Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
696 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #32933 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #13493 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #13380 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #13304 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #12680 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#131 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#466 | C22 | 1 | 6 | yes | — |
| 9 | #2615 | — | 2 | 2 | no | — |
| 10 | C2×#366 | C22 | 1 | 3 | yes | — |
| 11 | C2×#357 | C22 | 1 | 6 | yes | — |
| 12 | C2×C22⋊C16 | C22 | 1 | 3 | yes | — |
| 13 | C2×C22⋊C16 | C22 | 1 | 3 | yes | — |
| 14 | #1895 | — | 2 | 6 | no | — |
| 15 | #1808 | C22 | 1 | 6 | yes | — |
| 16 | #1380 | C22 | 1 | 3 | yes | — |
| 17 | #1380 | C22 | 1 | 3 | yes | — |
| 18 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 19 | C22×D8 | C23 | 1 | 2 | yes | — |
| 20 | C2×#131 | — | 4 | 3 | no | — |
| 21 | C2×#131 | — | 4 | 3 | no | — |
| 22 | C22×M16 | D4 | 1 | 3 | yes | — |
| 23 | C23×C8 | C23 | 1 | 3 | yes | — |
| 24 | C2×#121 | — | 2 | 6 | no | — |
| 25 | C2×D16 | — | 4 | 3 | no | — |
| 26 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 27 | C22×C16 | C23 | 1 | 3 | yes | — |
| 28 | #466 | — | 4 | 6 | no | — |
| 29 | #466 | — | 4 | 6 | no | — |
| 30 | C2×#107 | C23 | 1 | 6 | yes | — |
| 31 | C2×#102 | — | 2 | 6 | no | — |
| 32 | #393 | — | 4 | 2 | no | — |
| 33 | #392 | — | 4 | 6 | no | — |
| 34 | #380 | — | 2 | 6 | no | — |
| 35 | #366 | — | 4 | 3 | no | — |
| 36 | #357 | — | 4 | 6 | no | — |
| 37 | C22⋊C16 | — | 4 | 3 | no | — |
| 38 | C22⋊C16 | — | 4 | 3 | no | — |
| 39 | #340 | — | 2 | 6 | no | — |
| 40 | C2×#41 | — | 4 | 3 | no | — |
| 41 | C2×#41 | — | 4 | 3 | no | — |
| 42 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 43 | C22×D4 | — | 2 | 2 | no | — |
| 44 | C22×D4 | — | 4 | 6 | no | — |
| 45 | C22×D4 | D8 | 1 | 3 | yes | — |
| 46 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 47 | C2×D8 | — | 2 | 6 | no | — |
| 48 | C2×D8 | — | 2 | 6 | no | — |
| 49 | #131 | — | 4 | 12 | no | — |
| 50 | #131 | — | 4 | 4 | no | — |
| 51 | C2×M16 | — | 4 | 3 | no | — |
| 52 | C2×M16 | — | 4 | 6 | no | — |
| 53 | C22×C8 | — | 2 | 3 | no | — |
| 54 | C22×C8 | — | 2 | 3 | no | — |
| 55 | C22×C8 | — | 2 | 6 | no | — |
| 56 | C22×C8 | — | 2 | 6 | no | — |
| 57 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 58 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 59 | #121 | — | 4 | 12 | no | — |
| 60 | #121 | — | 4 | 12 | no | — |
| 61 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 62 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 63 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 64 | D16 | — | 4 | 4 | no | — |
| 65 | C2×C16 | — | 2 | 6 | no | — |
| 66 | C2×C16 | — | 2 | 3 | no | — |
| 67 | C2×C16 | — | 2 | 3 | no | — |
| 68 | #107 | — | 2 | 6 | no | — |
| 69 | #107 | — | 2 | 6 | no | — |
| 70 | #102 | — | 4 | 12 | no | — |
| 71 | #95 | — | 4 | 6 | no | — |
| 72 | C24 | — | 4 | 2 | no | — |
| 73 | C24 | C2×D8 | 1 | 3 | yes | — |
| 74 | #41 | — | 4 | 4 | no | — |
| 75 | #41 | — | 4 | 12 | no | — |
| 76 | C2×Q8 | — | 4 | 3 | no | — |
