Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
328 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 66 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 | #33803 | C2 | 1 | 2 | yes | maximal |
| 2 | #19618 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11865 | C2 | 1 | 2 | yes | maximal |
| 4 | #9859 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7196 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #3170 | — | 2 | 1 | no | — |
| 7 | #3170 | C22 | 1 | 2 | yes | — |
| 8 | #3160 | — | 2 | 1 | no | — |
| 9 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2255 | — | 2 | 2 | no | — |
| 11 | #2245 | — | 2 | 2 | no | — |
| 12 | #2237 | — | 2 | 4 | no | — |
| 13 | #2227 | — | 2 | 2 | no | — |
| 14 | #2225 | — | 2 | 2 | no | — |
| 15 | #2163 | C22 | 1 | 2 | yes | — |
| 16 | #1837 | C22 | 1 | 1 | yes | characteristic |
| 17 | #1723 | — | 2 | 2 | no | — |
| 18 | #1723 | — | 2 | 2 | no | — |
| 19 | #1493 | — | 2 | 1 | no | — |
| 20 | #1493 | — | 2 | 1 | no | — |
| 21 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1350 | — | 2 | 4 | no | — |
| 23 | #533 | — | 2 | 2 | no | — |
| 24 | #533 | — | 2 | 1 | no | — |
| 25 | #500 | — | 2 | 1 | no | — |
| 26 | #500 | — | 2 | 1 | no | — |
| 27 | #500 | — | 2 | 1 | no | — |
| 28 | #500 | D4 | 1 | 1 | yes | characteristic |
| 29 | #500 | D4 | 1 | 1 | yes | characteristic |
| 30 | #500 | D4 | 1 | 2 | yes | — |
| 31 | #499 | — | 2 | 2 | no | — |
| 32 | #499 | — | 4 | 2 | no | — |
| 33 | #498 | — | 2 | 2 | no | — |
| 34 | #498 | — | 2 | 1 | no | — |
| 35 | #494 | — | 2 | 1 | no | — |
| 36 | #494 | D4 | 1 | 2 | yes | — |
| 37 | #487 | — | 2 | 2 | no | — |
| 38 | #487 | — | 2 | 2 | no | — |
| 39 | #467 | — | 2 | 2 | no | — |
| 40 | #467 | — | 2 | 2 | no | — |
| 41 | #467 | — | 4 | 2 | no | — |
| 42 | C4×Q16 | — | 2 | 2 | no | — |
| 43 | C4×Q16 | — | 2 | 2 | no | — |
| 44 | C4×Q16 | — | 4 | 2 | no | — |
| 45 | #427 | — | 2 | 2 | no | — |
| 46 | #427 | — | 4 | 2 | no | — |
| 47 | #423 | — | 2 | 1 | no | — |
| 48 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 51 | #367 | — | 2 | 4 | no | — |
| 52 | #366 | — | 2 | 4 | no | — |
| 53 | C4⋊C16 | — | 2 | 2 | no | — |
| 54 | #358 | — | 2 | 4 | no | — |
| 55 | #358 | — | 2 | 4 | no | — |
| 56 | C22⋊C16 | D4 | 1 | 4 | yes | — |
| 57 | #346 | — | 2 | 2 | no | — |
| 58 | #341 | — | 2 | 2 | no | — |
| 59 | Q8⋊C8 | — | 4 | 2 | no | — |
| 60 | Q8⋊C8 | — | 4 | 2 | no | — |
| 61 | #322 | — | 2 | 2 | no | — |
| 62 | C2×Q16 | — | 2 | 2 | no | — |
| 63 | C2×Q16 | — | 2 | 2 | no | — |
| 64 | C2×Q16 | — | 4 | 2 | no | — |
| 65 | C2×M16 | — | 4 | 1 | no | — |
| 66 | C22×C8 | — | 2 | 1 | no | — |
| 67 | C22×C8 | — | 2 | 1 | no | — |
| 68 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 69 | C4⋊Q8 | — | 2 | 2 | no | — |
| 70 | C4⋊Q8 | — | 2 | 2 | no | — |
| 71 | C4⋊Q8 | — | 2 | 1 | no | — |
| 72 | C4⋊Q8 | — | 2 | 1 | no | — |
| 73 | #125 | — | 4 | 1 | no | — |
| 74 | #125 | — | 4 | 2 | no | — |
| 75 | C22⋊Q8 | — | 4 | 2 | no | — |
| 76 | C22⋊Q8 | — | 4 | 1 | no | — |
| 77 | C22⋊Q8 | — | 4 | 1 | no | — |
| 78 | C22⋊Q8 | — | 4 | 2 | no | — |
| 79 | C4×Q8 | — | 4 | 2 | no | — |
| 80 | C4×Q8 | — | 4 | 1 | no | — |
| 81 | C4×Q8 | — | 4 | 1 | no | — |
| 82 | C4×Q8 | — | 4 | 2 | no | — |
| 83 | #117 | — | 4 | 1 | no | — |
| 84 | #117 | — | 4 | 2 | no | — |
| 85 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C16 | — | 2 | 4 | no | — |
| 87 | #107 | — | 2 | 2 | no | — |
| 88 | #107 | — | 2 | 2 | no | — |
| 89 | #107 | — | 2 | 2 | no | — |
| 90 | #107 | — | 2 | 2 | no | — |
| 91 | #107 | — | 4 | 2 | no | — |
| 92 | #107 | — | 4 | 2 | no | — |
| 93 | #107 | — | 4 | 2 | no | — |
| 94 | #106 | — | 4 | 4 | no | — |
| 95 | #106 | — | 4 | 2 | no | — |
| 96 | C4⋊C8 | — | 4 | 1 | no | — |
