Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1386 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 231 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 | C3×#64367 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#64157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#88812 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11723663 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #11723660 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10595450 | — | 3 | 2 | no | maximal |
| 7 | #10595445 | — | 3 | 1 | no | maximal |
| 8 | C33×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×Q8×C3⋊D3 | — | 2 | 2 | no | — |
| 10 | C3×#3830 | — | 2 | 6 | no | — |
| 11 | C3×#3828 | C22 | 1 | 6 | yes | — |
| 12 | C2×C3×#634 | C22 | 1 | 6 | yes | — |
| 13 | C3×#3748 | C22 | 1 | 3 | yes | — |
| 14 | C2×C3×C3⋊(C3⋊C8) | C22 | 1 | 3 | yes | — |
| 15 | #88812 | C22 | 1 | 3 | yes | — |
| 16 | #88812 | C22 | 1 | 6 | yes | — |
| 17 | #88811 | C22 | 1 | 6 | yes | — |
| 18 | #88810 | C22 | 1 | 1 | yes | characteristic |
| 19 | #64367 | D3 | 1 | 2 | yes | complemented |
| 20 | C3×#1010 | — | 3 | 2 | no | — |
| 21 | C3×#1010 | — | 3 | 2 | no | — |
| 22 | C3×#1010 | — | 6 | 2 | no | — |
| 23 | #64157 | D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C3×#948 | — | 3 | 2 | no | — |
| 25 | C3×#948 | — | 6 | 1 | no | — |
| 26 | C2×#3719 | — | 3 | 6 | no | — |
| 27 | C2×#3718 | — | 3 | 3 | no | — |
| 28 | #63833 | — | 3 | 6 | no | — |
| 29 | #63831 | — | 3 | 6 | no | — |
| 30 | #63829 | — | 3 | 6 | no | — |
| 31 | #63827 | — | 3 | 6 | no | — |
| 32 | #63822 | — | 3 | 3 | no | — |
| 33 | C33×Q8 | D4 | 1 | 1 | yes | characteristic |
| 34 | C33×D4 | D4 | 1 | 3 | yes | — |
| 35 | C2×C33×C4 | C23 | 1 | 3 | yes | — |
| 36 | C3×#638 | — | 2 | 6 | no | — |
| 37 | C2×C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 38 | C3×C4⋊(C3⋊D3) | C23 | 1 | 2 | yes | — |
| 39 | C3×C4×C3⋊D3 | — | 2 | 6 | no | — |
| 40 | C3×#634 | — | 2 | 6 | no | — |
| 41 | C3×#634 | C23 | 1 | 6 | yes | — |
| 42 | C3×C3⋊(C3⋊C8) | C23 | 1 | 1 | yes | characteristic |
| 43 | C3×C3⋊(C3⋊C8) | C23 | 1 | 3 | yes | — |
| 44 | #5838 | — | 9 | 2 | no | — |
| 45 | #5753 | — | 9 | 1 | no | — |
| 46 | C32×#41 | — | 2 | 1 | no | — |
| 47 | C32×#41 | — | 2 | 2 | no | — |
| 48 | C32×#41 | — | 4 | 1 | no | — |
| 49 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 50 | C32×#41 | C2×D3 | 1 | 2 | yes | — |
| 51 | Q8×C3⋊D3 | — | 2 | 2 | no | — |
| 52 | #3830 | — | 2 | 6 | no | — |
| 53 | #3828 | C2×D3 | 1 | 6 | yes | — |
