Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
500 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #26507 | C2 | 1 | 4 | yes | maximal |
| 2 | #10601 | C2 | 1 | 4 | yes | maximal |
| 3 | #10576 | C2 | 1 | 2 | yes | maximal |
| 4 | #9825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #9594 | C2 | 1 | 2 | yes | maximal |
| 6 | #9066 | C2 | 1 | 2 | yes | maximal |
| 7 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2571 | C22 | 1 | 2 | yes | — |
| 9 | #2369 | C22 | 1 | 2 | yes | — |
| 10 | #2101 | — | 2 | 4 | no | — |
| 11 | #2086 | — | 2 | 4 | no | — |
| 12 | #1990 | C22 | 1 | 4 | yes | — |
| 13 | #1987 | C22 | 1 | 4 | yes | — |
| 14 | #1949 | — | 2 | 4 | no | — |
| 15 | #1948 | C22 | 1 | 4 | yes | — |
| 16 | #1939 | C22 | 1 | 4 | yes | — |
| 17 | #1920 | — | 2 | 4 | no | — |
| 18 | #1845 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×#107 | C22 | 1 | 2 | yes | — |
| 20 | C2×#335 | C22 | 1 | 2 | yes | — |
| 21 | C2×#333 | C22 | 1 | 2 | yes | — |
| 22 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1296 | C22 | 1 | 2 | yes | — |
| 24 | #1279 | C22 | 1 | 4 | yes | — |
| 25 | C22×Q16 | — | 4 | 1 | no | — |
| 26 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 27 | #494 | — | 4 | 1 | no | — |
| 28 | #494 | Q8 | 1 | 4 | yes | — |
| 29 | C2×#107 | — | 4 | 1 | no | — |
| 30 | C2×#107 | D4 | 1 | 2 | yes | — |
| 31 | C2×#107 | D4 | 1 | 2 | yes | — |
| 32 | C2×#106 | — | 2 | 4 | no | — |
| 33 | C2×C4⋊C8 | — | 2 | 4 | no | — |
| 34 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 35 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 36 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 37 | #400 | — | 4 | 2 | no | — |
| 38 | #391 | — | 4 | 2 | no | — |
| 39 | #389 | — | 2 | 4 | no | — |
| 40 | #389 | — | 2 | 4 | no | — |
| 41 | #384 | C23 | 1 | 2 | yes | — |
| 42 | #382 | — | 2 | 4 | no | — |
| 43 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 44 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 45 | #340 | — | 2 | 4 | no | — |
| 46 | #340 | — | 2 | 4 | no | — |
| 47 | #340 | — | 2 | 4 | no | — |
| 48 | #340 | — | 2 | 4 | no | — |
| 49 | #340 | — | 2 | 4 | no | — |
| 50 | #340 | D4 | 1 | 4 | yes | — |
| 51 | #336 | — | 2 | 4 | no | — |
| 52 | #335 | — | 4 | 2 | no | — |
| 53 | #333 | — | 4 | 2 | no | — |
| 54 | C82 | — | 4 | 1 | no | — |
| 55 | C22×Q8 | — | 2 | 2 | no | — |
| 56 | C2×Q16 | — | 4 | 4 | no | — |
| 57 | C2×Q16 | — | 8 | 2 | no | — |
| 58 | C2×Q16 | — | 8 | 2 | no | — |
| 59 | C22×C8 | — | 2 | 1 | no | — |
| 60 | C22×C8 | — | 2 | 4 | no | — |
| 61 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 62 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 63 | C4⋊Q8 | — | 2 | 2 | no | — |
| 64 | C4⋊Q8 | — | 2 | 4 | no | — |
| 65 | C4⋊Q8 | C2×Q8 | 1 | 4 | yes | — |
| 66 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 67 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 68 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 69 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 70 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 71 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 72 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 73 | C2×C42 | — | 2 | 4 | no | — |
| 74 | C2×C42 | — | 4 | 2 | no | — |
| 75 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 76 | #107 | — | 2 | 4 | no | — |
| 77 | #107 | — | 4 | 2 | no | — |
| 78 | #107 | — | 8 | 2 | no | — |
| 79 | #106 | — | 4 | 8 | no | — |
| 80 | C4⋊C8 | — | 2 | 4 | no | — |
| 81 | C4⋊C8 | — | 2 | 4 | no | — |
| 82 | C4⋊C8 | — | 4 | 8 | no | — |
| 83 | C4×C8 | — | 2 | 2 | no | — |
| 84 | C4×C8 | — | 2 | 1 | no | — |
| 85 | C4×C8 | — | 2 | 2 | no | — |
| 86 | C4×C8 | C2×Q8 | 1 | 2 | yes | — |
| 87 | #95 | — | 2 | 4 | no | — |
