Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
4778 conjugacy classes of subgroups; only the 807 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 807 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 | #22601995 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #22597742 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #22597732 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #22596804 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C71×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11765866 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #17986668 | — | 2 | 12 | no | — |
| 8 | #17986660 | — | 2 | 12 | no | — |
| 9 | #17986563 | C22 | 1 | 18 | yes | complemented |
| 10 | #17986506 | C22 | 1 | 18 | yes | complemented |
| 11 | #17986477 | C22 | 1 | 3 | yes | complemented |
| 12 | #17986189 | C22 | 1 | 16 | yes | complemented |
| 13 | #17986186 | C22 | 1 | 48 | yes | complemented |
| 14 | #17985973 | C22 | 1 | 36 | yes | complemented |
| 15 | #585×C71 | — | 2 | 12 | no | — |
| 16 | #585×C71 | C22 | 1 | 12 | yes | — |
| 17 | C2×2−1+4×C71 | — | 2 | 12 | no | — |
| 18 | C2×2−1+4×C71 | C22 | 1 | 3 | yes | — |
| 19 | C2×#11765866 | — | 2 | 12 | no | — |
| 20 | C22×D284 | C23 | 1 | 3 | yes | — |
| 21 | C22×C4×D71 | C23 | 1 | 3 | yes | — |
| 22 | #14556124 | — | 2 | 18 | no | — |
| 23 | #14556117 | — | 2 | 18 | no | — |
| 24 | #14556109 | C23 | 1 | 12 | yes | — |
| 25 | #14556104 | — | 2 | 18 | no | — |
| 26 | #14556102 | — | 2 | 36 | no | — |
| 27 | #14556099 | — | 2 | 18 | no | — |
| 28 | #14556096 | C23 | 1 | 36 | yes | — |
| 29 | Q8×Dic71 | — | 2 | 12 | no | — |
| 30 | C4×C22⋊D71 | — | 2 | 36 | no | — |
| 31 | #14556087 | C23 | 1 | 18 | yes | — |
| 32 | #14556043 | C23 | 1 | 48 | yes | — |
| 33 | 2−1+4×C71 | — | 2 | 48 | no | — |
| 34 | 2−1+4×C71 | — | 2 | 36 | no | — |
| 35 | 2−1+4×C71 | — | 2 | 24 | no | — |
| 36 | H(5,𝔽2)×C71 | — | 2 | 16 | no | — |
| 37 | H(5,𝔽2)×C71 | — | 2 | 36 | no | — |
| 38 | H(5,𝔽2)×C71 | C23 | 1 | 16 | yes | — |
| 39 | C2×#41×C71 | — | 2 | 48 | no | — |
| 40 | C2×#41×C71 | — | 2 | 72 | no | — |
| 41 | C2×#41×C71 | D4 | 1 | 12 | yes | complemented |
| 42 | C2×#41×C71 | C23 | 1 | 18 | yes | — |
| 43 | C22×Q8×C71 | — | 2 | 16 | no | — |
| 44 | C22×Q8×C71 | D4 | 1 | 12 | yes | complemented |
| 45 | C22×Q8×C71 | C23 | 1 | 1 | yes | characteristic |
| 46 | C23×D71 | C24 | 1 | 3 | yes | — |
| 47 | #11765866 | — | 4 | 16 | no | — |
| 48 | C2×D284 | — | 2 | 18 | no | — |
| 49 | C2×C22⋊D71 | — | 2 | 36 | no | — |
| 50 | C22×Dic71 | C24 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×D71 | — | 2 | 18 | no | — |
