Subgroups — #449

Order 64.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#121	C₂	1	1	yes	maximal, characteristic, complemented
5	#103	C₂	1	1	yes	maximal, characteristic, complemented
6	#102	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#41	—	2	2	no	—
9	#41	—	2	2	no	—
10	C₂×Q₈	C₂²	1	1	yes	characteristic
11	C₂×D₄	—	2	1	no	—
12	C₂×D₄	—	2	1	no	—
13	C₂×D₄	C₂²	1	1	yes	characteristic
14	C₂×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄	—	2	1	no	—
16	C₂²×C₄	C₂²	1	1	yes	characteristic
17	SD₁₆	—	2	2	no	—
18	D₈	—	2	2	no	—
19	C₂×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄	—	2	1	no	—
23	C₂³	—	2	1	no	—
24	C₂³	—	2	1	no	—
25	C₂³	D₄	1	1	yes	characteristic
26	Q₈	—	2	1	no	—
27	Q₈	D₄	1	2	yes	complemented
28	D₄	—	2	4	no	—
29	D₄	—	2	1	no	—
30	D₄	—	2	1	no	—
31	D₄	—	2	1	no	—
32	D₄	—	4	2	no	—
33	D₄	D₄	1	2	yes	complemented
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	D₄	1	1	yes	characteristic
40	C₂×C₄	C₂³	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	—	2	2	no	—
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	—	4	1	no	—
48	C₂²	—	4	1	no	—
49	C₂²	—	4	1	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	1	no	—
52	C₂²	C₂×D₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	2	no	—
55	C₄	—	4	1	no	—
56	C₄	C₂×D₄	1	1	yes	characteristic
57	C₄	C₂×D₄	1	1	yes	characteristic
58	C₂	—	4	1	no	—
59	C₂	—	4	2	no	—
60	C₂	—	8	1	no	—
61	C₂	#120	1	1	yes	characteristic
62	C₂	#135	1	1	yes	characteristic
63	C₂	AGL(1,ℤ/8)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#41	C₂	semidirect
2. C₂×SD₁₆	C₂	semidirect
3. C₂×D₈	C₂	semidirect
4. #121	C₂	semidirect
5. #103	C₂	semidirect
6. #102	C₂	semidirect
7. C₂²⋊C₈	C₂	semidirect
27. Q₈	D₄	semidirect
33. D₄	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41	2
C₂×SD₁₆	2
C₂×D₈	2
#121	2
#103	2
#102	2
C₂²⋊C₈	2