no | — |
| 236 | C3×C4 | — | 24 | 1 | no | — |
| 237 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 238 | Dic3 | — | 18 | 3 | no | — |
| 239 | Dic3 | — | 18 | 1 | no | — |
| 240 | C32 | — | 4 | 1 | no | — |
| 241 | C32 | — | 8 | 1 | no | — |
| 242 | C32 | #577⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 243 | C23 | — | 6 | 1 | no | — |
| 244 | Q8 | — | 27 | 1 | no | — |
| 245 | Q8 | — | 54 | 1 | no | — |
| 246 | Q8 | #638⋊C3 | 1 | 1 | yes | characteristic |
| 247 | Q8 | (C3⋊D3)⋊A4 | 1 | 1 | yes | characteristic |
| 248 | D4 | — | 3 | 1 | no | — |
| 249 | D4 | — | 3 | 1 | no | — |
| 250 | D4 | — | 6 | 1 | no | — |
| 251 | D4 | — | 6 | 1 | no | — |
| 252 | D4 | — | 9 | 1 | no | — |
| 253 | D4 | — | 54 | 1 | no | — |
| 254 | C2×C4 | — | 3 | 1 | no | — |
| 255 | C2×C4 | — | 6 | 1 | no | — |
| 256 | C2×C4 | — | 54 | 1 | no | — |
| 257 | C2×C4 | — | 54 | 1 | no | — |
| 258 | C8 | — | 9 | 1 | no | — |
| 259 | C8 | — | 27 | 1 | no | — |
| 260 | C2×C3 | — | 3 | 1 | no | — |
| 261 | C2×C3 | — | 6 | 3 | no | — |
| 262 | C2×C3 | — | 6 | 1 | no | — |
| 263 | C2×C3 | — | 12 | 3 | no | — |
| 264 | C2×C3 | — | 12 | 1 | no | — |
| 265 | C2×C3 | — | 12 | 1 | no | — |
| 266 | C2×C3 | — | 24 | 1 | no | — |
| 267 | C2×C3 | — | 144 | 1 | no | — |
| 268 | C2×C3 | A4×#67 | 1 | 1 | yes | characteristic |
| 269 | D3 | — | 12 | 1 | no | — |
| 270 | D3 | — | 36 | 1 | no | — |
| 271 | C22 | — | 3 | 1 | no | — |
| 272 | C22 | — | 6 | 1 | no | — |
| 273 | C22 | — | 18 | 1 | no | — |
| 274 | C22 | — | 24 | 1 | no | — |
| 275 | C4 | — | 2 | 1 | no | — |
| 276 | C4 | — | 3 | 1 | no | — |
| 277 | C4 | — | 54 | 1 | no | — |
| 278 | C4 | C2×(C3⋊D3)⋊A4 | 1 | 1 | yes | characteristic |
| 279 | C3 | — | 3 | 1 | no | — |
| 280 | C3 | — | 12 | 1 | no | — |
| 281 | C3 | — | 24 | 1 | no | — |
| 282 | C3 | #5836⋊C3 | 1 | 1 | yes | characteristic |
| 283 | C2 | — | 6 | 1 | no | — |
| 284 | C2 | — | 12 | 1 | no | — |
| 285 | C2 | — | 36 | 1 | no | — |
| 286 | C2 | #638⋊A4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722577 | C2 | semidirect |
| 2. #11722438 | C2 | semidirect |
| 3. #11720767 | C2 | semidirect |
| 4. #10596307 | C3 | semidirect |
| 10. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 13. C3×#994 | D3 | semidirect |
| 17. #64157 | C2×C3 | semidirect |
| 149. H(5,𝔽2) | (C3⋊D3)⋊C3 | semidirect |
| 242. C32 | #577⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722577 | 2 |
| #11722438 | 2 |
| #11720767 | 2 |
| #10596307 | 3 |
| #5836⋊C3 | 3 |
| #3755⋊C3 | 4 |