Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
648 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #21461260 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #21461131 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #21460513 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16840781 | C22 | 1 | 3 | yes | complemented |
| 5 | #16840707 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | #16840616 | — | 2 | 3 | no | — |
| 7 | #16840613 | — | 2 | 3 | no | — |
| 8 | Q8⋊#92420 | — | 2 | 3 | no | — |
| 9 | D4⋊#92420 | — | 2 | 3 | no | — |
| 10 | #16840512 | C4 | 1 | 3 | yes | complemented |
| 11 | #16840481 | — | 2 | 3 | no | — |
| 12 | #16840443 | — | 2 | 3 | no | — |
| 13 | #16840376 | — | 2 | 3 | no | — |
| 14 | #16840347 | — | 2 | 3 | no | — |
| 15 | #16840291 | C22 | 1 | 3 | yes | — |
| 16 | C61×H(3,𝔽4) | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C4⋊#92420 | — | 2 | 3 | no | — |
| 18 | #14519834 | — | 2 | 3 | no | — |
| 19 | C4×#92420 | — | 2 | 3 | no | — |
| 20 | (C2×#6923)⋊C4 | — | 2 | 3 | no | — |
| 21 | #14519775 | — | 2 | 3 | no | — |
| 22 | #14519771 | — | 2 | 1 | no | — |
| 23 | #14519770 | — | 2 | 3 | no | — |
| 24 | #14519770 | — | 2 | 3 | no | — |
| 25 | #14519770 | — | 2 | 3 | no | — |
| 26 | #14519770 | — | 2 | 3 | no | — |
| 27 | #14519769 | C2×C4 | 1 | 3 | yes | — |
| 28 | #14519764 | — | 2 | 3 | no | — |
| 29 | #14519763 | — | 2 | 3 | no | — |
| 30 | #14519759 | — | 2 | 3 | no | — |
| 31 | #14519758 | — | 2 | 3 | no | — |
| 32 | #14519758 | — | 2 | 3 | no | — |
| 33 | #14519754 | — | 2 | 3 | no | — |
| 34 | #14519754 | — | 2 | 3 | no | — |
| 35 | #14519754 | — | 2 | 3 | no | — |
| 36 | #14519753 | D4 | 1 | 3 | yes | — |
| 37 | #14519749 | — | 2 | 3 | no | — |
| 38 | #14519748 | D4 | 1 | 3 | yes | — |
| 39 | #14519746 | — | 2 | 1 | no | — |
| 40 | Q8×Dic61 | D4 | 1 | 3 | yes | — |
| 41 | C4×Dic122 | — | 2 | 3 | no | — |
| 42 | D4×Dic61 | — | 2 | 3 | no | — |
| 43 | D4×Dic61 | D4 | 1 | 3 | yes | — |
| 44 | #14519737 | — | 2 | 3 | no | — |
| 45 | #14519733 | — | 2 | 3 | no | — |
| 46 | #14519733 | — | 2 | 3 | no | — |
| 47 | C2×C22⋊Dic61 | — | 2 | 3 | no | — |
| 48 | C42×D61 | C23 | 1 | 1 | yes | characteristic |
| 49 | #124×C61 | — | 2 | 3 | no | — |
| 50 | #124×C61 | C2×C4 | 1 | 3 | yes | — |
| 51 | #120×C61 | — | 2 | 3 | no | — |
| 52 | C24×C61 | — | 2 | 1 | no | — |
| 53 | C22×#6923 | — | 4 | 1 | no | — |
| 54 | C2×Q8×C61 | C22⋊C4 | 1 | 3 | yes | — |
| 55 | C2×D4×C61 | — | 2 | 3 | no | — |
| 56 | C2×D4×C61 | C22⋊C4 | 1 | 3 | yes | — |
