Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
654 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 178 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 | #23675 | C2 | 1 | 6 | yes | maximal |
| 2 | #11337 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11331 | C2 | 1 | 3 | yes | maximal |
| 4 | #11312 | C2 | 1 | 2 | yes | maximal |
| 5 | #11297 | C2 | 1 | 3 | yes | maximal |
| 6 | #2911 | C22 | 1 | 3 | yes | — |
| 7 | C2×#428 | C22 | 1 | 6 | yes | — |
| 8 | C22×#107 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2905 | C22 | 1 | 3 | yes | — |
| 10 | #2090 | — | 2 | 12 | no | — |
| 11 | #2089 | — | 2 | 12 | no | — |
| 12 | #1742 | C22 | 1 | 3 | yes | — |
| 13 | C2×#343 | C22 | 1 | 6 | yes | — |
| 14 | #1738 | C22 | 1 | 6 | yes | — |
| 15 | C2×#341 | C22 | 1 | 3 | yes | — |
| 16 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 17 | #1729 | C22 | 1 | 3 | yes | — |
| 18 | C22×M16 | C23 | 1 | 6 | yes | — |
| 19 | C22×M16 | C23 | 1 | 3 | yes | — |
| 20 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×#117 | C23 | 1 | 3 | yes | — |
| 22 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 23 | #428 | — | 2 | 12 | no | — |
| 24 | #428 | — | 2 | 6 | no | — |
| 25 | #428 | — | 2 | 6 | no | — |
| 26 | #428 | — | 4 | 6 | no | — |
| 27 | #427 | — | 4 | 3 | no | — |
| 28 | C2×#107 | — | 2 | 6 | no | — |
| 29 | C2×#107 | — | 4 | 3 | no | — |
| 30 | C2×#106 | — | 2 | 6 | no | — |
| 31 | #424 | — | 2 | 12 | no | — |
| 32 | #423 | — | 4 | 3 | no | — |
| 33 | #343 | — | 2 | 12 | no | — |
| 34 | #343 | D4 | 1 | 6 | yes | — |
| 35 | #343 | D4 | 1 | 6 | yes | — |
| 36 | #341 | — | 2 | 12 | no | — |
| 37 | #341 | — | 2 | 6 | no | — |
| 38 | #338 | Q8 | 1 | 4 | yes | — |
| 39 | #336 | — | 4 | 3 | no | — |
| 40 | C23×C4 | — | 2 | 2 | no | — |
| 41 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 42 | C2×M16 | — | 2 | 6 | no | — |
| 43 | C2×M16 | — | 2 | 3 | no | — |
| 44 | C2×M16 | — | 2 | 6 | no | — |
| 45 | C2×M16 | — | 2 | 6 | no | — |
| 46 | C2×M16 | — | 2 | 6 | no | — |
| 47 | C2×M16 | — | 2 | 6 | no | — |
| 48 | C2×M16 | — | 2 | 6 | no | — |
| 49 | C2×M16 | — | 2 | 6 | no | — |
| 50 | C2×M16 | C2×D4 | 1 | 12 | yes | — |
| 51 | C2×M16 | C2×Q8 | 1 | 6 | yes | — |
| 52 | C22×C8 | — | 2 | 6 | no | — |
| 53 | C22×C8 | — | 2 | 3 | no | — |
| 54 | C22×C8 | — | 2 | 3 | no | — |
| 55 | C22×C8 | — | 2 | 3 | no | — |
| 56 | C22×C8 | #41 | 1 | 2 | yes | — |
| 57 | #117 | — | 2 | 6 | no | — |
| 58 | #117 | — | 2 | 6 | no | — |
| 59 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 60 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 61 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 62 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 63 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 64 | C2×C42 | — | 2 | 3 | no | — |
| 65 | #107 | — | 4 | 3 | no | — |
| 66 | #107 | — | 4 | 3 | no | — |
| 67 | #107 | — | 4 | 12 | no | — |
| 68 | #107 | — | 4 | 4 | no | — |
| 69 | #106 | — | 4 | 6 | no | — |
| 70 | #106 | — | 4 | 12 | no | — |
| 71 | C4⋊C8 | — | 4 | 12 | no | — |
