Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
43 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #446 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #346 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C4×Q8 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×C16 | — | 2 | 1 | no | — |
| 6 | C2×C16 | — | 2 | 1 | no | — |
| 7 | C4⋊C8 | — | 2 | 1 | no | — |
| 8 | C4⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 9 | #97 | — | 2 | 1 | no | — |
| 10 | C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×Q8 | C8 | 1 | 1 | yes | characteristic |
| 12 | C2×C8 | — | 2 | 1 | no | — |
| 13 | C2×C8 | D4 | 1 | 1 | yes | characteristic |
| 14 | C2×C8 | D4 | 1 | 1 | yes | characteristic |
| 15 | C4⋊C4 | — | 2 | 1 | no | — |
| 16 | C4⋊C4 | C8 | 1 | 1 | yes | characteristic |
| 17 | C42 | — | 2 | 1 | no | — |
| 18 | C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | C16 | — | 2 | 2 | no | — |
| 20 | C16 | — | 4 | 1 | no | — |
| 21 | Q8 | — | 4 | 1 | no | — |
| 22 | C2×C4 | — | 2 | 1 | no | — |
| 23 | C2×C4 | — | 2 | 1 | no | — |
| 24 | C2×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 25 | C2×C4 | C2×C8 | 1 | 1 | yes | characteristic |
| 26 | C2×C4 | M16 | 1 | 1 | yes | characteristic |
| 27 | C8 | — | 2 | 1 | no | — |
| 28 | C8 | — | 4 | 1 | no | — |
| 29 | C8 | SD16 | 1 | 1 | yes | characteristic |
| 30 | C8 | Q16 | 1 | 1 | yes | characteristic |
| 31 | C22 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 32 | C4 | — | 2 | 1 | no | — |
| 33 | C4 | — | 2 | 1 | no | — |
| 34 | C4 | — | 4 | 1 | no | — |
| 35 | C4 | — | 4 | 1 | no | — |
| 36 | C4 | #103 | 1 | 1 | yes | characteristic |
| 37 | C4 | #104 | 1 | 1 | yes | characteristic |
| 38 | C2 | Q8⋊C8 | 1 | 1 | yes | characteristic |
| 39 | C2 | #349 | 1 | 1 | yes | characteristic |
| 40 | C2 | #350 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle, prime radical | C22 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #446 | 2 |
| C4⋊C16 | 2 |
| #346 | 2 |