| 57 | C2×C22⋊D61 | — | 2 | 3 | no | — |
| 58 | C22×Dic61 | — | 2 | 3 | no | — |
| 59 | C22×Dic61 | — | 2 | 3 | no | — |
| 60 | C2×C4×D61 | C2×D4 | 1 | 3 | yes | — |
| 61 | C2×Dic122 | — | 2 | 3 | no | — |
| 62 | C2×#92420 | — | 4 | 3 | no | — |
| 63 | C22⋊C4×C61 | — | 2 | 3 | no | — |
| 64 | C22⋊C4×C61 | — | 2 | 3 | no | — |
| 65 | C22⋊C4×C61 | — | 2 | 3 | no | — |
| 66 | C42×C61 | — | 2 | 1 | no | — |
| 67 | C42×C61 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 69 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 70 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 71 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 72 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 73 | #11758198 | — | 2 | 3 | no | — |
| 74 | C4⋊Dic61 | — | 2 | 3 | no | — |
| 75 | C4⋊Dic61 | C22⋊C4 | 1 | 3 | yes | — |
| 76 | C4⋊Dic61 | C22⋊C4 | 1 | 3 | yes | — |
| 77 | #11758196 | — | 2 | 3 | no | — |
| 78 | #11758196 | — | 2 | 3 | no | — |
| 79 | #11758196 | — | 2 | 3 | no | — |
| 80 | #11758196 | — | 2 | 3 | no | — |
| 81 | #11758196 | — | 2 | 3 | no | — |
| 82 | C4×Dic61 | — | 2 | 3 | no | — |
| 83 | C4×Dic61 | — | 2 | 3 | no | — |
| 84 | C4×Dic61 | C2×D4 | 1 | 3 | yes | — |
| 85 | C23×C61 | — | 2 | 3 | no | — |
| 86 | C23×C61 | — | 8 | 3 | no | — |
| 87 | C22×D61 | #120 | 1 | 1 | yes | characteristic |
| 88 | C2×#6923 | — | 8 | 3 | no | — |
| 89 | Q8×C61 | — | 4 | 3 | no | — |
| 90 | D4×C61 | — | 4 | 3 | no | — |
| 91 | D4×C61 | — | 8 | 3 | no | — |
| 92 | C2×C4×C61 | — | 2 | 3 | no | — |
| 93 | C2×C4×C61 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 94 | C22⋊D61 | — | 8 | 3 | no | — |
| 95 | C2×Dic61 | — | 2 | 3 | no | — |
| 96 | C2×Dic61 | — | 2 | 3 | no | — |
| 97 | C2×Dic61 | — | 8 | 3 | no | — |
| 98 | C2×Dic61 | — | 8 | 3 | no | — |
| 99 | C2×Dic61 | #120 | 1 | 3 | yes | — |
| 100 | C4×D61 | — | 4 | 3 | no | — |
| 101 | Dic122 | — | 8 | 3 | no | — |
| 102 | #92420 | — | 8 | 3 | no | — |
| 103 | #8514 | — | 61 | 1 | no | maximal, Sylow |
| 104 | C22×C61 | — | 4 | 3 | no | — |
| 105 | C22×C61 | — | 4 | 3 | no | — |
| 106 | C22×C61 | — | 4 | 3 | no | — |
| 107 | C22×C61 | — | 16 | 2 | no | — |
| 108 | C22×C61 | #379 | 1 | 1 | yes | characteristic |
| 109 | C2×D61 | — | 2 | 3 | no | — |
| 110 | #6923 | — | 16 | 1 | no | — |
| 111 | C4×C61 | — | 2 | 3 | no | — |
| 112 | C4×C61 | — | 4 | 3 | no | — |
| 113 | Dic61 | — | 2 | 3 | no | — |
| 114 | Dic61 | — | 4 | 3 | no | — |
| 115 | Dic61 | — | 4 | 3 | no | — |
| 116 | #3568 | — | 61 | 1 | no | — |
| 117 | #1521 | — | 61 | 3 | no | — |
