Subgroups — #1670

Order 128.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#494	C₂	1	1	yes	maximal, characteristic
2	#475	C₂	1	2	yes	maximal
3	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	#335	C₂	1	1	yes	maximal, characteristic
5	#332	C₂	1	2	yes	maximal
6	C₂×Q₁₆	—	2	1	no	—
7	C₂×Q₁₆	—	2	1	no	—
8	C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄×Q₈	C₂²	1	1	yes	characteristic
10	#106	—	2	2	no	—
11	C₄⋊C₈	—	2	1	no	—
12	C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	2	yes	—
14	#103	—	2	2	no	—
15	C₄×C₈	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	—	2	2	no	—
18	C₂×Q₈	D₄	1	1	yes	characteristic
19	Q₁₆	—	4	2	no	—
20	Q₁₆	—	4	2	no	—
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₂×C₈	D₄	1	1	yes	characteristic
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	D₄	1	1	yes	characteristic
29	C₄²	—	2	1	no	—
30	C₄²	C₂³	1	1	yes	characteristic
31	Q₈	—	2	1	no	—
32	Q₈	—	4	2	no	—
33	Q₈	—	4	2	no	—
34	Q₈	Q₁₆	1	2	yes	complemented
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	4	2	no	—
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄	#41	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	4	2	no	—
43	C₈	—	4	1	no	—
44	C₈	SD₁₆	1	2	yes	—
45	C₂²	#121	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	—	8	2	no	—
50	C₄	C₂×SD₁₆	1	1	yes	characteristic
51	C₄	C₂×Q₁₆	1	1	yes	characteristic
52	C₄	#135	1	1	yes	characteristic
53	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂	#460	1	1	yes	characteristic
55	C₂	#467	1	1	yes	characteristic
56	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
34. Q₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#494	2
#475	2
C₈×Q₈	2
#335	2
#332	2