Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
249 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 56 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 | #32695 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13541 | C2 | 1 | 2 | yes | maximal |
| 3 | #10662 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7133 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7127 | C2 | 1 | 2 | yes | maximal |
| 6 | #2567 | — | 2 | 1 | no | — |
| 7 | #2086 | — | 2 | 1 | no | — |
| 8 | #2057 | — | 2 | 2 | no | — |
| 9 | #1985 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1944 | C22 | 1 | 2 | yes | — |
| 11 | #1920 | — | 2 | 2 | no | — |
| 12 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1571 | — | 2 | 1 | no | — |
| 14 | #1487 | — | 2 | 1 | no | — |
| 15 | #1387 | — | 2 | 2 | no | — |
| 16 | #1383 | — | 2 | 2 | no | — |
| 17 | #1377 | C22 | 1 | 2 | yes | — |
| 18 | #1275 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 20 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×M32 | — | 2 | 2 | no | — |
| 22 | #431 | — | 2 | 1 | no | — |
| 23 | C2×#108 | — | 2 | 2 | no | — |
| 24 | C2×#107 | — | 2 | 1 | no | — |
| 25 | C2×#107 | D4 | 1 | 2 | yes | — |
| 26 | C2×#106 | D4 | 1 | 2 | yes | — |
| 27 | #423 | — | 2 | 1 | no | — |
| 28 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 29 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 30 | #395 | — | 4 | 1 | no | — |
| 31 | #395 | — | 4 | 1 | no | — |
| 32 | #391 | — | 4 | 1 | no | — |
| 33 | #389 | — | 2 | 2 | no | — |
| 34 | #389 | — | 2 | 2 | no | — |
| 35 | #384 | — | 4 | 1 | no | — |
| 36 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 37 | #340 | — | 2 | 2 | no | — |
| 38 | #340 | — | 2 | 1 | no | — |
| 39 | #340 | — | 2 | 2 | no | — |
| 40 | #336 | — | 2 | 2 | no | — |
| 41 | #335 | — | 4 | 2 | no | — |
| 42 | Q8⋊C8 | — | 4 | 2 | no | — |
| 43 | C22×Q8 | — | 4 | 1 | no | — |
| 44 | C22×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 45 | C2×M16 | — | 4 | 1 | no | — |
| 46 | C22×C8 | — | 2 | 1 | no | — |
| 47 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 48 | C4⋊Q8 | — | 8 | 1 | no | — |
| 49 | C4⋊Q8 | — | 8 | 1 | no | — |
| 50 | C4×Q8 | — | 4 | 1 | no | — |
| 51 | C4×Q8 | — | 4 | 1 | no | — |
| 52 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 53 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 54 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 55 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 56 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 57 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 58 | C2×C4⋊C4 | SD16 | 1 | 1 | yes | characteristic |
| 59 | C2×C42 | — | 2 | 1 | no | — |
| 60 | C2×C42 | — | 4 | 1 | no | — |
| 61 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | M32 | — | 4 | 2 | no | — |
| 63 | M32 | — | 4 | 2 | no | — |
| 64 | C2×C16 | — | 4 | 2 | no | — |
| 65 | #108 | — | 4 | 4 | no | — |
| 66 | #107 | — | 4 | 2 | no | — |
| 67 | #107 | — | 8 | 1 | no | — |
| 68 | #106 | — | 2 | 2 | no | — |
| 69 | #106 | Q16 | 1 | 4 | yes | — |
| 70 | C4⋊C8 | — | 4 | 1 | no | — |
| 71 | C4⋊C8 | — | 4 | 1 | no | — |
| 72 | C4⋊C8 | — | 4 | 1 | no | — |
| 73 | C4⋊C8 | — | 4 | 1 | no | — |
| 74 | C22⋊C8 | — | 4 | 1 | no | — |
| 75 | #97 | — | 4 | 1 | no | — |
| 76 | C4×C8 | — | 2 | 2 | no | — |