| 118 | #1482 | — | 61 | 3 | no | — |
| 119 | C2×C61 | — | 8 | 3 | no | — |
| 120 | C2×C61 | #1794 | 1 | 3 | yes | — |
| 121 | D61 | — | 4 | 1 | no | — |
| 122 | #563 | — | 61 | 3 | no | — |
| 123 | H(3,𝔽4) | #6923 | 1 | 1 | yes | characteristic, complemented |
| 124 | #556 | — | 122 | 3 | no | — |
| 125 | #551 | — | 122 | 3 | no | — |
| 126 | #542 | — | 122 | 3 | no | — |
| 127 | #538 | — | 122 | 3 | no | — |
| 128 | #529 | — | 61 | 3 | no | — |
| 129 | #431 | — | 61 | 3 | no | — |
| 130 | #383 | — | 61 | 1 | no | — |
| 131 | #328 | — | 122 | 3 | no | — |
| 132 | #327 | — | 122 | 3 | no | — |
| 133 | Q8⋊C8 | — | 122 | 3 | no | — |
| 134 | D4⋊C8 | — | 122 | 3 | no | — |
| 135 | C61 | #8514 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 136 | C4⋊Q8 | — | 122 | 3 | no | — |
| 137 | #126 | — | 122 | 1 | no | — |
| 138 | #126 | — | 122 | 3 | no | — |
| 139 | #126 | — | 122 | 3 | no | — |
| 140 | #126 | — | 122 | 3 | no | — |
| 141 | #126 | — | 122 | 3 | no | — |
| 142 | #125 | — | 61 | 3 | no | — |
| 143 | #124 | — | 2 | 3 | no | — |
| 144 | #124 | — | 122 | 3 | no | — |
| 145 | #124 | — | 122 | 3 | no | — |
| 146 | #124 | C2×#6923 | 1 | 3 | yes | — |
| 147 | #123 | — | 122 | 3 | no | — |
| 148 | #123 | — | 122 | 3 | no | — |
| 149 | #123 | — | 122 | 3 | no | — |
| 150 | C22⋊Q8 | — | 61 | 3 | no | — |
| 151 | C22⋊Q8 | — | 122 | 3 | no | — |
| 152 | C22⋊Q8 | — | 122 | 3 | no | — |
| 153 | C22⋊Q8 | — | 122 | 3 | no | — |
| 154 | #121 | — | 61 | 3 | no | — |
| 155 | #121 | — | 122 | 3 | no | — |
| 156 | #120 | — | 2 | 3 | no | — |
| 157 | #120 | — | 122 | 1 | no | — |
| 158 | C4×Q8 | — | 61 | 3 | no | — |
| 159 | C4×Q8 | — | 122 | 3 | no | — |
| 160 | C4×D4 | — | 61 | 3 | no | — |
| 161 | C4×D4 | — | 122 | 3 | no | — |
| 162 | C4×D4 | — | 122 | 3 | no | — |
| 163 | #117 | — | 122 | 3 | no | — |
| 164 | #117 | — | 122 | 3 | no | — |
| 165 | C2×C22⋊C4 | — | 122 | 3 | no | — |
| 166 | C2×C42 | — | 61 | 1 | no | — |
| 167 | C4⋊C8 | — | 122 | 3 | no | — |
| 168 | C22⋊C8 | — | 122 | 3 | no | — |
| 169 | C4×C8 | — | 122 | 3 | no | — |
| 170 | #95 | — | 122 | 3 | no | — |
| 171 | C24 | — | 2 | 1 | no | — |
| 172 | C2×Q8 | — | 122 | 3 | no | — |
| 173 | C2×Q8 | C22⋊#6923 | 1 | 3 | yes | — |
| 174 | C2×D4 | — | 2 | 3 | no | — |
| 175 | C2×D4 | — | 122 | 3 | no | — |
| 176 | C2×D4 | C22⋊#6923 | 1 | 3 | yes | — |
| 177 | C22×C4 | — | 61 | 3 | no | — |
| 178 | C22×C4 | — | 122 | 3 | no | — |
| 179 | C22×C4 | — | 122 | 3 | no | — |
| 180 | C22×C4 | — | 244 | 1 | no | — |
| 181 | C2×C8 | — | 244 | 3 | no | — |