| 54 | C2×#634 | C2×D3 | 1 | 6 | yes | — |
| 55 | C3×D3×Q8 | — | 6 | 4 | no | — |
| 56 | C3×D3×Q8 | — | 6 | 2 | no | — |
| 57 | C3×D3×Q8 | — | 6 | 2 | no | — |
| 58 | C3×#237 | — | 6 | 12 | no | — |
| 59 | C3×#237 | — | 6 | 6 | no | — |
| 60 | C3×#237 | — | 6 | 6 | no | — |
| 61 | C3×#235 | — | 3 | 6 | no | — |
| 62 | C3×#235 | — | 3 | 6 | no | — |
| 63 | C3×#235 | — | 6 | 6 | no | — |
| 64 | C2×C3×Dic6 | — | 3 | 6 | no | — |
| 65 | C2×C3×Dic6 | — | 3 | 6 | no | — |
| 66 | C2×C3×Dic6 | — | 6 | 6 | no | — |
| 67 | #3748 | C2×D3 | 1 | 3 | yes | — |
| 68 | C2×C3⋊(C3⋊C8) | C2×D3 | 1 | 3 | yes | — |
| 69 | C3×#208 | — | 3 | 6 | no | — |
| 70 | C3×#208 | — | 6 | 3 | no | — |
| 71 | C2×C3×C3⋊C8 | — | 3 | 6 | no | — |
| 72 | C2×C3×C3⋊C8 | — | 6 | 3 | no | — |
| 73 | #3719 | — | 3 | 6 | no | — |
| 74 | #3719 | — | 3 | 12 | no | — |
| 75 | #3718 | — | 3 | 3 | no | — |
| 76 | #3718 | — | 3 | 6 | no | — |
| 77 | #3717 | — | 3 | 6 | no | — |
| 78 | #3716 | — | 3 | 6 | no | — |
| 79 | #3715 | — | 3 | 6 | no | — |
| 80 | #3714 | — | 3 | 2 | no | — |
| 81 | #3713 | — | 3 | 1 | no | — |
| 82 | C22×C33 | C2×D4 | 1 | 3 | yes | — |
| 83 | C2×C3×C3⋊D3 | — | 2 | 2 | no | — |
| 84 | C33×C4 | C2×D4 | 1 | 3 | yes | — |
| 85 | C33×C4 | C24 | 1 | 1 | yes | characteristic |
| 86 | C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 87 | C3×2−1+4 | — | 9 | 2 | no | — |
| 88 | #1010 | — | 3 | 2 | no | — |
| 89 | #1010 | — | 3 | 2 | no | — |
| 90 | #1010 | — | 6 | 2 | no | — |
| 91 | C3×#131 | — | 9 | 1 | no | — |
| 92 | #951 | — | 9 | 6 | no | — |
| 93 | #950 | — | 9 | 6 | no | — |
| 94 | #948 | — | 3 | 2 | no | — |
| 95 | #948 | — | 6 | 1 | no | — |
| 96 | C2×#216 | — | 9 | 6 | no | — |
| 97 | #942 | — | 9 | 6 | no | — |
| 98 | #909 | — | 9 | 6 | no | — |
| 99 | C2×Dic12 | — | 9 | 3 | no | — |
| 100 | #904 | — | 9 | 3 | no | — |
| 101 | C32×Q8 | — | 2 | 1 | no | — |
| 102 | C32×Q8 | — | 2 | 2 | no | — |
| 103 | C32×Q8 | — | 4 | 1 | no | — |
| 104 | C32×Q8 | D12 | 1 | 1 | yes | characteristic |
| 105 | C32×Q8 | #67 | 1 | 2 | yes | — |
| 106 | C32×D4 | — | 2 | 3 | no | — |
| 107 | C32×D4 | — | 2 | 6 | no | — |
| 108 | C32×D4 | — | 4 | 3 | no | — |
| 109 | C32×D4 | D12 | 1 | 3 | yes | — |
| 110 | C32×D4 | #67 | 1 | 6 | yes | — |
| 111 | C2×C32×C4 | — | 2 | 3 | no | — |
| 112 | C2×C32×C4 | — | 2 | 6 | no | — |
| 113 | C2×C32×C4 | — | 4 | 3 | no | — |
| 114 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 115 | C2×C32×C4 | C22×D3 | 1 | 6 | yes | — |
| 116 | #638 | — | 2 | 6 | no | — |
| 117 | C2×C3⋊Dic3 | — | 2 | 6 | no | — |
