Subgroups — #1565

Order 128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#335	C₂	1	2	yes	maximal, complemented
5	#334	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	C₄	1	1	yes	characteristic
7	C₂×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₈	—	2	1	no	—
9	C₂×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊C₈	—	2	1	no	—
11	C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊C₈	C₂²	1	2	yes	—
13	C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈	—	2	1	no	—
15	#97	C₄	1	2	yes	—
16	C₄×C₈	—	2	1	no	—
17	C₄×C₈	C₂²	1	2	yes	—
18	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₄	Q₈	1	1	yes	characteristic
21	M₁₆	—	2	2	no	—
22	M₁₆	C₈	1	4	yes	complemented
23	C₂×C₈	—	2	1	no	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	—	4	2	no	—
27	C₂×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈	C₂×C₄	1	2	yes	—
29	C₄²	C₂×C₄	1	1	yes	characteristic
30	C₄²	D₄	1	1	yes	characteristic
31	C₄²	Q₈	1	1	yes	characteristic
32	C₄²	C₂³	1	1	yes	characteristic
33	C₂³	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄	C₂×C₈	1	2	yes	—
41	C₂×C₄	M₁₆	1	2	yes	—
42	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₈	—	2	2	no	—
46	C₈	—	4	1	no	—
47	C₈	—	4	2	no	—
48	C₈	—	4	1	no	—
49	C₈	C₂×C₈	1	4	yes	complemented
50	C₂²	—	2	1	no	—
51	C₂²	—	2	1	no	—
52	C₂²	C₄⋊C₈	1	2	yes	—
53	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄	—	2	1	no	—
55	C₄	—	2	1	no	—
56	C₄	—	2	1	no	—
57	C₄	C₄⋊C₈	1	2	yes	—
58	C₄	C₂²×C₈	1	1	yes	characteristic
59	C₄	C₂×M₁₆	1	1	yes	characteristic
60	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄	#137	1	1	yes	characteristic
62	C₂	—	2	2	no	—
63	C₂	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂	#428	1	1	yes	characteristic
65	C₂	#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₄	C₂²×C₈
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₂×C₄⋊C₄
prime radical	C₂³	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #335	C₂	semidirect
5. #334	C₂	semidirect
22. M₁₆	C₈	semidirect
49. C₈	C₂×C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#431	2
C₂×C₄⋊C₈	2
C₄×M₁₆	2
#335	2
#334	2