Subgroups — #3184

Order 128.

223 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532	C₂	1	2	yes	maximal, complemented
2	#498	C₂	1	1	yes	maximal, characteristic, complemented
3	#493	C₂	1	1	yes	maximal, characteristic, complemented
4	#486	C₂	1	2	yes	maximal, complemented
5	#469	C₂	1	4	yes	maximal, complemented
6	#442	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	—	2	2	no	—
9	C₂×D₈	C₂²	1	4	yes	—
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#127	C₂²	1	2	yes	—
13	#124	—	2	2	no	—
14	#121	—	2	2	no	—
15	#121	C₂²	1	4	yes	—
16	C₄×D₄	—	2	2	no	—
17	C₄×D₄	C₂²	1	4	yes	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#106	C₂²	1	4	yes	—
20	#102	C₂²	1	8	yes	—
21	#97	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₄×C₈	C₂²	1	1	yes	characteristic
24	#41	—	4	8	no	—
25	C₂×Q₈	—	2	2	no	—
26	C₂×D₄	—	2	4	no	—
27	C₂×D₄	—	2	2	no	—
28	C₂×D₄	—	2	2	no	—
29	C₂×D₄	C₂³	1	4	yes	—
30	C₂²×C₄	—	2	4	no	—
31	C₂²×C₄	D₄	1	2	yes	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	D₈	—	2	8	no	—
34	M₁₆	—	2	4	no	—
35	C₂×C₈	C₂³	1	2	yes	—
36	C₂×C₈	C₂³	1	2	yes	—
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	C₂³	1	4	yes	—
39	C₂²⋊C₄	—	2	4	no	—
40	C₂²⋊C₄	—	2	4	no	—
41	C₄²	D₄	1	2	yes	—
42	C₄²	C₂³	1	1	yes	characteristic
43	C₄²	C₂³	1	1	yes	characteristic
44	C₂³	—	2	4	no	—
45	C₂³	C₂×D₄	1	1	yes	characteristic
46	Q₈	—	4	4	no	—
47	D₄	—	2	8	no	—
48	D₄	—	4	8	no	—
49	D₄	—	4	4	no	—
50	D₄	—	4	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	4	8	no	—
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂⁴	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	#41	1	4	yes	complemented
61	C₂²	—	2	1	no	—
62	C₂²	—	2	2	no	—
63	C₂²	—	4	4	no	—
64	C₂²	—	4	4	no	—
65	C₂²	—	4	4	no	—
66	C₂²	C₂²×D₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	2	no	—
69	C₄	—	4	4	no	—
70	C₄	AGL(1,ℤ/8)	1	4	yes	—
71	C₄	C₂×#41	1	2	yes	—
72	C₂	—	4	1	no	—
73	C₂	—	8	4	no	—
74	C₂	#532	1	1	yes	characteristic
75	C₂	C₂×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #532	C₂	semidirect
2. #498	C₂	semidirect
3. #493	C₂	semidirect
4. #486	C₂	semidirect
5. #469	C₂	semidirect
6. #442	C₂	semidirect
7. C₄×M₁₆	C₂	semidirect
60. C₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#532	2
#498	2
#493	2
#486	2
#469	2
#442	2
C₄×M₁₆	2