| 97 | C4⋊C8 | — | 4 | 1 | no | — |
| 98 | #103 | — | 4 | 4 | no | — |
| 99 | #103 | — | 4 | 2 | no | — |
| 100 | #103 | — | 4 | 2 | no | — |
| 101 | C4×C8 | — | 2 | 1 | no | — |
| 102 | C4×C8 | — | 2 | 1 | no | — |
| 103 | C4×C8 | D8 | 1 | 2 | yes | — |
| 104 | C4×C8 | D8 | 1 | 2 | yes | — |
| 105 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 106 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 108 | C2×Q8 | — | 4 | 2 | no | — |
| 109 | C2×Q8 | — | 4 | 2 | no | — |
| 110 | C2×Q8 | — | 4 | 1 | no | — |
| 111 | C2×Q8 | — | 4 | 1 | no | — |
| 112 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 114 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 115 | Q16 | — | 4 | 4 | no | — |
| 116 | Q16 | — | 8 | 2 | no | — |
| 117 | Q16 | — | 8 | 4 | no | — |
| 118 | M16 | — | 8 | 2 | no | — |
| 119 | C2×C8 | — | 2 | 4 | no | — |
| 120 | C2×C8 | — | 2 | 2 | no | — |
| 121 | C2×C8 | — | 2 | 2 | no | — |
| 122 | C2×C8 | — | 4 | 2 | no | — |
| 123 | C2×C8 | — | 8 | 1 | no | — |
| 124 | C2×C8 | — | 8 | 1 | no | — |
| 125 | C2×C8 | #120 | 1 | 2 | yes | — |
| 126 | C2×C8 | #121 | 1 | 2 | yes | — |
| 127 | C4⋊C4 | — | 4 | 2 | no | — |
| 128 | C4⋊C4 | — | 4 | 1 | no | — |
| 129 | C4⋊C4 | — | 4 | 1 | no | — |
| 130 | C4⋊C4 | — | 4 | 2 | no | — |
| 131 | C4⋊C4 | — | 4 | 2 | no | — |
| 132 | C4⋊C4 | — | 4 | 2 | no | — |
| 133 | C4⋊C4 | — | 4 | 1 | no | — |
| 134 | C4⋊C4 | — | 4 | 1 | no | — |
| 135 | C4⋊C4 | — | 8 | 1 | no | — |
| 136 | C4⋊C4 | — | 8 | 1 | no | — |
| 137 | C4⋊C4 | — | 8 | 2 | no | — |
| 138 | C4⋊C4 | — | 8 | 2 | no | — |
| 139 | C22⋊C4 | — | 8 | 1 | no | — |
| 140 | C22⋊C4 | — | 8 | 2 | no | — |
| 141 | C42 | — | 8 | 2 | no | — |
| 142 | C42 | — | 8 | 1 | no | — |
| 143 | C42 | #120 | 1 | 1 | yes | characteristic |
| 144 | C42 | #121 | 1 | 1 | yes | characteristic |
| 145 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 146 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 147 | C16 | — | 4 | 4 | no | — |
| 148 | C23 | #496 | 1 | 1 | yes | characteristic |
| 149 | Q8 | — | 8 | 4 | no | — |
| 150 | Q8 | — | 8 | 2 | no | — |
| 151 | Q8 | — | 8 | 2 | no | — |
| 152 | Q8 | — | 8 | 2 | no | — |
| 153 | Q8 | — | 8 | 2 | no | — |
| 154 | C2×C4 | — | 2 | 2 | no | — |
| 155 | C2×C4 | — | 2 | 2 | no | — |
| 156 | C2×C4 | — | 4 | 1 | no | — |
| 157 | C2×C4 | — | 8 | 1 | no | — |
| 158 | C2×C4 | — | 8 | 2 | no | — |
| 159 | C2×C4 | — | 8 | 1 | no | — |
| 160 | C2×C4 | — | 8 | 2 | no | — |
| 161 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 162 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 163 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 164 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 165 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 166 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 167 | C8 | — | 2 | 4 | no | — |
| 168 | C8 | — | 2 | 2 | no | — |
| 169 | C8 | — | 2 | 2 | no | — |
| 170 | C8 | — | 4 | 2 | no | — |
| 171 | C8 | — | 16 | 1 | no | — |
| 172 | C22 | — | 2 | 1 | no | — |
| 173 | C22 | — | 2 | 1 | no | — |
| 174 | C22 | — | 2 | 1 | no | — |
| 175 | C22 | #2011 | 1 | 1 | yes | characteristic |
| 176 | C4 | — | 2 | 1 | no | — |
| 177 | C4 | — | 2 | 1 | no | — |
| 178 | C4 | — | 2 | 1 | no | — |
| 179 | C4 | — | 2 | 1 | no | — |
| 180 | C4 | — | 16 | 1 | no | — |
| 181 | C4 | — | 16 | 2 | no | — |
| 182 | C4 | — | 16 | 1 | no | — |
| 183 | C4 | — | 16 | 2 | no | — |
| 184 | C4 | #2009 | 1 | 2 | yes | — |
| 185 | C4 | #2214 | 1 | 2 | yes | — |
| 186 | C2 | — | 4 | 1 | no | — |
| 187 | C2 | #10225 | 1 | 1 | yes | characteristic |
| 188 | C2 | #13333 | 1 | 1 | yes | characteristic |
| 189 | C2 | #13340 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33803 | 2 |
| #19618 | 2 |
| #11865 | 2 |
| #9859 | 2 |
| #7196 | 2 |