| 52 | C4⋊Dic71 | — | 2 | 18 | no | — |
| 53 | Dic71⋊C4 | — | 2 | 12 | no | — |
| 54 | C22⋊Dic71 | — | 2 | 12 | no | — |
| 55 | C22⋊Dic71 | C24 | 1 | 12 | yes | — |
| 56 | (C2×D71)⋊C4 | — | 2 | 36 | no | — |
| 57 | C4×Dic71 | — | 2 | 18 | no | — |
| 58 | #41×C71 | — | 2 | 144 | no | — |
| 59 | #41×C71 | — | 2 | 48 | no | — |
| 60 | #41×C71 | — | 2 | 48 | no | — |
| 61 | #41×C71 | — | 2 | 48 | no | — |
| 62 | #41×C71 | — | 2 | 72 | no | — |
| 63 | C2×Q8×C71 | — | 2 | 72 | no | — |
| 64 | C2×Q8×C71 | — | 2 | 48 | no | — |
| 65 | C2×Q8×C71 | C2×D4 | 1 | 18 | yes | — |
| 66 | C2×Q8×C71 | C2×D4 | 1 | 12 | yes | — |
| 67 | C2×D4×C71 | — | 2 | 72 | no | — |
| 68 | C2×D4×C71 | — | 2 | 48 | no | — |
| 69 | C2×D4×C71 | C2×D4 | 1 | 18 | yes | — |
| 70 | C2×D4×C71 | C24 | 1 | 12 | yes | — |
| 71 | C22×C4×C71 | — | 2 | 48 | no | — |
| 72 | C22×C4×C71 | C2×D4 | 1 | 36 | yes | — |
| 73 | C22×C4×C71 | C24 | 1 | 3 | yes | — |
| 74 | C23×C71 | — | 2 | 16 | no | — |
| 75 | C23×C71 | C22×D4 | 1 | 12 | yes | — |
| 76 | C23×C71 | C25 | 1 | 1 | yes | characteristic |
| 77 | C22×D71 | — | 2 | 9 | no | — |
| 78 | C22×D71 | — | 8 | 3 | no | — |
| 79 | Q8×C71 | — | 2 | 16 | no | — |
| 80 | Q8×C71 | — | 2 | 36 | no | — |
| 81 | Q8×C71 | #120 | 1 | 16 | yes | — |
| 82 | D4×C71 | — | 2 | 48 | no | — |
| 83 | D4×C71 | — | 2 | 24 | no | — |
| 84 | D4×C71 | — | 2 | 36 | no | — |
| 85 | C2×C4×C71 | — | 2 | 48 | no | — |
| 86 | C2×C4×C71 | — | 2 | 72 | no | — |
| 87 | C2×C4×C71 | C22×D4 | 1 | 12 | yes | — |
| 88 | C2×C4×C71 | C22×D4 | 1 | 18 | yes | — |
| 89 | C22⋊D71 | — | 8 | 36 | no | — |
| 90 | C2×Dic71 | — | 2 | 12 | no | — |
| 91 | C2×Dic71 | — | 2 | 3 | no | — |
| 92 | D284 | — | 4 | 12 | no | — |
| 93 | C4×D71 | — | 4 | 12 | no | — |
| 94 | C22×C71 | — | 2 | 12 | no | — |
| 95 | C22×C71 | — | 4 | 1 | no | — |
| 96 | C22×C71 | — | 4 | 12 | no | — |
| 97 | C22×C71 | — | 8 | 16 | no | — |
| 98 | C22×C71 | C23×D4 | 1 | 3 | yes | — |
| 99 | C2×D71 | — | 4 | 18 | no | — |
| 100 | C2×D71 | — | 4 | 3 | no | — |
| 101 | C4×C71 | — | 2 | 12 | no | — |
| 102 | C4×C71 | C2×#120 | 1 | 12 | yes | — |
| 103 | Dic71 | — | 4 | 12 | no | — |
| 104 | Dic71 | — | 4 | 1 | no | — |
| 105 | #60402 | — | 71 | 1 | no | maximal, Sylow |
| 106 | C2×C71 | — | 2 | 3 | no | — |
| 107 | C2×C71 | — | 4 | 12 | no | — |
| 108 | C2×C71 | C22×#120 | 1 | 1 | yes | characteristic |
| 109 | D71 | — | 8 | 3 | no | — |
| 110 | 2−1+6 | D71 | 1 | 1 | yes | characteristic, complemented |
| 111 | #3487 | — | 71 | 3 | no | — |