| 88 | #95 | — | 4 | 4 | no | — |
| 89 | #95 | — | 4 | 2 | no | — |
| 90 | #95 | — | 4 | 4 | no | — |
| 91 | C2×Q8 | — | 2 | 4 | no | — |
| 92 | C2×Q8 | — | 4 | 2 | no | — |
| 93 | C2×Q8 | — | 8 | 2 | no | — |
| 94 | C2×Q8 | — | 8 | 2 | no | — |
| 95 | C22×C4 | — | 2 | 4 | no | — |
| 96 | C22×C4 | — | 2 | 4 | no | — |
| 97 | C22×C4 | — | 4 | 2 | no | — |
| 98 | C22×C4 | — | 4 | 2 | no | — |
| 99 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 100 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 101 | Q16 | — | 8 | 8 | no | — |
| 102 | C2×C8 | — | 2 | 2 | no | — |
| 103 | C2×C8 | — | 2 | 2 | no | — |
| 104 | C2×C8 | — | 2 | 2 | no | — |
| 105 | C2×C8 | — | 2 | 2 | no | — |
| 106 | C2×C8 | — | 2 | 2 | no | — |
| 107 | C2×C8 | — | 2 | 2 | no | — |
| 108 | C2×C8 | — | 2 | 2 | no | — |
| 109 | C2×C8 | — | 2 | 2 | no | — |
| 110 | C2×C8 | — | 4 | 4 | no | — |
| 111 | C2×C8 | — | 4 | 4 | no | — |
| 112 | C2×C8 | — | 4 | 4 | no | — |
| 113 | C2×C8 | #135 | 1 | 4 | yes | — |
| 114 | C4⋊C4 | — | 2 | 8 | no | — |
| 115 | C4⋊C4 | — | 2 | 4 | no | — |
| 116 | C4⋊C4 | — | 2 | 4 | no | — |
| 117 | C4⋊C4 | — | 2 | 4 | no | — |
| 118 | C4⋊C4 | — | 2 | 4 | no | — |
| 119 | C4⋊C4 | — | 4 | 4 | no | — |
| 120 | C4⋊C4 | — | 4 | 4 | no | — |
| 121 | C4⋊C4 | — | 4 | 4 | no | — |
| 122 | C4⋊C4 | — | 8 | 2 | no | — |
| 123 | C4⋊C4 | — | 16 | 2 | no | — |
| 124 | C4⋊C4 | — | 16 | 2 | no | — |
| 125 | C42 | — | 4 | 4 | no | — |
| 126 | C42 | — | 4 | 4 | no | — |
| 127 | C42 | — | 16 | 2 | no | — |
| 128 | C42 | C22×Q8 | 1 | 1 | yes | characteristic |
| 129 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 130 | C42 | 2−1+4 | 1 | 2 | yes | — |
| 131 | C23 | D42 | 1 | 1 | yes | characteristic |
| 132 | Q8 | — | 8 | 4 | no | — |
| 133 | Q8 | — | 16 | 2 | no | — |
| 134 | C2×C4 | — | 2 | 1 | no | — |
| 135 | C2×C4 | — | 2 | 1 | no | — |
| 136 | C2×C4 | — | 4 | 4 | no | — |
| 137 | C2×C4 | — | 4 | 4 | no | — |
| 138 | C2×C4 | — | 4 | 4 | no | — |
| 139 | C2×C4 | — | 4 | 4 | no | — |
| 140 | C2×C4 | — | 4 | 4 | no | — |
| 141 | C2×C4 | — | 4 | 4 | no | — |
| 142 | C2×C4 | — | 4 | 4 | no | — |
| 143 | C2×C4 | — | 8 | 2 | no | — |
| 144 | C2×C4 | — | 8 | 2 | no | — |
| 145 | C2×C4 | — | 8 | 4 | no | — |
| 146 | C2×C4 | — | 8 | 2 | no | — |
| 147 | C2×C4 | #537 | 1 | 2 | yes | — |
| 148 | C2×C4 | #543 | 1 | 1 | yes | characteristic |
| 149 | C2×C4 | D4×Q8 | 1 | 2 | yes | — |
| 150 | C2×C4 | #563 | 1 | 1 | yes | characteristic |
| 151 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 152 | C2×C4 | #575 | 1 | 2 | yes | — |
| 153 | C2×C4 | #578 | 1 | 2 | yes | — |
| 154 | C2×C4 | #578 | 1 | 2 | yes | — |
| 155 | C8 | — | 2 | 4 | no | — |
| 156 | C8 | — | 2 | 4 | no | — |
| 157 | C8 | — | 4 | 2 | no | — |
| 158 | C8 | — | 8 | 4 | no | — |
| 159 | C22 | #2683 | 1 | 1 | yes | characteristic |
| 160 | C22 | #3277 | 1 | 2 | yes | — |
| 161 | C22 | #3283 | 1 | 2 | yes | — |
| 162 | C22 | #3291 | 1 | 1 | yes | characteristic |
| 163 | C22 | #3292 | 1 | 1 | yes | characteristic |
| 164 | C4 | — | 2 | 1 | no | — |
| 165 | C4 | — | 2 | 1 | no | — |
| 166 | C4 | — | 8 | 4 | no | — |
| 167 | C4 | — | 8 | 4 | no | — |
| 168 | C4 | — | 8 | 4 | no | — |
| 169 | C4 | — | 16 | 2 | no | — |
| 170 | C4 | #3250 | 1 | 2 | yes | — |
| 171 | C4 | #3261 | 1 | 2 | yes | — |
| 172 | C4 | #3383 | 1 | 2 | yes | — |
| 173 | C4 | #3390 | 1 | 2 | yes | — |
| 174 | C2 | #19958 | 1 | 2 | yes | — |
| 175 | C2 | #20019 | 1 | 1 | yes | characteristic |
| 176 | C2 | #20319 | 1 | 1 | yes | characteristic |
| 177 | C2 | #31236 | 1 | 2 | yes | — |
| 178 | C2 | #31254 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26507 | 2 |
| #10601 | 2 |
| #10576 | 2 |
| #9825 | 2 |
| #9594 | 2 |
| #9066 | 2 |