Subgroups — #1858

Order 128.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#568	C₂	1	1	yes	maximal, characteristic
2	#430	C₂	1	2	yes	maximal, complemented
3	#413	C₂	1	1	yes	maximal, characteristic
4	#408	C₂	1	1	yes	maximal, characteristic
5	#341	C₂	1	2	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	—	2	4	no	—
8	C₂×M₁₆	—	2	2	no	—
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	#108	—	2	4	no	—
14	#101	C₄	1	2	yes	—
15	#100	C₄	1	2	yes	—
16	C₂²⋊C₈	—	2	2	no	—
17	#41	—	2	2	no	—
18	#41	—	2	2	no	—
19	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄	Q₈	1	2	yes	—
22	C₂²×C₄	D₄	1	2	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	M₁₆	—	2	4	no	—
25	M₁₆	—	2	2	no	—
26	M₁₆	—	2	2	no	—
27	M₁₆	—	2	2	no	—
28	M₁₆	C₂×C₄	1	4	yes	—
29	M₁₆	D₄	1	4	yes	—
30	C₂×C₈	—	2	4	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂³	C₄⋊C₄	1	2	yes	—
37	C₂³	C₂×Q₈	1	1	yes	characteristic
38	Q₈	—	2	2	no	—
39	D₄	—	2	4	no	—
40	D₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄	C₂×D₄	1	2	yes	—
46	C₂×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄	#41	1	2	yes	—
48	C₈	—	2	4	no	—
49	C₈	—	2	4	no	—
50	C₈	—	4	1	no	—
51	C₈	—	4	1	no	—
52	C₂²	—	2	2	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²	C₂²⋊Q₈	1	2	yes	—
57	C₄	—	2	2	no	—
58	C₄	C₄×D₄	1	2	yes	—
59	C₄	#120	1	1	yes	characteristic
60	C₄	#123	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	2	2	no	—
63	C₂	—	4	2	no	—
64	C₂	#385	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#120
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#385
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×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
2. #430	C₂	semidirect
5. #341	C₂	semidirect

Maximal subgroups

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

subgroup	index
#568	2
#430	2
#413	2
#408	2
#341	2