Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
652 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 152 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 | #59924 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #9637 | C2 | 1 | 3 | yes | maximal |
| 3 | #8587 | C2 | 1 | 3 | yes | maximal |
| 4 | #2811 | C4 | 1 | 4 | yes | — |
| 5 | #2567 | — | 2 | 6 | no | — |
| 6 | #2551 | — | 2 | 3 | no | — |
| 7 | #2369 | — | 2 | 3 | no | — |
| 8 | C4×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 9 | #1842 | C22 | 1 | 3 | yes | — |
| 10 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#333 | — | 2 | 3 | no | — |
| 12 | C2×Q8⋊C8 | — | 2 | 6 | no | — |
| 13 | #1296 | — | 2 | 3 | no | — |
| 14 | C2×C4⋊Q8 | — | 2 | 6 | no | — |
| 15 | C2×C4⋊Q8 | D4 | 1 | 3 | yes | — |
| 16 | C2×C4×Q8 | D4 | 1 | 6 | yes | — |
| 17 | C2×C4⋊C8 | — | 2 | 6 | no | — |
| 18 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 19 | C2×C4×C8 | — | 2 | 3 | no | — |
| 20 | #395 | — | 2 | 4 | no | — |
| 21 | #395 | — | 2 | 12 | no | — |
| 22 | #391 | — | 2 | 12 | no | — |
| 23 | #391 | C2×C4 | 1 | 6 | yes | — |
| 24 | #389 | — | 2 | 12 | no | — |
| 25 | #389 | — | 2 | 6 | no | — |
| 26 | #384 | — | 2 | 4 | no | — |
| 27 | #384 | D4 | 1 | 3 | yes | — |
| 28 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 29 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 30 | C43 | C23 | 1 | 1 | yes | characteristic |
| 31 | #333 | — | 4 | 6 | no | — |
| 32 | Q8⋊C8 | — | 4 | 12 | no | — |
| 33 | C22×Q8 | — | 2 | 4 | no | — |
| 34 | C22×Q8 | C22⋊C4 | 1 | 6 | yes | — |
| 35 | C22×C8 | — | 4 | 1 | no | — |
| 36 | C22×C8 | — | 4 | 3 | no | — |
| 37 | C4⋊Q8 | — | 2 | 6 | no | — |
| 38 | C4⋊Q8 | — | 8 | 6 | no | — |
| 39 | C4⋊Q8 | — | 8 | 6 | no | — |
| 40 | C4⋊Q8 | SD16 | 1 | 6 | yes | — |
| 41 | C4⋊Q8 | Q16 | 1 | 6 | yes | — |
| 42 | C4×Q8 | — | 4 | 6 | no | — |
| 43 | C4×Q8 | — | 4 | 6 | no | — |
| 44 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 45 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 46 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 47 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 48 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 49 | C2×C42 | — | 2 | 6 | no | — |
| 50 | C2×C42 | — | 2 | 4 | no | — |
| 51 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 52 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 53 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 54 | C4⋊C8 | — | 4 | 6 | no | — |
| 55 | C4⋊C8 | — | 4 | 6 | no | — |
| 56 | C4⋊C8 | — | 4 | 6 | no | — |
| 57 | C4×C8 | — | 2 | 6 | no | — |
| 58 | C4×C8 | — | 4 | 3 | no | — |