| 118 | C4⋊(C3⋊D3) | C22×D3 | 1 | 2 | yes | — |
| 119 | C4×C3⋊D3 | — | 2 | 6 | no | — |
| 120 | #634 | — | 2 | 6 | no | — |
| 121 | #634 | C22×D3 | 1 | 6 | yes | — |
| 122 | C3×#67 | — | 6 | 12 | no | — |
| 123 | C3×#67 | — | 6 | 6 | no | — |
| 124 | C3×#67 | — | 6 | 6 | no | — |
| 125 | C2×C3×Dic3 | — | 6 | 12 | no | — |
| 126 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 127 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 128 | C3×D12 | — | 3 | 2 | no | — |
| 129 | C3×D12 | — | 3 | 2 | no | — |
| 130 | C3×D12 | — | 6 | 2 | no | — |
| 131 | C3×C4×D3 | — | 6 | 12 | no | — |
| 132 | C3×C4×D3 | — | 6 | 6 | no | — |
| 133 | C3×C4×D3 | — | 6 | 6 | no | — |
| 134 | C3×Dic6 | — | 3 | 6 | no | — |
| 135 | C3×Dic6 | — | 3 | 6 | no | — |
| 136 | C3×Dic6 | — | 6 | 6 | no | — |
| 137 | C3×Dic6 | — | 6 | 12 | no | — |
| 138 | C3×Dic6 | — | 6 | 6 | no | — |
| 139 | C3×Dic6 | — | 6 | 6 | no | — |
| 140 | C3⋊(C3⋊C8) | C22×D3 | 1 | 1 | yes | characteristic |
| 141 | C3⋊(C3⋊C8) | C22×D3 | 1 | 3 | yes | — |
| 142 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 143 | C3×C3⋊C8 | — | 3 | 6 | no | — |
| 144 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 145 | C3×C3⋊C8 | — | 6 | 3 | no | — |
| 146 | #578 | — | 27 | 1 | no | Sylow |
| 147 | C2×C33 | — | 2 | 3 | no | — |
| 148 | C2×C33 | C22×D4 | 1 | 1 | yes | characteristic |
| 149 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 150 | C3×#41 | — | 2 | 1 | no | — |
| 151 | C3×#41 | — | 2 | 2 | no | — |
| 152 | C3×#41 | — | 4 | 1 | no | — |
| 153 | C3×#41 | — | 9 | 6 | no | — |
| 154 | C3×#41 | — | 18 | 6 | no | — |
| 155 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 156 | C3×#41 | D32 | 1 | 2 | yes | — |
| 157 | C2×C3×Q8 | — | 9 | 6 | no | — |
| 158 | C2×C3×Q8 | — | 18 | 2 | no | — |
| 159 | D3×Q8 | — | 6 | 4 | no | — |
| 160 | D3×Q8 | — | 6 | 2 | no | — |
| 161 | D3×Q8 | — | 6 | 2 | no | — |
| 162 | #237 | — | 6 | 12 | no | — |
| 163 | #237 | — | 6 | 6 | no | — |
| 164 | #237 | — | 6 | 6 | no | — |
| 165 | #235 | — | 3 | 6 | no | — |
| 166 | #235 | — | 3 | 6 | no | — |
| 167 | #235 | — | 6 | 6 | no | — |
| 168 | C2×Dic6 | — | 3 | 6 | no | — |
| 169 | C2×Dic6 | — | 3 | 6 | no | — |
| 170 | C2×Dic6 | — | 6 | 6 | no | — |
| 171 | C3×M16 | — | 9 | 3 | no | — |
| 172 | C2×C3×C8 | — | 9 | 3 | no | — |
| 173 | #216 | — | 9 | 12 | no | — |
| 174 | #216 | — | 9 | 6 | no | — |
| 175 | #215 | — | 9 | 6 | no | — |
| 176 | #214 | — | 9 | 6 | no | — |
| 177 | #213 | — | 9 | 2 | no | — |
| 178 | #208 | — | 3 | 6 | no | — |
| 179 | #208 | — | 6 | 3 | no | — |
