Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
523 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 523 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 | #19604 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #18090 | C2 | 1 | 4 | yes | maximal |
| 3 | #9727 | C2 | 1 | 4 | yes | maximal |
| 4 | #9068 | C2 | 1 | 4 | yes | maximal |
| 5 | #8293 | C2 | 1 | 2 | yes | maximal |
| 6 | #3170 | C22 | 1 | 4 | yes | — |
| 7 | #3160 | C22 | 1 | 2 | yes | — |
| 8 | #2986 | C22 | 1 | 4 | yes | — |
| 9 | C4×#108 | C22 | 1 | 2 | yes | — |
| 10 | #1713 | C4 | 1 | 2 | yes | — |
| 11 | #1710 | C4 | 1 | 2 | yes | — |
| 12 | #1707 | C4 | 1 | 4 | yes | — |
| 13 | #1593 | — | 2 | 4 | no | — |
| 14 | #1584 | — | 2 | 8 | no | — |
| 15 | #1579 | — | 2 | 4 | no | — |
| 16 | #1564 | C22 | 1 | 2 | yes | — |
| 17 | #1526 | C22 | 1 | 4 | yes | — |
| 18 | #1485 | C22 | 1 | 8 | yes | — |
| 19 | #1451 | C22 | 1 | 4 | yes | — |
| 20 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1278 | C22 | 1 | 4 | yes | — |
| 22 | #533 | C23 | 1 | 4 | yes | — |
| 23 | #500 | C2×C4 | 1 | 4 | yes | — |
| 24 | #500 | C2×C4 | 1 | 4 | yes | — |
| 25 | #499 | — | 4 | 2 | no | — |
| 26 | #498 | C2×C4 | 1 | 4 | yes | — |
| 27 | #494 | C2×C4 | 1 | 4 | yes | — |
| 28 | #487 | C2×C4 | 1 | 8 | yes | — |
| 29 | #467 | — | 2 | 4 | no | — |
| 30 | #467 | — | 2 | 4 | no | — |
| 31 | #446 | — | 2 | 8 | no | — |
| 32 | C8×Q8 | — | 2 | 8 | no | — |
| 33 | C4×Q16 | — | 2 | 4 | no | — |
| 34 | C4×Q16 | — | 2 | 4 | no | — |
| 35 | #433 | — | 4 | 4 | no | — |
| 36 | #431 | C23 | 1 | 4 | yes | — |
| 37 | C2×#108 | — | 2 | 2 | no | — |
| 38 | #427 | — | 4 | 2 | no | — |
| 39 | C4×M16 | C23 | 1 | 4 | yes | — |
| 40 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 42 | #341 | — | 2 | 2 | no | — |
| 43 | #335 | D4 | 1 | 4 | yes | — |
| 44 | #334 | D4 | 1 | 4 | yes | — |
| 45 | #330 | — | 4 | 4 | no | — |
| 46 | Q8⋊C8 | — | 2 | 8 | no | — |
| 47 | Q8⋊C8 | — | 2 | 8 | no | — |
| 48 | #322 | — | 2 | 4 | no | — |
| 49 | #322 | — | 2 | 4 | no | — |
| 50 | C82 | C2×C4 | 1 | 2 | yes | — |
| 51 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C2×Q16 | — | 2 | 4 | no | — |
| 54 | C2×Q16 | — | 2 | 4 | no | — |
| 55 | C2×M16 | — | 2 | 4 | no | — |
| 56 | C22×C8 | — | 2 | 1 | no | — |
| 57 | C22×C8 | #41 | 1 | 2 | yes | — |
| 58 | C22×C8 | #41 | 1 | 2 | yes | — |
| 59 | C4⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 60 | #125 | — | 2 | 4 | no | — |
| 61 | C22⋊Q8 | — | 2 | 8 | no | — |
| 62 | C4×Q8 | — | 2 | 8 | no | — |
| 63 | #117 | — | 2 | 4 | no | — |
| 64 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 65 | #108 | — | 4 | 4 | no | — |
| 66 | #107 | — | 2 | 4 | no | — |
| 67 | #107 | — | 2 | 4 | no | — |
| 68 | #107 | — | 4 | 4 | no | — |
| 69 | #106 | — | 4 | 4 | no | — |
| 70 | C4⋊C8 | — | 4 | 8 | no | — |
| 71 | C4⋊C8 | C2×D4 | 1 | 8 | yes | — |
| 72 | #103 | — | 2 | 8 | no | — |
| 73 | #103 | — | 2 | 8 | no | — |
| 74 | C22⋊C8 | — | 2 | 4 | no | — |
