Subgroups — #2988

Order 128.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈	C₂	1	2	yes	maximal, complemented
3	C₈×D₄	C₂	1	6	yes	maximal, complemented
4	#431	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂×#41	C₄	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	6	yes	—
8	C₄×Q₈	C₄	1	1	yes	characteristic
9	C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₄×D₄	C₄	1	3	yes	—
11	C₄×D₄	C₂²	1	3	yes	—
12	#117	C₄	1	3	yes	—
13	C₂×C₄²	C₂²	1	3	yes	—
14	C₄⋊C₈	C₂²	1	3	yes	—
15	C₄⋊C₈	C₂²	1	6	yes	—
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈	C₂²	1	6	yes	—
18	C₄×C₈	C₂²	1	6	yes	—
19	#41	C₈	1	8	yes	complemented
20	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄	C₂×C₄	1	3	yes	—
22	C₂²×C₄	C₂×C₄	1	3	yes	—
23	C₂²×C₄	C₂×C₄	1	3	yes	—
24	C₂²×C₄	C₂³	1	3	yes	—
25	C₂×C₈	—	4	6	no	—
26	C₂×C₈	C₂³	1	2	yes	—
27	C₂×C₈	C₂³	1	6	yes	—
28	C₄⋊C₄	C₂×C₄	1	3	yes	—
29	C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄	C₂×C₄	1	6	yes	—
31	C₄²	C₂×C₄	1	3	yes	—
32	C₄²	C₂×C₄	1	3	yes	—
33	C₄²	C₂³	1	3	yes	—
34	C₄²	C₂³	1	1	yes	characteristic
35	C₂³	C₂²×C₄	1	3	yes	—
36	Q₈	C₂×C₈	1	4	yes	complemented
37	D₄	C₂×C₈	1	12	yes	complemented
38	C₂×C₄	—	2	3	no	—
39	C₂×C₄	—	2	3	no	—
40	C₂×C₄	—	4	3	no	—
41	C₂×C₄	C₂×C₈	1	12	yes	complemented
42	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄	C₂²×C₄	1	3	yes	—
44	C₂×C₄	C₂²×C₄	1	3	yes	—
45	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	C₂²×C₄	1	3	yes	—
47	C₂×C₄	C₂⁴	1	1	yes	characteristic
48	C₈	—	2	6	no	—
49	C₈	—	2	2	no	—
50	C₂²	—	2	3	no	—
51	C₂²	—	2	3	no	—
52	C₂²	C₂²×C₈	1	6	yes	—
53	C₂²	C₂³×C₄	1	1	yes	characteristic
54	C₄	—	2	3	no	—
55	C₄	—	2	1	no	—
56	C₄	—	2	3	no	—
57	C₄	C₂²×C₈	1	2	yes	—
58	C₄	C₂²×C₈	1	6	yes	—
59	C₄	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂	—	2	6	no	—
62	C₂	#520	1	1	yes	characteristic
63	C₂	C₂³×C₈	1	1	yes	characteristic
64	C₂	#568	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator	C₂	C₂³×C₈
socle	C₂²	C₂³×C₄
prime radical	C₂×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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. C₈×Q₈	C₂	semidirect
3. C₈×D₄	C₂	semidirect
4. #431	C₂	semidirect
5. C₂×C₄⋊C₈	C₂	semidirect
19. #41	C₈	semidirect
36. Q₈	C₂×C₈	semidirect
37. D₄	C₂×C₈	semidirect
41. C₂×C₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41	2
C₈×Q₈	2
C₈×D₄	2
#431	2
C₂×C₄⋊C₈	2