| 180 | C2×C3⋊C8 | — | 3 | 6 | no | — |
| 181 | C2×C3⋊C8 | — | 6 | 3 | no | — |
| 182 | Dic12 | — | 9 | 3 | no | — |
| 183 | Dic12 | — | 9 | 6 | no | — |
| 184 | D24 | — | 9 | 1 | no | — |
| 185 | #204 | — | 9 | 6 | no | — |
| 186 | C22×C32 | — | 2 | 3 | no | — |
| 187 | C22×C32 | — | 2 | 6 | no | — |
| 188 | C22×C32 | — | 4 | 3 | no | — |
| 189 | C22×C32 | C2×D12 | 1 | 3 | yes | — |
| 190 | C22×C32 | C2×#67 | 1 | 6 | yes | — |
| 191 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 192 | C2×C3×D3 | — | 6 | 4 | no | — |
| 193 | C2×C3×D3 | — | 6 | 2 | no | — |
| 194 | C2×C3×D3 | — | 6 | 2 | no | — |
| 195 | C32×C4 | — | 2 | 1 | no | — |
| 196 | C32×C4 | — | 2 | 2 | no | — |
| 197 | C32×C4 | — | 2 | 3 | no | — |
| 198 | C32×C4 | — | 2 | 6 | no | — |
| 199 | C32×C4 | — | 4 | 3 | no | — |
| 200 | C32×C4 | — | 4 | 1 | no | — |
| 201 | C32×C4 | C2×D12 | 1 | 3 | yes | — |
| 202 | C32×C4 | C2×#67 | 1 | 6 | yes | — |
| 203 | C32×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 204 | C32×C4 | C23×D3 | 1 | 2 | yes | — |
| 205 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 206 | C3×Dic3 | — | 6 | 12 | no | — |
| 207 | C3×Dic3 | — | 6 | 6 | no | — |
| 208 | C3×Dic3 | — | 6 | 6 | no | — |
| 209 | 2−1+4 | — | 9 | 2 | no | — |
| 210 | #137 | — | 27 | 6 | no | — |
| 211 | #135 | — | 27 | 3 | no | — |
| 212 | C2×Q16 | — | 27 | 3 | no | — |
| 213 | #131 | — | 9 | 1 | no | — |
| 214 | C33 | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 215 | C3×Q8 | — | 2 | 1 | no | — |
| 216 | C3×Q8 | — | 2 | 2 | no | — |
| 217 | C3×Q8 | — | 4 | 1 | no | — |
| 218 | C3×Q8 | — | 9 | 6 | no | — |
| 219 | C3×Q8 | — | 18 | 6 | no | — |
| 220 | C3×Q8 | #625 | 1 | 1 | yes | characteristic |
| 221 | C3×Q8 | #626 | 1 | 2 | yes | — |
| 222 | C3×D4 | — | 2 | 3 | no | — |
| 223 | C3×D4 | — | 2 | 6 | no | — |
| 224 | C3×D4 | — | 4 | 3 | no | — |
| 225 | C3×D4 | — | 9 | 2 | no | — |
| 226 | C3×D4 | — | 18 | 6 | no | — |
| 227 | C3×D4 | #625 | 1 | 3 | yes | — |
| 228 | C3×D4 | #626 | 1 | 6 | yes | — |
| 229 | C2×C3×C4 | — | 2 | 3 | no | — |
| 230 | C2×C3×C4 | — | 2 | 6 | no | — |
| 231 | C2×C3×C4 | — | 4 | 3 | no | — |
| 232 | C2×C3×C4 | — | 18 | 6 | no | — |
| 233 | C2×C3×C4 | — | 18 | 6 | no | — |
| 234 | C2×C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 235 | C2×C3×C4 | C2×D32 | 1 | 6 | yes | — |
| 236 | #67 | — | 6 | 12 | no | — |
| 237 | #67 | — | 6 | 6 | no | — |
| 238 | #67 | — | 6 | 6 | no | — |
| 239 | C2×Dic3 | — | 6 | 12 | no | — |
| 240 | C2×Dic3 | — | 6 | 6 | no | — |
| 241 | C2×Dic3 | — | 6 | 6 | no | — |
