Subgroups — #3131

Order 128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#532	C₂	1	1	yes	maximal, characteristic
3	#488	C₂	1	4	yes	maximal, complemented
4	#487	C₂	1	2	yes	maximal, complemented
5	#486	C₂	1	2	yes	maximal, complemented
6	#416	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	C₂²×C₈	C₂²	1	2	yes	—
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#127	C₂²	1	1	yes	characteristic
13	#125	C₂²	1	2	yes	—
14	#124	C₂²	1	2	yes	—
15	C₂²⋊Q₈	—	2	2	no	—
16	#121	—	2	2	no	—
17	C₄×Q₈	—	2	2	no	—
18	C₄×D₄	—	2	2	no	—
19	#117	C₂²	1	2	yes	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	#103	C₂²	1	8	yes	—
22	#102	C₂²	1	8	yes	—
23	C₄×C₈	C₂²	1	2	yes	—
24	C₄×C₈	C₂²	1	2	yes	—
25	#41	—	2	8	no	—
26	C₂×Q₈	—	2	2	no	—
27	C₂×Q₈	C₂³	1	2	yes	—
28	C₂×D₄	—	2	2	no	—
29	C₂×D₄	—	2	2	no	—
30	C₂×D₄	C₂³	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	D₄	1	2	yes	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	C₂³	1	4	yes	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	C₂³	1	4	yes	—
40	C₂²⋊C₄	—	2	2	no	—
41	C₂²⋊C₄	—	2	4	no	—
42	C₄²	—	2	2	no	—
43	C₄²	D₄	1	2	yes	—
44	C₄²	C₂³	1	1	yes	characteristic
45	C₄²	C₂³	1	1	yes	characteristic
46	C₂³	—	2	2	no	—
47	C₂³	C₂×D₄	1	1	yes	characteristic
48	Q₈	—	2	4	no	—
49	Q₈	—	4	4	no	—
50	D₄	—	2	4	no	—
51	D₄	—	4	4	no	—
52	D₄	—	4	4	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄	#41	1	4	yes	—
61	C₂×C₄	C₂⁴	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	4	2	no	—
65	C₂²	—	4	4	no	—
66	C₂²	#124	1	2	yes	—
67	C₂²	C₂²×D₄	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	4	2	no	—
70	C₄	—	4	4	no	—
71	C₄	#124	1	2	yes	—
72	C₄	#135	1	4	yes	—
73	C₄	C₂×#41	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	—	8	2	no	—
76	C₂	C₂×#124	1	1	yes	characteristic
77	C₂	C₂×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
prime radical	#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #533	C₂	semidirect
3. #488	C₂	semidirect
4. #487	C₂	semidirect
5. #486	C₂	semidirect
6. #416	C₂	semidirect
7. C₂×C₄×C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#532	2
#488	2
#487	2
#486	2
#416	2
C₂×C₄×C₈	2