| 182 | C4⋊C4 | — | 61 | 3 | no | — |
| 183 | C4⋊C4 | — | 61 | 3 | no | — |
| 184 | C4⋊C4 | — | 122 | 3 | no | — |
| 185 | C4⋊C4 | — | 122 | 3 | no | — |
| 186 | C4⋊C4 | — | 122 | 3 | no | — |
| 187 | C4⋊C4 | — | 122 | 3 | no | — |
| 188 | C4⋊C4 | — | 122 | 3 | no | — |
| 189 | C4⋊C4 | — | 122 | 3 | no | — |
| 190 | C22⋊C4 | — | 2 | 3 | no | — |
| 191 | C22⋊C4 | — | 2 | 3 | no | — |
| 192 | C22⋊C4 | — | 2 | 3 | no | — |
| 193 | C22⋊C4 | — | 122 | 3 | no | — |
| 194 | C22⋊C4 | — | 122 | 3 | no | — |
| 195 | C22⋊C4 | — | 122 | 3 | no | — |
| 196 | C22⋊C4 | — | 122 | 3 | no | — |
| 197 | C22⋊C4 | — | 122 | 3 | no | — |
| 198 | C22⋊C4 | — | 122 | 3 | no | — |
| 199 | C42 | — | 2 | 1 | no | — |
| 200 | C42 | — | 61 | 3 | no | — |
| 201 | C42 | — | 122 | 3 | no | — |
| 202 | C42 | — | 122 | 3 | no | — |
| 203 | C42 | C22×#6923 | 1 | 1 | yes | characteristic |
| 204 | C23 | — | 2 | 3 | no | — |
| 205 | C23 | — | 8 | 3 | no | — |
| 206 | C23 | — | 61 | 1 | no | — |
| 207 | Q8 | — | 4 | 3 | no | — |
| 208 | Q8 | — | 488 | 3 | no | — |
| 209 | D4 | — | 4 | 3 | no | — |
| 210 | D4 | — | 8 | 3 | no | — |
| 211 | D4 | — | 488 | 3 | no | — |
| 212 | C2×C4 | — | 2 | 3 | no | — |
| 213 | C2×C4 | — | 61 | 3 | no | — |
| 214 | C2×C4 | — | 122 | 3 | no | — |
| 215 | C2×C4 | — | 122 | 3 | no | — |
| 216 | C2×C4 | — | 488 | 3 | no | — |
| 217 | C2×C4 | — | 488 | 3 | no | — |
| 218 | C2×C4 | — | 488 | 3 | no | — |
| 219 | C2×C4 | — | 244 | 3 | no | — |
| 220 | C2×C4 | C2×C22⋊#6923 | 1 | 3 | yes | — |
| 221 | C8 | — | 488 | 3 | no | — |
| 222 | C22 | — | 4 | 3 | no | — |
| 223 | C22 | — | 4 | 3 | no | — |
| 224 | C22 | — | 4 | 3 | no | — |
| 225 | C22 | — | 16 | 2 | no | — |
| 226 | C22 | — | 122 | 3 | no | — |
| 227 | C22 | #16840703 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 2 | 3 | no | — |
| 229 | C4 | — | 4 | 3 | no | — |
| 230 | C4 | — | 122 | 3 | no | — |
| 231 | C4 | — | 976 | 1 | no | — |
| 232 | C4 | — | 244 | 3 | no | — |
| 233 | C4 | — | 244 | 3 | no | — |
| 234 | C2 | — | 8 | 3 | no | — |
| 235 | C2 | — | 244 | 1 | no | — |
| 236 | C2 | #21461667 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21461260 | C2 | semidirect |
| 2. #21461131 | C2 | semidirect |
| 4. #16840781 | C22 | semidirect |
| 5. #16840707 | C22 | semidirect |
| 10. #16840512 | C4 | semidirect |
| 16. C61×H(3,𝔽4) | C4 | semidirect |
| 123. H(3,𝔽4) | #6923 | semidirect |
| 135. C61 | #8514 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21461260 | 2 |
| #21461131 | 2 |
| #21460513 | 2 |
| #8514 | 61 |