Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
690 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 186 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 | #33310 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33290 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #30279 | C2 | 1 | 2 | yes | maximal |
| 4 | #30220 | C2 | 1 | 2 | yes | maximal |
| 5 | #10314 | C2 | 1 | 8 | yes | maximal |
| 6 | #8133 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×#500 | C22 | 1 | 4 | yes | — |
| 8 | C2×#498 | C22 | 1 | 4 | yes | — |
| 9 | C2×#487 | C22 | 1 | 4 | yes | — |
| 10 | #2868 | C22 | 1 | 2 | yes | — |
| 11 | #2862 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2859 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2095 | — | 2 | 8 | no | — |
| 14 | #2043 | — | 2 | 8 | no | — |
| 15 | #1991 | — | 2 | 2 | no | — |
| 16 | #1990 | C22 | 1 | 4 | yes | — |
| 17 | #1956 | C22 | 1 | 8 | yes | — |
| 18 | #1844 | — | 2 | 2 | no | — |
| 19 | #1844 | — | 2 | 2 | no | — |
| 20 | #1835 | — | 2 | 2 | no | — |
| 21 | #1744 | C22 | 1 | 2 | yes | — |
| 22 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1273 | C22 | 1 | 4 | yes | — |
| 24 | C2×C4⋊Q8 | — | 2 | 4 | no | — |
| 25 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 26 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 27 | C2×#125 | — | 2 | 1 | no | — |
| 28 | #500 | — | 4 | 8 | no | — |
| 29 | #498 | — | 4 | 4 | no | — |
| 30 | #498 | — | 4 | 4 | no | — |
| 31 | #487 | — | 4 | 8 | no | — |
| 32 | C2×#107 | — | 2 | 8 | no | — |
| 33 | C2×#106 | — | 2 | 8 | no | — |
| 34 | C2×#103 | — | 2 | 8 | no | — |
| 35 | C2×#103 | — | 2 | 16 | no | — |
| 36 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 37 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 38 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 39 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 40 | #391 | — | 2 | 4 | no | — |
| 41 | #389 | — | 2 | 4 | no | — |
| 42 | #384 | — | 2 | 4 | no | — |
| 43 | #384 | — | 2 | 2 | no | — |
| 44 | #384 | — | 2 | 2 | no | — |
| 45 | #384 | C23 | 1 | 2 | yes | — |
| 46 | #382 | — | 2 | 1 | no | — |
| 47 | #382 | — | 2 | 1 | no | — |
| 48 | #382 | — | 2 | 1 | no | — |
| 49 | C43 | C23 | 1 | 1 | yes | characteristic |
| 50 | #340 | — | 2 | 8 | no | — |
| 51 | #336 | D4 | 1 | 8 | yes | — |
| 52 | C22×Q8 | — | 2 | 4 | no | — |
| 53 | C22×Q8 | — | 2 | 2 | no | — |
| 54 | C22×C8 | C2×D4 | 1 | 8 | yes | — |
| 55 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 56 | C4⋊Q8 | — | 2 | 8 | no | — |
| 57 | C4⋊Q8 | — | 2 | 8 | no | — |
| 58 | C4⋊Q8 | — | 2 | 8 | no | — |
| 59 | C4⋊Q8 | — | 8 | 8 | no | — |
| 60 | #125 | — | 4 | 4 | no | — |
| 61 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 62 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 63 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 64 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 65 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 66 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 67 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 68 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 69 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 70 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 71 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 74 | #107 | — | 4 | 16 | no | — |