| 75 | #97 | — | 2 | 4 | no | — |
| 76 | #97 | — | 4 | 4 | no | — |
| 77 | C4×C8 | — | 2 | 4 | no | — |
| 78 | C4×C8 | — | 2 | 4 | no | — |
| 79 | C4×C8 | — | 2 | 1 | no | — |
| 80 | C4×C8 | — | 4 | 4 | no | — |
| 81 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 84 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 85 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 86 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 87 | C2×Q8 | — | 2 | 8 | no | — |
| 88 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 89 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 90 | Q16 | — | 4 | 8 | no | — |
| 91 | Q16 | — | 8 | 4 | no | — |
| 92 | M16 | — | 4 | 4 | no | — |
| 93 | M16 | — | 4 | 4 | no | — |
| 94 | C2×C8 | — | 2 | 4 | no | — |
| 95 | C2×C8 | — | 2 | 1 | no | — |
| 96 | C2×C8 | — | 2 | 1 | no | — |
| 97 | C2×C8 | — | 2 | 4 | no | — |
| 98 | C2×C8 | — | 2 | 8 | no | — |
| 99 | C2×C8 | — | 4 | 2 | no | — |
| 100 | C2×C8 | — | 4 | 4 | no | — |
| 101 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 102 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 103 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 104 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 105 | C4⋊C4 | — | 2 | 8 | no | — |
| 106 | C4⋊C4 | — | 2 | 8 | no | — |
| 107 | C4⋊C4 | — | 4 | 8 | no | — |
| 108 | C22⋊C4 | — | 4 | 4 | no | — |
| 109 | C42 | — | 4 | 4 | no | — |
| 110 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 111 | C42 | C22×D4 | 1 | 2 | yes | — |
| 112 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 113 | C23 | #559 | 1 | 1 | yes | characteristic |
| 114 | Q8 | — | 4 | 8 | no | — |
| 115 | Q8 | — | 4 | 8 | no | — |
| 116 | C2×C4 | — | 2 | 2 | no | — |
| 117 | C2×C4 | — | 2 | 2 | no | — |
| 118 | C2×C4 | — | 2 | 2 | no | — |
| 119 | C2×C4 | — | 4 | 8 | no | — |
| 120 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 121 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 122 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 123 | C2×C4 | #550 | 1 | 2 | yes | — |
| 124 | C8 | — | 2 | 2 | no | — |
| 125 | C8 | — | 2 | 2 | no | — |
| 126 | C8 | — | 2 | 2 | no | — |
| 127 | C8 | — | 2 | 2 | no | — |
| 128 | C8 | — | 2 | 2 | no | — |
| 129 | C8 | — | 4 | 8 | no | — |
| 130 | C8 | — | 8 | 4 | no | — |
| 131 | C8 | #444 | 1 | 4 | yes | — |
| 132 | C22 | — | 2 | 2 | no | — |
| 133 | C22 | — | 2 | 1 | no | — |
| 134 | C22 | #2366 | 1 | 1 | yes | characteristic |
| 135 | C4 | — | 2 | 2 | no | — |
| 136 | C4 | — | 2 | 1 | no | — |
| 137 | C4 | — | 8 | 8 | no | — |
| 138 | C4 | C2×#444 | 1 | 2 | yes | — |
| 139 | C4 | #2957 | 1 | 2 | yes | — |
| 140 | C4 | #3293 | 1 | 2 | yes | — |
| 141 | C2 | — | 4 | 1 | no | — |
| 142 | C2 | #23180 | 1 | 2 | yes | — |
| 143 | C2 | #23200 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C23 | #559 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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 |
|---|---|
| #19604 | 2 |
| #18090 | 2 |
| #9727 | 2 |
| #9068 | 2 |
| #8293 | 2 |