| 77 | C2×D4 | — | 4 | 6 | no | — |
| 78 | C2×D4 | — | 4 | 3 | no | — |
| 79 | C2×D4 | — | 4 | 6 | no | — |
| 80 | C2×D4 | — | 4 | 6 | no | — |
| 81 | C2×D4 | — | 8 | 12 | no | — |
| 82 | C2×D4 | — | 16 | 6 | no | — |
| 83 | C2×D4 | — | 16 | 6 | no | — |
| 84 | C22×C4 | — | 2 | 6 | no | — |
| 85 | C22×C4 | — | 2 | 3 | no | — |
| 86 | C22×C4 | — | 2 | 6 | no | — |
| 87 | C22×C4 | — | 2 | 3 | no | — |
| 88 | C22×C4 | — | 4 | 6 | no | — |
| 89 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 90 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 91 | D8 | — | 4 | 6 | no | — |
| 92 | D8 | — | 4 | 2 | no | — |
| 93 | M16 | — | 4 | 12 | no | — |
| 94 | C2×C8 | — | 2 | 6 | no | — |
| 95 | C2×C8 | — | 2 | 3 | no | — |
| 96 | C2×C8 | — | 4 | 6 | no | — |
| 97 | C2×C8 | — | 4 | 3 | no | — |
| 98 | C2×C8 | — | 4 | 3 | no | — |
| 99 | C2×C8 | H(5,𝔽2) | 1 | 3 | yes | — |
| 100 | C2×C8 | H(5,𝔽2) | 1 | 3 | yes | — |
| 101 | C4⋊C4 | — | 4 | 6 | no | — |
| 102 | C4⋊C4 | — | 4 | 6 | no | — |
| 103 | C22⋊C4 | — | 8 | 12 | no | — |
| 104 | C22⋊C4 | — | 16 | 6 | no | — |
| 105 | C16 | — | 4 | 3 | no | — |
| 106 | C16 | — | 4 | 1 | no | — |
| 107 | C23 | — | 2 | 6 | no | — |
| 108 | C23 | — | 2 | 3 | no | — |
| 109 | C23 | — | 2 | 3 | no | — |
| 110 | C23 | — | 2 | 6 | no | — |
| 111 | C23 | — | 2 | 3 | no | — |
| 112 | C23 | — | 8 | 6 | no | — |
| 113 | C23 | — | 8 | 2 | no | — |
| 114 | C23 | — | 8 | 6 | no | — |
| 115 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 116 | Q8 | — | 4 | 4 | no | — |
| 117 | D4 | — | 4 | 12 | no | — |
| 118 | D4 | — | 8 | 6 | no | — |
| 119 | D4 | — | 8 | 2 | no | — |
| 120 | D4 | — | 16 | 24 | no | — |
| 121 | C2×C4 | — | 2 | 6 | no | — |
| 122 | C2×C4 | — | 2 | 3 | no | — |
| 123 | C2×C4 | — | 4 | 6 | no | — |
| 124 | C2×C4 | — | 4 | 3 | no | — |
| 125 | C2×C4 | — | 4 | 3 | no | — |
| 126 | C2×C4 | — | 8 | 6 | no | — |
| 127 | C2×C4 | — | 8 | 6 | no | — |
| 128 | C2×C4 | — | 8 | 6 | no | — |
| 129 | C2×C4 | #534 | 1 | 3 | yes | — |
| 130 | C2×C4 | #575 | 1 | 3 | yes | — |
| 131 | C8 | — | 4 | 3 | no | — |
| 132 | C8 | #560 | 1 | 3 | yes | — |
| 133 | C8 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 134 | C22 | — | 4 | 3 | no | — |
| 135 | C22 | — | 4 | 6 | no | — |
| 136 | C22 | — | 4 | 3 | no | — |
| 137 | C22 | — | 4 | 6 | no | — |
| 138 | C22 | — | 4 | 3 | no | — |
| 139 | C22 | — | 16 | 2 | no | — |
| 140 | C22 | — | 16 | 6 | no | — |
| 141 | C22 | — | 16 | 6 | no | — |
| 142 | C22 | #3186 | 1 | 3 | yes | — |
| 143 | C22 | #3415 | 1 | 3 | yes | — |
| 144 | C22 | #3417 | 1 | 1 | yes | characteristic |
| 145 | C4 | — | 4 | 3 | no | — |
| 146 | C4 | — | 16 | 6 | no | — |
| 147 | C4 | #2812 | 1 | 1 | yes | characteristic |
| 148 | C4 | #3197 | 1 | 3 | yes | — |
| 149 | C2 | — | 8 | 3 | no | — |
| 150 | C2 | — | 32 | 2 | no | — |
| 151 | C2 | #32706 | 1 | 1 | yes | characteristic |
| 152 | C2 | #33765 | 1 | 3 | yes | — |
| 153 | C2 | #33769 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #32933 | C2 | semidirect |
| 2. #13493 | C2 | semidirect |
| 3. #13380 | C2 | semidirect |
| 4. #13304 | C2 | semidirect |
| 5. #12680 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #32933 | 2 |
| #13493 | 2 |
| #13380 | 2 |
| #13304 | 2 |
| #12680 | 2 |