Subgroups — #421

Order 64.

77 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×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#104	C₂	1	4	yes	maximal, complemented
5	#41	—	2	1	no	—
6	#41	—	2	1	no	—
7	#41	C₄	1	2	yes	complemented
8	#41	C₂²	1	2	yes	—
9	C₂×Q₈	C₄	1	1	yes	characteristic, complemented
10	C₂×D₄	—	2	1	no	—
11	C₂×D₄	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₄	—	2	1	no	—
13	C₂²×C₄	C₂²	1	1	yes	characteristic
14	M₁₆	—	2	1	no	—
15	M₁₆	C₂²	1	2	yes	complemented
16	C₂×C₈	—	2	1	no	—
17	C₄⋊C₄	—	2	1	no	—
18	C₂²⋊C₄	—	2	1	no	—
19	C₄²	C₂²	1	2	yes	complemented
20	C₂³	—	2	1	no	—
21	C₂³	D₄	1	1	yes	characteristic
22	Q₈	—	2	1	no	—
23	Q₈	C₂×C₄	1	2	yes	—
24	D₄	—	2	2	no	—
25	D₄	—	2	2	no	—
26	D₄	—	2	1	no	—
27	D₄	C₂×C₄	1	2	yes	—
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄	D₄	1	1	yes	characteristic
35	C₂×C₄	D₄	1	1	yes	characteristic
36	C₂×C₄	D₄	1	1	yes	characteristic
37	C₂×C₄	C₂³	1	1	yes	characteristic
38	C₈	—	2	2	no	—
39	C₂²	—	2	2	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	4	2	no	—
42	C₂²	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²	C₂×D₄	1	1	yes	characteristic
45	C₄	—	2	2	no	—
46	C₄	—	4	2	no	—
47	C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₄	C₂²×C₄	1	1	yes	characteristic
50	C₄	C₂×D₄	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	—	2	1	no	—
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	C₂×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄
commutator	C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄	C₂³
socle	C₂	C₂×C₂²⋊C₄
prime radical	C₂×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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×M₁₆	C₂	semidirect
3. #117	C₂	semidirect
4. #104	C₂	semidirect
7. #41	C₄	semidirect
9. C₂×Q₈	C₄	semidirect
11. C₂×D₄	C₄	semidirect
15. M₁₆	C₂²	semidirect
19. C₄²	C₂²	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×M₁₆	2
#117	2
#104	2