Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
478 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 96 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 | #37613 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #9871 | C2 | 1 | 3 | yes | maximal |
| 3 | #7602 | C2 | 1 | 3 | yes | maximal |
| 4 | #2845 | C4 | 1 | 1 | yes | characteristic |
| 5 | #2822 | C4 | 1 | 3 | yes | — |
| 6 | #2655 | — | 2 | 6 | no | — |
| 7 | #2585 | — | 2 | 6 | no | — |
| 8 | C4×#125 | C22 | 1 | 3 | yes | — |
| 9 | #1831 | C22 | 1 | 3 | yes | — |
| 10 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1279 | — | 2 | 6 | no | — |
| 12 | #1271 | — | 2 | 6 | no | — |
| 13 | C2×#125 | C2×C4 | 1 | 3 | yes | — |
| 14 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 15 | C2×C4×C8 | — | 2 | 3 | no | — |
| 16 | #401 | — | 2 | 2 | no | — |
| 17 | #400 | — | 2 | 6 | no | — |
| 18 | #400 | — | 2 | 6 | no | — |
| 19 | #398 | — | 2 | 6 | no | — |
| 20 | #398 | — | 2 | 6 | no | — |
| 21 | #398 | — | 2 | 6 | no | — |
| 22 | #389 | — | 2 | 6 | no | — |
| 23 | #389 | D4 | 1 | 6 | yes | — |
| 24 | #387 | — | 2 | 6 | no | — |
| 25 | #387 | — | 2 | 6 | no | — |
| 26 | #387 | — | 2 | 6 | no | — |
| 27 | #387 | — | 2 | 6 | no | — |
| 28 | #382 | C2×C4 | 1 | 3 | yes | — |
| 29 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 30 | C4×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 31 | C43 | C23 | 1 | 1 | yes | characteristic |
| 32 | #336 | — | 2 | 6 | no | — |
| 33 | C22×C8 | — | 4 | 1 | no | — |
| 34 | C22×C8 | — | 4 | 3 | no | — |
| 35 | #125 | — | 2 | 12 | no | — |
| 36 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 37 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 38 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 39 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 40 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 41 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 42 | C2×C42 | — | 2 | 6 | no | — |
| 43 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 44 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 45 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 46 | C4⋊C8 | — | 4 | 6 | no | — |
| 47 | C4×C8 | — | 2 | 6 | no | — |
| 48 | C4×C8 | — | 4 | 3 | no | — |
| 49 | #95 | — | 2 | 2 | no | — |
| 50 | #95 | — | 2 | 6 | no | — |
| 51 | #95 | — | 2 | 6 | no | — |
| 52 | #95 | — | 2 | 6 | no | — |
| 53 | #95 | — | 2 | 6 | no | — |
| 54 | #95 | — | 2 | 6 | no | — |
| 55 | #95 | — | 2 | 6 | no | — |
| 56 | C22×C4 | — | 2 | 6 | no | — |
| 57 | C22×C4 | — | 2 | 6 | no | — |
| 58 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 59 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 60 | C22×C4 | #120 | 1 | 3 | yes | — |
| 61 | C2×C8 | — | 4 | 6 | no | — |
| 62 | C2×C8 | — | 8 | 6 | no | — |
| 63 | C2×C8 | — | 8 | 3 | no | — |
| 64 | C4⋊C4 | — | 2 | 12 | no | — |
| 65 | C4⋊C4 | — | 4 | 6 | no | — |
| 66 | C4⋊C4 | — | 4 | 12 | no | — |
| 67 | C4⋊C4 | — | 4 | 6 | no | — |
| 68 | C4⋊C4 | — | 4 | 6 | no | — |
| 69 | C4⋊C4 | — | 8 | 6 | no | — |
| 70 | C4⋊C4 | — | 8 | 6 | no | — |
| 71 | C42 | — | 2 | 3 | no | — |
| 72 | C42 | — | 2 | 3 | no | — |
| 73 | C42 | — | 4 | 3 | no | — |
| 74 | C42 | — | 8 | 6 | no | — |
| 75 | C42 | #131 | 1 | 4 | yes | — |
| 76 | C23 | #379 | 1 | 1 | yes | characteristic |
| 77 | C2×C4 | — | 2 | 3 | no | — |
| 78 | C2×C4 | — | 2 | 6 | no | — |
| 79 | C2×C4 | — | 2 | 3 | no | — |
| 80 | C2×C4 | — | 2 | 3 | no | — |
| 81 | C2×C4 | — | 4 | 6 | no | — |
| 82 | C2×C4 | — | 4 | 6 | no | — |
| 83 | C2×C4 | — | 4 | 6 | no | — |
| 84 | C2×C4 | — | 4 | 6 | no | — |
| 85 | C2×C4 | — | 4 | 6 | no | — |
| 86 | C2×C4 | — | 4 | 6 | no | — |
| 87 | C2×C4 | #408 | 1 | 6 | yes | — |
| 88 | C2×C4 | #454 | 1 | 3 | yes | — |
| 89 | C2×C4 | #456 | 1 | 3 | yes | — |
| 90 | C8 | — | 4 | 4 | no | — |
| 91 | C8 | — | 16 | 3 | no | — |
| 92 | C22 | #1780 | 1 | 1 | yes | characteristic |
| 93 | C22 | #1793 | 1 | 3 | yes | — |
| 94 | C22 | #1794 | 1 | 3 | yes | — |
| 95 | C4 | — | 2 | 3 | no | — |
| 96 | C4 | — | 2 | 3 | no | — |
| 97 | C4 | — | 2 | 6 | no | — |
| 98 | C4 | — | 4 | 1 | no | — |
| 99 | C4 | — | 8 | 6 | no | — |
| 100 | C4 | — | 8 | 6 | no | — |
| 101 | C2 | #8321 | 1 | 3 | yes | — |
| 102 | C2 | #8529 | 1 | 3 | yes | — |
| 103 | C2 | #8533 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #37613 | 2 |
| #9871 | 2 |
| #7602 | 2 |