| 77 | #95 | — | 4 | 1 | no | — |
| 78 | #95 | — | 4 | 2 | no | — |
| 79 | #95 | — | 4 | 2 | no | — |
| 80 | C2×Q8 | — | 4 | 2 | no | — |
| 81 | C2×Q8 | — | 4 | 1 | no | — |
| 82 | C2×Q8 | — | 8 | 2 | no | — |
| 83 | C2×Q8 | — | 16 | 2 | no | — |
| 84 | C22×C4 | — | 2 | 1 | no | — |
| 85 | C22×C4 | — | 2 | 1 | no | — |
| 86 | C22×C4 | — | 4 | 2 | no | — |
| 87 | C22×C4 | — | 4 | 2 | no | — |
| 88 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 89 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 90 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 91 | M16 | — | 8 | 2 | no | — |
| 92 | C2×C8 | — | 2 | 2 | no | — |
| 93 | C2×C8 | — | 2 | 2 | no | — |
| 94 | C2×C8 | — | 4 | 2 | no | — |
| 95 | C2×C8 | — | 4 | 1 | no | — |
| 96 | C2×C8 | — | 4 | 2 | no | — |
| 97 | C2×C8 | — | 8 | 1 | no | — |
| 98 | C2×C8 | C2×Q16 | 1 | 2 | yes | — |
| 99 | C2×C8 | #137 | 1 | 2 | yes | — |
| 100 | C4⋊C4 | — | 2 | 4 | no | — |
| 101 | C4⋊C4 | — | 4 | 1 | no | — |
| 102 | C4⋊C4 | — | 4 | 2 | no | — |
| 103 | C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C4⋊C4 | — | 4 | 2 | no | — |
| 105 | C4⋊C4 | — | 8 | 1 | no | — |
| 106 | C4⋊C4 | — | 8 | 2 | no | — |
| 107 | C4⋊C4 | — | 16 | 2 | no | — |
| 108 | C4⋊C4 | — | 16 | 2 | no | — |
| 109 | C42 | — | 2 | 1 | no | — |
| 110 | C42 | — | 2 | 1 | no | — |
| 111 | C42 | — | 4 | 1 | no | — |
| 112 | C42 | — | 4 | 1 | no | — |
| 113 | C42 | — | 8 | 2 | no | — |
| 114 | C16 | — | 8 | 2 | no | — |
| 115 | C23 | #450 | 1 | 1 | yes | characteristic |
| 116 | Q8 | — | 4 | 2 | no | — |
| 117 | Q8 | — | 8 | 1 | no | — |
| 118 | Q8 | — | 16 | 4 | no | — |
| 119 | C2×C4 | — | 2 | 2 | no | — |
| 120 | C2×C4 | — | 2 | 1 | no | — |
| 121 | C2×C4 | — | 2 | 1 | no | — |
| 122 | C2×C4 | — | 4 | 1 | no | — |
| 123 | C2×C4 | — | 4 | 1 | no | — |
| 124 | C2×C4 | — | 4 | 1 | no | — |
| 125 | C2×C4 | — | 4 | 1 | no | — |
| 126 | C2×C4 | — | 4 | 2 | no | — |
| 127 | C2×C4 | — | 4 | 1 | no | — |
| 128 | C2×C4 | — | 4 | 4 | no | — |
| 129 | C2×C4 | — | 8 | 2 | no | — |
| 130 | C2×C4 | — | 8 | 2 | no | — |
| 131 | C2×C4 | — | 8 | 2 | no | — |
| 132 | C2×C4 | — | 8 | 2 | no | — |
| 133 | C2×C4 | — | 8 | 2 | no | — |
| 134 | C2×C4 | #451 | 1 | 2 | yes | — |
| 135 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 136 | C2×C4 | #454 | 1 | 2 | yes | — |
| 137 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 138 | C8 | — | 2 | 2 | no | — |
| 139 | C8 | — | 2 | 2 | no | — |
| 140 | C8 | — | 8 | 2 | no | — |
| 141 | C8 | — | 8 | 1 | no | — |
| 142 | C8 | — | 16 | 1 | no | — |
| 143 | C22 | — | 2 | 1 | no | — |
| 144 | C22 | — | 2 | 1 | no | — |
| 145 | C22 | #1613 | 1 | 1 | yes | characteristic |
| 146 | C22 | #1637 | 1 | 2 | yes | — |
| 147 | C4 | — | 4 | 2 | no | — |
| 148 | C4 | — | 4 | 1 | no | — |
| 149 | C4 | — | 4 | 1 | no | — |
| 150 | C4 | — | 8 | 4 | no | — |
| 151 | C4 | — | 8 | 1 | no | — |
| 152 | C4 | — | 16 | 2 | no | — |
| 153 | C4 | #1617 | 1 | 2 | yes | — |
| 154 | C4 | #1633 | 1 | 1 | yes | characteristic |
| 155 | C4 | #1657 | 1 | 1 | yes | characteristic |
| 156 | C2 | — | 2 | 2 | no | — |
| 157 | C2 | #7829 | 1 | 1 | yes | characteristic |
| 158 | C2 | #12299 | 1 | 1 | yes | characteristic |
| 159 | C2 | #12301 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #32695 | 2 |
| #13541 | 2 |
| #10662 | 2 |
| #7133 | 2 |
| #7127 | 2 |