Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
371 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 64 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 | #30215 | C2 | 1 | 3 | yes | maximal |
| 2 | #7702 | C2 | 1 | 3 | yes | maximal |
| 3 | #7048 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #2862 | C22 | 1 | 3 | yes | — |
| 5 | #2074 | — | 2 | 6 | no | — |
| 6 | #2024 | — | 2 | 3 | no | — |
| 7 | #2024 | — | 2 | 3 | no | — |
| 8 | #1993 | — | 2 | 3 | no | — |
| 9 | #1960 | — | 2 | 3 | no | — |
| 10 | #1839 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1745 | C22 | 1 | 3 | yes | — |
| 12 | #1272 | — | 2 | 6 | no | — |
| 13 | C2×C4⋊Q8 | D4 | 1 | 3 | yes | — |
| 14 | C2×C4⋊Q8 | D4 | 1 | 3 | yes | — |
| 15 | C2×#103 | — | 2 | 6 | no | — |
| 16 | C2×#103 | — | 2 | 6 | no | — |
| 17 | C2×#97 | — | 2 | 3 | no | — |
| 18 | #391 | — | 2 | 3 | no | — |
| 19 | #391 | — | 2 | 3 | no | — |
| 20 | #391 | — | 2 | 3 | no | — |
| 21 | #391 | — | 2 | 3 | no | — |
| 22 | #382 | — | 2 | 3 | no | — |
| 23 | #382 | — | 2 | 3 | no | — |
| 24 | C43 | C23 | 1 | 1 | yes | characteristic |
| 25 | #340 | — | 2 | 6 | no | — |
| 26 | #340 | — | 2 | 6 | no | — |
| 27 | #336 | — | 2 | 3 | no | — |
| 28 | #336 | — | 2 | 3 | no | — |
| 29 | #336 | D4 | 1 | 6 | yes | — |
| 30 | C22×Q8 | — | 2 | 3 | no | — |
| 31 | C22×Q8 | — | 2 | 3 | no | — |
| 32 | C22×C8 | — | 2 | 6 | no | — |
| 33 | C22×C8 | — | 4 | 1 | no | — |
| 34 | C4⋊Q8 | — | 4 | 6 | no | — |
| 35 | C4⋊Q8 | — | 4 | 6 | no | — |
| 36 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 37 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 38 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 39 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 40 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 41 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 42 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 43 | #103 | — | 8 | 6 | no | — |
| 44 | #103 | — | 8 | 6 | no | — |
| 45 | #97 | — | 4 | 6 | no | — |
| 46 | #95 | — | 4 | 3 | no | — |
| 47 | #95 | — | 4 | 3 | no | — |
| 48 | #95 | — | 4 | 3 | no | — |
| 49 | #95 | — | 4 | 3 | no | — |
| 50 | C2×Q8 | — | 4 | 6 | no | — |
| 51 | C2×Q8 | — | 4 | 6 | no | — |
| 52 | C2×Q8 | — | 4 | 6 | no | — |
| 53 | C2×Q8 | — | 4 | 6 | no | — |
| 54 | C2×Q8 | — | 8 | 3 | no | — |
| 55 | C2×Q8 | — | 8 | 3 | no | — |
| 56 | C22×C4 | — | 4 | 3 | no | — |
| 57 | C22×C4 | — | 4 | 3 | no | — |
| 58 | C22×C4 | #120 | 1 | 3 | yes | — |
| 59 | C22×C4 | #121 | 1 | 3 | yes | — |
| 60 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 61 | C2×C8 | — | 4 | 6 | no | — |
| 62 | C2×C8 | — | 4 | 6 | no | — |
| 63 | C2×C8 | — | 4 | 6 | no | — |
| 64 | C2×C8 | — | 8 | 3 | no | — |
| 65 | C4⋊C4 | — | 4 | 6 | no | — |
| 66 | C4⋊C4 | — | 4 | 6 | no | — |
| 67 | C4⋊C4 | — | 4 | 6 | no | — |
| 68 | C4⋊C4 | — | 8 | 3 | no | — |
| 69 | C42 | — | 2 | 6 | no | — |
| 70 | C42 | — | 4 | 3 | no | — |
| 71 | C42 | — | 4 | 1 | no | — |
| 72 | C23 | #392 | 1 | 1 | yes | characteristic |
| 73 | Q8 | — | 8 | 12 | no | — |
| 74 | Q8 | — | 16 | 6 | no | — |
| 75 | C2×C4 | — | 2 | 3 | no | — |
| 76 | C2×C4 | — | 2 | 3 | no | — |
| 77 | C2×C4 | — | 2 | 3 | no | — |
| 78 | C2×C4 | — | 2 | 3 | no | — |
| 79 | C2×C4 | — | 2 | 3 | no | — |
| 80 | C2×C4 | — | 2 | 3 | no | — |
| 81 | C2×C4 | — | 8 | 3 | no | — |
| 82 | C2×C4 | — | 8 | 3 | no | — |
| 83 | C2×C4 | — | 8 | 3 | no | — |
| 84 | C2×C4 | — | 8 | 3 | no | — |
| 85 | C2×C4 | — | 8 | 3 | no | — |
| 86 | C2×C4 | — | 8 | 3 | no | — |
| 87 | C2×C4 | #455 | 1 | 6 | yes | — |
| 88 | C8 | — | 8 | 6 | no | — |
| 89 | C8 | — | 16 | 1 | no | — |
| 90 | C22 | #2004 | 1 | 3 | yes | — |
| 91 | C22 | #2009 | 1 | 3 | yes | — |
| 92 | C22 | #2017 | 1 | 1 | yes | characteristic |
| 93 | C4 | — | 2 | 6 | no | — |
| 94 | C4 | — | 4 | 3 | no | — |
| 95 | C4 | — | 4 | 1 | no | — |
| 96 | C4 | — | 16 | 3 | no | — |
| 97 | C4 | — | 16 | 3 | no | — |
| 98 | C2 | #10098 | 1 | 3 | yes | — |
| 99 | C2 | #10125 | 1 | 3 | yes | — |
| 100 | C2 | #10211 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #392 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ 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 |
|---|---|
| #30215 | 2 |
| #7702 | 2 |
| #7048 | 2 |