| 242 | D12 | — | 3 | 2 | no | — |
| 243 | D12 | — | 3 | 2 | no | — |
| 244 | D12 | — | 6 | 2 | no | — |
| 245 | C4×D3 | — | 6 | 12 | no | — |
| 246 | C4×D3 | — | 6 | 6 | no | — |
| 247 | C4×D3 | — | 6 | 6 | no | — |
| 248 | Dic6 | — | 3 | 6 | no | — |
| 249 | Dic6 | — | 3 | 6 | no | — |
| 250 | Dic6 | — | 6 | 6 | no | — |
| 251 | Dic6 | — | 6 | 12 | no | — |
| 252 | Dic6 | — | 6 | 6 | no | — |
| 253 | Dic6 | — | 6 | 6 | no | — |
| 254 | C3×C8 | — | 9 | 3 | no | — |
| 255 | C3×C8 | — | 9 | 1 | no | — |
| 256 | C3⋊C8 | — | 3 | 2 | no | — |
| 257 | C3⋊C8 | — | 3 | 6 | no | — |
| 258 | C3⋊C8 | — | 6 | 1 | no | — |
| 259 | C3⋊C8 | — | 6 | 3 | no | — |
| 260 | C2×C32 | — | 2 | 3 | no | — |
| 261 | C2×C32 | — | 2 | 6 | no | — |
| 262 | C2×C32 | — | 2 | 1 | no | — |
| 263 | C2×C32 | — | 2 | 2 | no | — |
| 264 | C2×C32 | — | 4 | 1 | no | — |
| 265 | C2×C32 | — | 4 | 3 | no | — |
| 266 | C2×C32 | — | 4 | 6 | no | — |
| 267 | C2×C32 | — | 8 | 3 | no | — |
| 268 | C2×C32 | C22×D12 | 1 | 1 | yes | characteristic |
| 269 | C2×C32 | C22×#67 | 1 | 2 | yes | — |
| 270 | C3⋊D3 | — | 4 | 2 | no | — |
| 271 | C3×D3 | — | 12 | 4 | no | — |
| 272 | C3×D3 | — | 12 | 2 | no | — |
| 273 | C3×D3 | — | 12 | 2 | no | — |
| 274 | #41 | — | 9 | 6 | no | — |
| 275 | #41 | — | 18 | 6 | no | — |
| 276 | #41 | #1113 | 1 | 1 | yes | characteristic |
| 277 | C2×Q8 | — | 9 | 6 | no | — |
| 278 | C2×Q8 | — | 18 | 2 | no | — |
| 279 | Q16 | — | 27 | 3 | no | — |
| 280 | Q16 | — | 27 | 6 | no | — |
| 281 | SD16 | — | 27 | 6 | no | — |
| 282 | D8 | — | 27 | 1 | no | — |
| 283 | M16 | — | 9 | 3 | no | — |
| 284 | C2×C8 | — | 9 | 3 | no | — |
| 285 | C22×C3 | — | 2 | 6 | no | — |
| 286 | C22×C3 | — | 2 | 3 | no | — |
| 287 | C22×C3 | — | 4 | 3 | no | — |
| 288 | C22×C3 | — | 18 | 2 | no | — |
| 289 | C22×C3 | C2×#625 | 1 | 3 | yes | — |
| 290 | C22×C3 | C2×#626 | 1 | 6 | yes | — |
| 291 | C2×D3 | — | 6 | 4 | no | — |
| 292 | C2×D3 | — | 6 | 2 | no | — |
| 293 | C2×D3 | — | 6 | 2 | no | — |
| 294 | C3×C4 | — | 2 | 1 | no | — |
| 295 | C3×C4 | — | 2 | 2 | no | — |
| 296 | C3×C4 | — | 2 | 6 | no | — |
| 297 | C3×C4 | — | 2 | 3 | no | — |
| 298 | C3×C4 | — | 4 | 1 | no | — |
| 299 | C3×C4 | — | 4 | 3 | no | — |
| 300 | C3×C4 | — | 18 | 6 | no | — |
| 301 | C3×C4 | C2×#625 | 1 | 3 | yes | — |
| 302 | C3×C4 | C2×#626 | 1 | 6 | yes | — |
| 303 | C3×C4 | C22×D32 | 1 | 1 | yes | characteristic |
| 304 | C3×C4 | C22×D32 | 1 | 2 | yes | — |
| 305 | Dic3 | — | 6 | 12 | no | — |
| 306 | Dic3 | — | 6 | 6 | no | — |
| 307 | Dic3 | — | 6 | 6 | no | — |
| 308 | C32 | — | 2 | 1 | no | — |
