Subgroups — #463

Order 64.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈	C₂	1	1	yes	maximal, characteristic
3	#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#103	C₂	1	1	yes	maximal, characteristic, complemented
7	#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈	C₂²	1	1	yes	characteristic
9	C₂×D₄	C₂²	1	1	yes	characteristic
10	C₂×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄	—	2	1	no	—
12	SD₁₆	—	2	2	no	—
13	D₈	—	2	2	no	—
14	C₂×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄	—	2	1	no	—
18	C₂²⋊C₄	—	2	1	no	—
19	C₂²⋊C₄	—	2	1	no	—
20	C₄²	C₂²	1	1	yes	characteristic
21	C₂³	—	2	1	no	—
22	C₂³	—	2	1	no	—
23	Q₈	—	2	1	no	—
24	Q₈	—	2	1	no	—
25	D₄	—	2	1	no	—
26	D₄	—	2	1	no	—
27	D₄	—	2	1	no	—
28	D₄	D₄	1	2	yes	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	4	2	no	—
32	C₂×C₄	D₄	1	1	yes	characteristic
33	C₂×C₄	D₄	1	1	yes	characteristic
34	C₂×C₄	C₂³	1	1	yes	characteristic
35	C₈	—	2	1	no	—
36	C₈	—	2	1	no	—
37	C₂²	—	2	2	no	—
38	C₂²	—	4	1	no	—
39	C₂²	—	4	1	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	—	4	1	no	—
43	C₂²	C₂×D₄	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	4	1	no	—
48	C₄	C₂×D₄	1	1	yes	characteristic
49	C₄	#41	1	1	yes	characteristic
50	C₂	—	4	2	no	—
51	C₂	—	8	1	no	—
52	C₂	#121	1	1	yes	characteristic
53	C₂	#135	1	1	yes	characteristic
54	C₂	AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	C₂×D₈	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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×SD₁₆	C₂	semidirect
3. #124	C₂	semidirect
4. C₄×D₄	C₂	semidirect
5. C₄⋊C₈	C₂	semidirect
6. #103	C₂	semidirect
7. #102	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆	2
C₂×D₈	2
#124	2
C₄×D₄	2
C₄⋊C₈	2
#103	2
#102	2