Subgroups — #452

Order 64.

76 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
2	C₂×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Q₈	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×Q₈	C₂²	1	1	yes	characteristic
12	C₂×D₄	—	2	1	no	—
13	C₂×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄	—	2	1	no	—
15	C₂²×C₄	C₂²	1	1	yes	characteristic
16	Q₁₆	—	2	2	no	—
17	SD₁₆	—	2	2	no	—
18	C₂×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₄	—	2	1	no	—
21	C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄	—	2	1	no	—
23	C₂³	—	2	1	no	—
24	C₂³	D₄	1	1	yes	characteristic
25	Q₈	—	2	1	no	—
26	Q₈	—	2	1	no	—
27	Q₈	—	2	1	no	—
28	Q₈	D₄	1	2	yes	—
29	D₄	—	2	4	no	—
30	D₄	—	2	1	no	—
31	D₄	D₄	1	2	yes	—
32	C₂×C₄	—	2	4	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	D₄	1	1	yes	characteristic
39	C₂×C₄	C₂³	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	—	2	1	no	—
42	C₂²	—	2	1	no	—
43	C₂²	—	2	2	no	—
44	C₂²	—	2	1	no	—
45	C₂²	—	2	1	no	—
46	C₂²	—	4	1	no	—
47	C₂²	—	4	1	no	—
48	C₂²	C₂×D₄	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	2	no	—
51	C₄	—	4	1	no	—
52	C₄	—	4	1	no	—
53	C₄	C₂×D₄	1	1	yes	characteristic
54	C₄	C₂×D₄	1	1	yes	characteristic
55	C₂	—	4	1	no	—
56	C₂	—	4	2	no	—
57	C₂	#120	1	1	yes	characteristic
58	C₂	#135	1	1	yes	characteristic
59	C₂	#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	C₂×#41	C₂
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
2. C₂×Q₁₆	C₂	semidirect
3. C₂×SD₁₆	C₂	semidirect
4. C₂²⋊Q₈	C₂	semidirect
5. #103	C₂	semidirect
6. #102	C₂	semidirect
7. C₂²⋊C₈	C₂	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×Q₁₆	2
C₂×SD₁₆	2
C₂²⋊Q₈	2
#103	2
#102	2
C₂²⋊C₈	2