| 309 | C32 | — | 2 | 2 | no | — |
| 310 | C32 | — | 4 | 1 | no | — |
| 311 | C32 | #5753 | 1 | 1 | yes | characteristic, complemented |
| 312 | C32 | #5838 | 1 | 2 | yes | complemented |
| 313 | Q8 | — | 9 | 6 | no | — |
| 314 | Q8 | — | 18 | 6 | no | — |
| 315 | Q8 | #6295 | 1 | 1 | yes | characteristic |
| 316 | D4 | — | 9 | 2 | no | — |
| 317 | D4 | — | 18 | 6 | no | — |
| 318 | D4 | #6295 | 1 | 3 | yes | — |
| 319 | C2×C4 | — | 18 | 6 | no | — |
| 320 | C2×C4 | — | 18 | 6 | no | — |
| 321 | C2×C4 | C2×#1113 | 1 | 3 | yes | — |
| 322 | C8 | — | 9 | 1 | no | — |
| 323 | C8 | — | 9 | 3 | no | — |
| 324 | C2×C3 | — | 2 | 3 | no | — |
| 325 | C2×C3 | — | 2 | 6 | no | — |
| 326 | C2×C3 | — | 2 | 2 | no | — |
| 327 | C2×C3 | — | 2 | 1 | no | — |
| 328 | C2×C3 | — | 4 | 6 | no | — |
| 329 | C2×C3 | — | 4 | 3 | no | — |
| 330 | C2×C3 | — | 4 | 1 | no | — |
| 331 | C2×C3 | — | 8 | 3 | no | — |
| 332 | C2×C3 | — | 36 | 2 | no | — |
| 333 | C2×C3 | C22×#625 | 1 | 1 | yes | characteristic |
| 334 | C2×C3 | C22×#626 | 1 | 2 | yes | — |
| 335 | D3 | — | 12 | 4 | no | — |
| 336 | D3 | — | 12 | 2 | no | — |
| 337 | D3 | — | 12 | 2 | no | — |
| 338 | C22 | — | 18 | 2 | no | — |
| 339 | C22 | C2×#6295 | 1 | 3 | yes | — |
| 340 | C4 | — | 18 | 6 | no | — |
| 341 | C4 | C2×#6295 | 1 | 3 | yes | — |
| 342 | C4 | C22×#1113 | 1 | 1 | yes | characteristic |
| 343 | C3 | — | 2 | 2 | no | — |
| 344 | C3 | — | 2 | 1 | no | — |
| 345 | C3 | — | 4 | 1 | no | — |
| 346 | C3 | #10595445 | 1 | 1 | yes | characteristic, complemented |
| 347 | C3 | #10595450 | 1 | 2 | yes | complemented |
| 348 | C2 | — | 2 | 3 | no | — |
| 349 | C2 | — | 36 | 2 | no | — |
| 350 | C2 | C22×#6295 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#6295 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×#1113 |
| Fitting | C33×#41 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | #41 | #1113 |
| hypocenter | C33 | #578 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#64367 | C2 | semidirect |
| 2. C3×#64157 | C2 | semidirect |
| 3. C2×#88812 | C2 | semidirect |
| 4. #11723663 | C2 | semidirect |
| 5. #11723660 | C2 | semidirect |
| 19. #64367 | D3 | semidirect |
| 23. #64157 | D3 | semidirect |
| 214. C33 | #578 | semidirect |
| 311. C32 | #5753 | semidirect |
| 312. C32 | #5838 | semidirect |
| 346. C3 | #10595445 | semidirect |
| 347. C3 | #10595450 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64367 | 2 |
| C3×#64157 | 2 |
| C2×#88812 | 2 |
| #11723663 | 2 |
| #11723660 | 2 |
| #10595450 | 3 |
| #10595445 | 3 |