Subgroups — #2961

Order 128.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	#446	C₂	1	8	yes	maximal, complemented
3	#424	C₂	1	2	yes	maximal
4	#423	C₂	1	1	yes	maximal, characteristic
5	#405	C₂	1	2	yes	maximal
6	C₂×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₄	1	2	yes	—
11	#125	C₄	1	2	yes	—
12	C₂²⋊Q₈	C₄	1	4	yes	—
13	C₄×Q₈	C₂²	1	4	yes	—
14	#117	C₂²	1	2	yes	—
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	C₂²	1	4	yes	—
17	C₄⋊C₈	C₂²	1	8	yes	—
18	#97	C₂²	1	4	yes	—
19	#97	C₂²	1	4	yes	—
20	C₄×C₈	C₂²	1	4	yes	—
21	C₂×Q₈	C₂×C₄	1	4	yes	—
22	C₂²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	M₁₆	—	2	4	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	Q₈	1	4	yes	complemented
27	C₂×C₈	C₂³	1	4	yes	—
28	C₂×C₈	C₂³	1	2	yes	—
29	C₂×C₈	C₂³	1	2	yes	—
30	C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₄²	C₂×C₄	1	1	yes	characteristic
35	C₄²	C₂×C₄	1	1	yes	characteristic
36	C₄²	C₂³	1	4	yes	—
37	C₄²	C₂³	1	2	yes	—
38	C₂³	C₂²×C₄	1	1	yes	characteristic
39	Q₈	—	4	4	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	C₂²×C₄	1	2	yes	—
42	C₂×C₄	C₂²×C₄	1	2	yes	—
43	C₂×C₄	C₂²×C₄	1	8	yes	—
44	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄	C₂×Q₈	1	2	yes	—
46	C₂×C₄	#41	1	2	yes	—
47	C₂×C₄	C₂⁴	1	1	yes	characteristic
48	C₈	—	2	2	no	—
49	C₈	—	2	4	no	—
50	C₈	C₂×Q₈	1	4	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄×Q₈	1	2	yes	—
53	C₂²	C₂³×C₄	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	2	2	no	—
56	C₄	—	2	8	no	—
57	C₄	C₄×Q₈	1	2	yes	—
58	C₄	C₂²×Q₈	1	1	yes	characteristic
59	C₄	C₂×#41	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂	#568	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	C₂³	C₂²×C₄
LCS	G ▷ C₂² ▷ 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. #446	C₂	semidirect
26. C₂×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#446	2
#424	2
#423	2
#405	2
C₂×#97	2