Subgroups — #2960

Order 128.

182 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	4	yes	maximal, complemented
3	C₈×Q₈	C₂	1	4	yes	maximal, complemented
4	#424	C₂	1	2	yes	maximal
5	#423	C₂	1	1	yes	maximal, characteristic
6	#405	C₂	1	2	yes	maximal
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₄	1	2	yes	—
12	#125	C₄	1	2	yes	—
13	C₂²⋊Q₈	C₄	1	4	yes	—
14	C₄×Q₈	C₂²	1	4	yes	—
15	#117	C₂²	1	2	yes	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₂²	1	4	yes	—
18	C₄⋊C₈	C₂²	1	8	yes	—
19	#97	C₂²	1	4	yes	—
20	C₄×C₈	C₂²	1	2	yes	—
21	C₄×C₈	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	4	yes	—
23	C₂×Q₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	M₁₆	—	2	4	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	Q₈	1	4	yes	complemented
29	C₂×C₈	C₂³	1	2	yes	—
30	C₂×C₈	C₂³	1	4	yes	—
31	C₂×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₄⋊C₄	C₂×C₄	1	8	yes	—
35	C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄	C₂×C₄	1	4	yes	—
37	C₄²	C₂×C₄	1	1	yes	characteristic
38	C₄²	C₂×C₄	1	1	yes	characteristic
39	C₄²	C₂³	1	2	yes	—
40	C₄²	C₂³	1	4	yes	—
41	C₂³	C₂²×C₄	1	1	yes	characteristic
42	Q₈	—	2	8	no	—
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	C₂²×C₄	1	2	yes	—
45	C₂×C₄	C₂²×C₄	1	8	yes	—
46	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄	C₂²×C₄	1	2	yes	—
48	C₂×C₄	C₂×Q₈	1	2	yes	—
49	C₂×C₄	#41	1	2	yes	—
50	C₂×C₄	C₂⁴	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	2	4	no	—
53	C₈	C₂×Q₈	1	2	yes	—
54	C₈	C₂×Q₈	1	2	yes	—
55	C₂²	—	2	2	no	—
56	C₂²	C₄×Q₈	1	2	yes	—
57	C₂²	C₂³×C₄	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	8	no	—
60	C₄	C₄×Q₈	1	2	yes	—
61	C₄	#131	1	4	yes	—
62	C₄	C₂²×Q₈	1	1	yes	characteristic
63	C₄	C₂×#41	1	1	yes	characteristic
64	C₂	—	2	2	no	—
65	C₂	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂	C₂×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂³
commutator, socle	C₂²	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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
3. C₈×Q₈	C₂	semidirect
28. 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
C₈×Q₈	2
#424	2
#423	2
#405	2
C₂×C₄×C₈	2