Subgroups — #1868

Order 128.

130 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	#424	C₂	1	2	yes	maximal, complemented
3	#410	C₂	1	1	yes	maximal, characteristic
4	#408	C₂	1	1	yes	maximal, characteristic
5	#344	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	C₂²×C₈	C₂²	1	1	yes	characteristic
12	#117	C₂²	1	2	yes	—
13	C₄⋊C₈	—	2	4	no	—
14	#99	C₄	1	4	yes	—
15	C₂²⋊C₈	—	2	2	no	—
16	#41	—	2	2	no	—
17	#41	—	2	2	no	—
18	C₂×Q₈	Q₈	1	1	yes	characteristic
19	C₂×D₄	C₂×C₄	1	2	yes	—
20	C₂×D₄	Q₈	1	1	yes	characteristic
21	C₂²×C₄	D₄	1	2	yes	—
22	C₂²×C₄	C₂³	1	1	yes	characteristic
23	M₁₆	—	2	4	no	—
24	M₁₆	—	2	2	no	—
25	M₁₆	—	2	2	no	—
26	M₁₆	—	2	2	no	—
27	C₂×C₈	—	2	4	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	D₄	1	4	yes	—
32	C₄⋊C₄	—	2	2	no	—
33	C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₄²	—	2	2	no	—
35	C₂³	C₄⋊C₄	1	2	yes	—
36	C₂³	C₂²×C₄	1	1	yes	characteristic
37	Q₈	—	2	2	no	—
38	D₄	—	2	4	no	—
39	D₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₄	C₂×D₄	1	2	yes	—
45	C₂×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄	#41	1	2	yes	—
48	C₈	—	2	4	no	—
49	C₈	—	4	1	no	—
50	C₈	—	4	1	no	—
51	C₂²	—	2	2	no	—
52	C₂²	—	4	1	no	—
53	C₂²	—	4	2	no	—
54	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²	C₄×D₄	1	2	yes	—
56	C₄	—	2	2	no	—
57	C₄	—	4	4	no	—
58	C₄	#120	1	1	yes	characteristic
59	C₄	C₂²⋊Q₈	1	2	yes	—
60	C₄	#123	1	1	yes	characteristic
61	C₂	—	2	2	no	—
62	C₂	—	2	1	no	—
63	C₂	—	4	2	no	—
64	C₂	#385	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#120
commutator	C₂³	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#385
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂³ ▷ 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. #424	C₂	semidirect
5. #344	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
#424	2
#410	2
#408	2
#344	2