Subgroups — #2098

Order 128.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430	C₂	1	1	yes	maximal, characteristic, complemented
2	#424	C₂	1	2	yes	maximal, complemented
3	#421	C₂	1	2	yes	maximal
4	#408	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	#117	C₂²	1	2	yes	—
11	#108	—	2	2	no	—
12	C₄⋊C₈	—	2	4	no	—
13	#104	—	2	4	no	—
14	C₂²⋊C₈	—	2	2	no	—
15	#41	—	2	2	no	—
16	#41	—	2	2	no	—
17	C₂×Q₈	Q₈	1	1	yes	characteristic
18	C₂×D₄	—	2	1	no	—
19	C₂×D₄	Q₈	1	1	yes	characteristic
20	C₂²×C₄	—	2	1	no	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	M₁₆	—	2	2	no	—
23	M₁₆	—	2	2	no	—
24	M₁₆	—	2	2	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	D₄	1	2	yes	—
29	C₄⋊C₄	—	2	2	no	—
30	C₂²⋊C₄	—	2	2	no	—
31	C₄²	D₄	1	4	yes	—
32	C₂³	—	2	1	no	—
33	C₂³	#41	1	1	yes	characteristic
34	Q₈	—	2	2	no	—
35	D₄	—	2	2	no	—
36	D₄	—	4	2	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	C₂×D₄	1	2	yes	—
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄	#41	1	2	yes	—
45	C₈	—	2	2	no	—
46	C₈	—	4	1	no	—
47	C₈	—	4	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	—	4	2	no	—
50	C₂²	—	4	1	no	—
51	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²	#123	1	2	yes	—
53	C₄	—	4	4	no	—
54	C₄	—	4	1	no	—
55	C₄	C₂²⋊Q₈	1	2	yes	—
56	C₄	#123	1	1	yes	characteristic
57	C₄	#127	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	2	1	no	—
60	C₂	—	8	1	no	—
61	C₂	#399	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#399
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #430	C₂	semidirect
2. #424	C₂	semidirect
5. H(3,ℤ/4)	C₂	semidirect

Maximal subgroups

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

subgroup	index
#430	2
#424	2
#421	2
#408	2
H(3,ℤ/4)	2