| 72 | C24 | 2−1+4 | 1 | 1 | yes | characteristic |
| 73 | C22×C4 | — | 2 | 3 | no | — |
| 74 | C22×C4 | — | 2 | 3 | no | — |
| 75 | C22×C4 | — | 4 | 6 | no | — |
| 76 | C22×C4 | — | 16 | 2 | no | — |
| 77 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 78 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 79 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 80 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 81 | C22×C4 | 2−1+4 | 1 | 3 | yes | — |
| 82 | C22×C4 | 2−1+4 | 1 | 3 | yes | — |
| 83 | M16 | — | 2 | 12 | no | — |
| 84 | M16 | — | 2 | 12 | no | — |
| 85 | M16 | — | 2 | 12 | no | — |
| 86 | M16 | — | 4 | 6 | no | — |
| 87 | M16 | — | 4 | 6 | no | — |
| 88 | M16 | — | 4 | 6 | no | — |
| 89 | C2×C8 | — | 2 | 6 | no | — |
| 90 | C2×C8 | — | 2 | 6 | no | — |
| 91 | C2×C8 | — | 2 | 12 | no | — |
| 92 | C2×C8 | — | 4 | 6 | no | — |
| 93 | C2×C8 | — | 4 | 3 | no | — |
| 94 | C2×C8 | — | 4 | 6 | no | — |
| 95 | C2×C8 | — | 4 | 1 | no | — |
| 96 | C2×C8 | — | 4 | 3 | no | — |
| 97 | C2×C8 | — | 4 | 3 | no | — |
| 98 | C4⋊C4 | — | 2 | 6 | no | — |
| 99 | C4⋊C4 | — | 2 | 6 | no | — |
| 100 | C4⋊C4 | — | 4 | 6 | no | — |
| 101 | C4⋊C4 | — | 4 | 2 | no | — |
| 102 | C22⋊C4 | — | 4 | 6 | no | — |
| 103 | C42 | — | 4 | 3 | no | — |
| 104 | C42 | — | 4 | 3 | no | — |
| 105 | C23 | — | 4 | 1 | no | — |
| 106 | C23 | — | 4 | 1 | no | — |
| 107 | C23 | #537 | 1 | 3 | yes | — |
| 108 | C23 | #557 | 1 | 1 | yes | characteristic |
| 109 | C23 | #564 | 1 | 3 | yes | — |
| 110 | C2×C4 | — | 4 | 3 | no | — |
| 111 | C2×C4 | — | 4 | 3 | no | — |
| 112 | C2×C4 | — | 8 | 3 | no | — |
| 113 | C2×C4 | — | 8 | 3 | no | — |
| 114 | C2×C4 | — | 8 | 2 | no | — |
| 115 | C2×C4 | — | 16 | 6 | no | — |
| 116 | C2×C4 | #536 | 1 | 6 | yes | — |
| 117 | C2×C4 | #539 | 1 | 3 | yes | — |
| 118 | C2×C4 | #544 | 1 | 3 | yes | — |
| 119 | C2×C4 | D42 | 1 | 3 | yes | — |
| 120 | C2×C4 | #547 | 1 | 3 | yes | — |
| 121 | C2×C4 | D4×Q8 | 1 | 3 | yes | — |
| 122 | C2×C4 | #560 | 1 | 1 | yes | characteristic |
| 123 | C2×C4 | #563 | 1 | 3 | yes | — |
| 124 | C2×C4 | #563 | 1 | 3 | yes | — |
| 125 | C8 | — | 4 | 2 | no | — |
| 126 | C8 | — | 4 | 6 | no | — |
| 127 | C8 | — | 4 | 12 | no | — |
| 128 | C22 | — | 2 | 3 | no | — |
| 129 | C22 | — | 2 | 3 | no | — |
| 130 | C22 | — | 4 | 1 | no | — |
| 131 | C22 | — | 4 | 3 | no | — |
| 132 | C22 | #2675 | 1 | 1 | yes | characteristic |
| 133 | C22 | #2726 | 1 | 3 | yes | — |
| 134 | C22 | #2822 | 1 | 3 | yes | — |
| 135 | C4 | — | 8 | 3 | no | — |
| 136 | C4 | — | 8 | 3 | no | — |
| 137 | C4 | — | 16 | 2 | no | — |
| 138 | C4 | #2677 | 1 | 3 | yes | — |
| 139 | C4 | #2680 | 1 | 3 | yes | — |
| 140 | C4 | #2834 | 1 | 1 | yes | characteristic |
| 141 | C4 | #2843 | 1 | 1 | yes | characteristic |
| 142 | C2 | — | 2 | 3 | no | — |
| 143 | C2 | — | 2 | 3 | no | — |
| 144 | C2 | #17114 | 1 | 1 | yes | characteristic |
| 145 | C2 | #30871 | 1 | 1 | yes | characteristic |
| 146 | C2 | #30872 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23675 | 2 |
| #11337 | 2 |
| #11331 | 2 |
| #11312 | 2 |
| #11297 | 2 |