| 75 | #106 | — | 4 | 16 | no | — |
| 76 | #103 | — | 4 | 16 | no | — |
| 77 | #103 | — | 8 | 16 | no | — |
| 78 | C4×C8 | — | 2 | 4 | no | — |
| 79 | C4×C8 | — | 2 | 8 | no | — |
| 80 | C4×C8 | — | 2 | 4 | no | — |
| 81 | C4×C8 | Q16 | 1 | 8 | yes | — |
| 82 | #95 | — | 4 | 2 | no | — |
| 83 | #95 | — | 4 | 2 | no | — |
| 84 | #95 | — | 4 | 2 | no | — |
| 85 | C2×Q8 | — | 4 | 4 | no | — |
| 86 | C2×Q8 | — | 4 | 8 | no | — |
| 87 | C2×Q8 | — | 4 | 16 | no | — |
| 88 | C2×Q8 | — | 4 | 8 | no | — |
| 89 | C22×C4 | — | 4 | 2 | no | — |
| 90 | C22×C4 | — | 4 | 2 | no | — |
| 91 | C22×C4 | — | 4 | 4 | no | — |
| 92 | C22×C4 | #127 | 1 | 2 | yes | — |
| 93 | C22×C4 | #127 | 1 | 2 | yes | — |
| 94 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 96 | C2×C8 | — | 2 | 4 | no | — |
| 97 | C2×C8 | — | 2 | 4 | no | — |
| 98 | C2×C8 | — | 2 | 8 | no | — |
| 99 | C2×C8 | — | 2 | 16 | no | — |
| 100 | C2×C8 | C2×Q16 | 1 | 8 | yes | — |
| 101 | C4⋊C4 | — | 4 | 4 | no | — |
| 102 | C4⋊C4 | — | 4 | 8 | no | — |
| 103 | C4⋊C4 | — | 4 | 4 | no | — |
| 104 | C4⋊C4 | — | 4 | 4 | no | — |
| 105 | C4⋊C4 | — | 4 | 4 | no | — |
| 106 | C4⋊C4 | — | 4 | 4 | no | — |
| 107 | C4⋊C4 | — | 4 | 8 | no | — |
| 108 | C4⋊C4 | — | 8 | 8 | no | — |
| 109 | C4⋊C4 | — | 8 | 8 | no | — |
| 110 | C4⋊C4 | — | 8 | 4 | no | — |
| 111 | C42 | — | 2 | 8 | no | — |
| 112 | C42 | C2×Q16 | 1 | 4 | yes | — |
| 113 | C42 | #137 | 1 | 4 | yes | — |
| 114 | C42 | H(5,𝔽2) | 1 | 4 | yes | — |
| 115 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 116 | Q8 | — | 8 | 8 | no | — |
| 117 | Q8 | — | 8 | 16 | no | — |
| 118 | C2×C4 | — | 2 | 4 | no | — |
| 119 | C2×C4 | — | 2 | 4 | no | — |
| 120 | C2×C4 | — | 8 | 2 | no | — |
| 121 | C2×C4 | — | 8 | 4 | no | — |
| 122 | C2×C4 | — | 8 | 8 | no | — |
| 123 | C2×C4 | — | 8 | 4 | no | — |
| 124 | C2×C4 | — | 8 | 4 | no | — |
| 125 | C2×C4 | — | 8 | 2 | no | — |
| 126 | C2×C4 | #494 | 1 | 4 | yes | — |
| 127 | C2×C4 | #497 | 1 | 4 | yes | — |
| 128 | C2×C4 | #535 | 1 | 2 | yes | — |
| 129 | C2×C4 | #535 | 1 | 4 | yes | — |
| 130 | C2×C4 | C22×Q16 | 1 | 2 | yes | — |
| 131 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 132 | C2×C4 | #577 | 1 | 8 | yes | — |
| 133 | C8 | — | 2 | 8 | no | — |
| 134 | C8 | — | 4 | 8 | no | — |
| 135 | C22 | #2657 | 1 | 1 | yes | characteristic |
| 136 | C22 | C2×#494 | 1 | 1 | yes | characteristic |
| 137 | C22 | C2×#497 | 1 | 1 | yes | characteristic |
| 138 | C22 | #3154 | 1 | 4 | yes | — |
| 139 | C4 | — | 2 | 4 | no | — |
| 140 | C4 | — | 2 | 4 | no | — |
| 141 | C4 | — | 16 | 4 | no | — |
| 142 | C4 | — | 16 | 2 | no | — |
| 143 | C4 | — | 16 | 2 | no | — |
| 144 | C4 | #3209 | 1 | 4 | yes | — |
| 145 | C4 | #3221 | 1 | 8 | yes | — |
| 146 | C2 | #19663 | 1 | 2 | yes | — |
| 147 | C2 | #19702 | 1 | 2 | yes | — |
| 148 | C2 | #30262 | 1 | 1 | yes | characteristic |
| 149 | C2 | #30273 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ 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 |
|---|---|
| #33310 | 2 |
| #33290 | 2 |
| #30279 | 2 |
| #30220 | 2 |
| #10314 | 2 |
| #8133 | 2 |