| 112 | #3027 | — | 71 | 12 | no | — |
| 113 | #3025 | — | 71 | 12 | no | — |
| 114 | #2884 | — | 71 | 3 | no | — |
| 115 | C71 | #60402 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 116 | #585 | — | 2 | 12 | no | — |
| 117 | #585 | C2×D71 | 1 | 12 | yes | — |
| 118 | C2×2−1+4 | — | 2 | 12 | no | — |
| 119 | C2×2−1+4 | C2×D71 | 1 | 3 | yes | — |
| 120 | C22×#41 | — | 71 | 1 | no | — |
| 121 | #550 | — | 142 | 12 | no | — |
| 122 | #548 | — | 142 | 12 | no | — |
| 123 | #535 | — | 71 | 18 | no | — |
| 124 | #525 | — | 71 | 18 | no | — |
| 125 | #519 | — | 71 | 3 | no | — |
| 126 | #458 | — | 71 | 16 | no | — |
| 127 | B(4,𝔽2) | — | 71 | 48 | no | — |
| 128 | #410 | — | 71 | 36 | no | — |
| 129 | 2−1+4 | — | 2 | 48 | no | — |
| 130 | 2−1+4 | — | 2 | 36 | no | — |
| 131 | 2−1+4 | — | 2 | 24 | no | — |
| 132 | H(5,𝔽2) | — | 2 | 16 | no | — |
| 133 | H(5,𝔽2) | — | 2 | 36 | no | — |
| 134 | H(5,𝔽2) | C22×D71 | 1 | 16 | yes | — |
| 135 | C2×#41 | — | 2 | 48 | no | — |
| 136 | C2×#41 | — | 2 | 72 | no | — |
| 137 | C2×#41 | — | 142 | 12 | no | — |
| 138 | C2×#41 | C22⋊D71 | 1 | 12 | yes | complemented |
| 139 | C2×#41 | C22×D71 | 1 | 18 | yes | — |
| 140 | C22×Q8 | — | 2 | 16 | no | — |
| 141 | C22×Q8 | C22⋊D71 | 1 | 12 | yes | complemented |
| 142 | C22×Q8 | C22×D71 | 1 | 1 | yes | characteristic |
| 143 | C22×D4 | — | 71 | 3 | no | — |
| 144 | C23×C4 | — | 71 | 3 | no | — |
| 145 | #127 | — | 142 | 18 | no | — |
| 146 | #124 | — | 142 | 18 | no | — |
| 147 | #123 | — | 71 | 12 | no | — |
| 148 | C22⋊Q8 | — | 142 | 18 | no | — |
| 149 | #121 | — | 142 | 18 | no | — |
| 150 | #121 | — | 142 | 36 | no | — |
| 151 | #120 | — | 71 | 36 | no | — |
| 152 | C4×Q8 | — | 142 | 12 | no | — |
| 153 | C4×D4 | — | 142 | 36 | no | — |
| 154 | #117 | — | 71 | 18 | no | — |
| 155 | #99 | — | 71 | 48 | no | — |
| 156 | C24 | — | 71 | 3 | no | — |
| 157 | #41 | — | 2 | 144 | no | — |
| 158 | #41 | — | 2 | 48 | no | — |
| 159 | #41 | — | 2 | 48 | no | — |
| 160 | #41 | — | 2 | 48 | no | — |
| 161 | #41 | — | 2 | 72 | no | — |
| 162 | #41 | — | 284 | 16 | no | — |
| 163 | C2×Q8 | — | 2 | 72 | no | — |
| 164 | C2×Q8 | — | 2 | 48 | no | — |
| 165 | C2×Q8 | C2×C22⋊D71 | 1 | 18 | yes | — |
| 166 | C2×Q8 | C2×C22⋊D71 | 1 | 12 | yes | — |
| 167 | C2×D4 | — | 2 | 72 | no | — |
| 168 | C2×D4 | — | 2 | 48 | no | — |
| 169 | C2×D4 | — | 142 | 36 | no | — |
| 170 | C2×D4 | — | 142 | 18 | no | — |
| 171 | C2×D4 | C2×C22⋊D71 | 1 | 18 | yes | — |
| 172 | C2×D4 | C23×D71 | 1 | 12 | yes | — |
| 173 | C22×C4 | — | 2 | 48 | no | — |
| 174 | C22×C4 | — | 71 | 1 | no | — |