| 59 | #95 | — | 2 | 12 | no | — |
| 60 | #95 | — | 2 | 12 | no | — |
| 61 | C2×Q8 | — | 2 | 6 | no | — |
| 62 | C2×Q8 | — | 4 | 6 | no | — |
| 63 | C2×Q8 | — | 4 | 6 | no | — |
| 64 | C2×Q8 | — | 8 | 12 | no | — |
| 65 | C2×Q8 | #103 | 1 | 12 | yes | — |
| 66 | C22×C4 | — | 2 | 6 | no | — |
| 67 | C22×C4 | — | 2 | 6 | no | — |
| 68 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 69 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 70 | C22×C4 | #120 | 1 | 3 | yes | — |
| 71 | C2×C8 | — | 4 | 6 | no | — |
| 72 | C2×C8 | — | 8 | 6 | no | — |
| 73 | C2×C8 | — | 8 | 3 | no | — |
| 74 | C4⋊C4 | — | 2 | 12 | no | — |
| 75 | C4⋊C4 | — | 2 | 12 | no | — |
| 76 | C4⋊C4 | — | 4 | 6 | no | — |
| 77 | C4⋊C4 | — | 4 | 6 | no | — |
| 78 | C4⋊C4 | — | 4 | 6 | no | — |
| 79 | C4⋊C4 | — | 8 | 6 | no | — |
| 80 | C4⋊C4 | — | 8 | 12 | no | — |
| 81 | C42 | — | 2 | 3 | no | — |
| 82 | C42 | — | 2 | 3 | no | — |
| 83 | C42 | — | 4 | 1 | no | — |
| 84 | C42 | — | 4 | 6 | no | — |
| 85 | C42 | — | 4 | 6 | no | — |
| 86 | C42 | — | 8 | 4 | no | — |
| 87 | C42 | C2×SD16 | 1 | 3 | yes | — |
| 88 | C42 | C2×Q16 | 1 | 3 | yes | — |
| 89 | C42 | #137 | 1 | 6 | yes | — |
| 90 | C23 | #379 | 1 | 1 | yes | characteristic |
| 91 | Q8 | — | 4 | 12 | no | — |
| 92 | Q8 | — | 8 | 6 | no | — |
| 93 | Q8 | — | 16 | 8 | no | — |
| 94 | C2×C4 | — | 2 | 3 | no | — |
| 95 | C2×C4 | — | 2 | 3 | no | — |
| 96 | C2×C4 | — | 2 | 3 | no | — |
| 97 | C2×C4 | — | 2 | 12 | no | — |
| 98 | C2×C4 | — | 4 | 6 | no | — |
| 99 | C2×C4 | — | 4 | 6 | no | — |
| 100 | C2×C4 | — | 4 | 6 | no | — |
| 101 | C2×C4 | — | 4 | 12 | no | — |
| 102 | C2×C4 | C2×#103 | 1 | 3 | yes | — |
| 103 | C2×C4 | #419 | 1 | 3 | yes | — |
| 104 | C2×C4 | #448 | 1 | 6 | yes | — |
| 105 | C2×C4 | #451 | 1 | 6 | yes | — |
| 106 | C2×C4 | #453 | 1 | 3 | yes | — |
| 107 | C2×C4 | #455 | 1 | 3 | yes | — |
| 108 | C8 | — | 4 | 4 | no | — |
| 109 | C8 | — | 16 | 3 | no | — |
| 110 | C22 | #1787 | 1 | 3 | yes | — |
| 111 | C22 | #1791 | 1 | 1 | yes | characteristic |
| 112 | C22 | #1792 | 1 | 3 | yes | — |
| 113 | C4 | — | 2 | 3 | no | — |
| 114 | C4 | — | 2 | 3 | no | — |
| 115 | C4 | — | 4 | 1 | no | — |
| 116 | C4 | — | 4 | 12 | no | — |
| 117 | C4 | — | 8 | 6 | no | — |
| 118 | C4 | #1623 | 1 | 3 | yes | — |
| 119 | C4 | #1633 | 1 | 3 | yes | — |
| 120 | C4 | #1645 | 1 | 6 | yes | — |
| 121 | C2 | #8355 | 1 | 3 | yes | — |
| 122 | C2 | #8451 | 1 | 3 | yes | — |
| 123 | C2 | #8530 | 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 ▷ 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 |
|---|---|
| #59924 | 2 |
| #9637 | 2 |
| #8587 | 2 |