Subgroups — #2080

Order 128.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#424	C₂	1	1	yes	maximal, characteristic, complemented
2	#421	C₂	1	1	yes	maximal, characteristic
3	#416	C₂	1	1	yes	maximal, characteristic
4	#413	C₂	1	1	yes	maximal, characteristic
5	#344	C₂	1	1	yes	maximal, characteristic, complemented
6	#341	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	#117	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	—	2	2	no	—
16	#104	—	2	2	no	—
17	#103	—	2	1	no	—
18	#102	—	2	1	no	—
19	#101	—	2	1	no	—
20	#100	—	2	1	no	—
21	#41	—	2	1	no	—
22	#41	—	2	1	no	—
23	#41	—	2	1	no	—
24	#41	—	2	1	no	—
25	C₂×Q₈	D₄	1	1	yes	characteristic
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	D₄	1	1	yes	characteristic
28	C₂²×C₄	—	2	1	no	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	M₁₆	—	2	1	no	—
31	M₁₆	—	2	1	no	—
32	M₁₆	—	2	1	no	—
33	M₁₆	—	2	1	no	—
34	M₁₆	—	2	1	no	—
35	M₁₆	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₂²⋊C₄	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	D₄	1	2	yes	—
48	C₂³	—	2	1	no	—
49	C₂³	#41	1	1	yes	characteristic
50	Q₈	—	2	1	no	—
51	Q₈	—	2	1	no	—
52	D₄	—	2	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	D₄	—	4	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	#41	1	1	yes	characteristic
64	C₂×C₄	#41	1	1	yes	characteristic
65	C₂×C₄	#41	1	1	yes	characteristic
66	C₂×C₄	#41	1	1	yes	characteristic
67	C₈	—	4	1	no	—
68	C₈	—	4	1	no	—
69	C₈	—	4	1	no	—
70	C₈	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#123	1	1	yes	characteristic
75	C₂²	#123	1	1	yes	characteristic
76	C₂²	#126	1	1	yes	characteristic
77	C₄	—	4	2	no	—
78	C₄	—	4	1	no	—
79	C₄	—	8	1	no	—
80	C₄	#121	1	1	yes	characteristic
81	C₄	#123	1	1	yes	characteristic
82	C₄	#124	1	1	yes	characteristic
83	C₄	#126	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	#397	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#397
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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
1. #424	C₂	semidirect
5. #344	C₂	semidirect
6. #341	C₂	semidirect
7. H(3,ℤ/4)	C₂	semidirect

Maximal subgroups

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

subgroup	index
#424	2
#421	2
#416	2
#413	2
#344	2
#341	2
H(3,ℤ/4)	2