| 175 | C22×C4 | — | 142 | 18 | no | — |
| 176 | C22×C4 | C2×C22⋊D71 | 1 | 36 | yes | — |
| 177 | C22×C4 | C23×D71 | 1 | 3 | yes | — |
| 178 | C4⋊C4 | — | 142 | 12 | no | — |
| 179 | C4⋊C4 | — | 142 | 18 | no | — |
| 180 | C22⋊C4 | — | 71 | 12 | no | — |
| 181 | C22⋊C4 | — | 142 | 12 | no | — |
| 182 | C22⋊C4 | — | 142 | 36 | no | — |
| 183 | C42 | — | 142 | 18 | no | — |
| 184 | C23 | — | 2 | 16 | no | — |
| 185 | C23 | — | 568 | 3 | no | — |
| 186 | C23 | — | 142 | 9 | no | — |
| 187 | C23 | C22×C22⋊D71 | 1 | 12 | yes | — |
| 188 | C23 | C24×D71 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 2 | 16 | no | — |
| 190 | Q8 | — | 2 | 36 | no | — |
| 191 | Q8 | #14556097 | 1 | 16 | yes | — |
| 192 | D4 | — | 2 | 48 | no | — |
| 193 | D4 | — | 2 | 24 | no | — |
| 194 | D4 | — | 2 | 36 | no | — |
| 195 | D4 | — | 284 | 12 | no | — |
| 196 | D4 | — | 568 | 36 | no | — |
| 197 | C2×C4 | — | 2 | 48 | no | — |
| 198 | C2×C4 | — | 2 | 72 | no | — |
| 199 | C2×C4 | — | 284 | 12 | no | — |
| 200 | C2×C4 | — | 142 | 12 | no | — |
| 201 | C2×C4 | — | 142 | 3 | no | — |
| 202 | C2×C4 | C22×C22⋊D71 | 1 | 12 | yes | — |
| 203 | C2×C4 | C22×C22⋊D71 | 1 | 18 | yes | — |
| 204 | C22 | — | 2 | 12 | no | — |
| 205 | C22 | — | 4 | 1 | no | — |
| 206 | C22 | — | 4 | 12 | no | — |
| 207 | C22 | — | 8 | 16 | no | — |
| 208 | C22 | — | 284 | 18 | no | — |
| 209 | C22 | — | 284 | 3 | no | — |
| 210 | C22 | C23×C22⋊D71 | 1 | 3 | yes | — |
| 211 | C4 | — | 2 | 12 | no | — |
| 212 | C4 | — | 284 | 12 | no | — |
| 213 | C4 | — | 284 | 1 | no | — |
| 214 | C4 | C2×#14556097 | 1 | 12 | yes | — |
| 215 | C2 | — | 2 | 3 | no | — |
| 216 | C2 | — | 4 | 12 | no | — |
| 217 | C2 | — | 568 | 3 | no | — |
| 218 | C2 | C22×#14556097 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22601995 | C2 | semidirect |
| 2. #22597742 | C2 | semidirect |
| 3. #22597732 | C2 | semidirect |
| 4. #22596804 | C2 | semidirect |
| 5. C71×2−1+6 | C2 | semidirect |
| 6. C22×#11765866 | C22 | semidirect |
| 9. #17986563 | C22 | semidirect |
| 10. #17986506 | C22 | semidirect |
| 11. #17986477 | C22 | semidirect |
| 12. #17986189 | C22 | semidirect |
| 13. #17986186 | C22 | semidirect |
| 14. #17985973 | C22 | semidirect |
| 41. C2×#41×C71 | D4 | semidirect |
| 44. C22×Q8×C71 | D4 | semidirect |
| 110. 2−1+6 | D71 | semidirect |
| 115. C71 | #60402 | semidirect |
| 138. C2×#41 | C22⋊D71 | semidirect |
| 141. C22×Q8 | C22⋊D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22601995 | 2 |
| #22597742 | 2 |
| #22597732 | 2 |
| #22596804 | 2 |
| C71×2−1+6 | 2 |
| #60402 | 71 |