Subgroups — #2981

Order 128.

201 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532	C₂	1	1	yes	maximal, characteristic
2	#436	C₂	1	2	yes	maximal, complemented
3	#435	C₂	1	4	yes	maximal, complemented
4	C₈×D₄	C₂	1	2	yes	maximal, complemented
5	#432	C₂	1	1	yes	maximal, characteristic, complemented
6	#431	C₂	1	1	yes	maximal, characteristic, complemented
7	#408	C₂	1	4	yes	maximal, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	C₂×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₈	C₂²	1	4	yes	—
11	C₄⋊Q₈	C₄	1	1	yes	characteristic
12	#127	C₄	1	1	yes	characteristic
13	#124	C₄	1	2	yes	—
14	#121	C₄	1	2	yes	—
15	#121	C₄	1	2	yes	—
16	C₄×D₄	C₂²	1	2	yes	—
17	C₄×D₄	C₂²	1	2	yes	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	C₄⋊C₈	C₂²	1	2	yes	—
20	C₄⋊C₈	C₂²	1	2	yes	—
21	C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂²⋊C₈	C₂²	1	4	yes	—
24	C₂²⋊C₈	C₂²	1	4	yes	—
25	#97	C₂²	1	2	yes	—
26	C₄×C₈	C₂²	1	1	yes	characteristic
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	#41	—	4	4	no	—
29	C₂×Q₈	C₂×C₄	1	2	yes	—
30	C₂×D₄	C₂×C₄	1	2	yes	—
31	C₂×D₄	C₂×C₄	1	2	yes	—
32	C₂×D₄	C₂×C₄	1	4	yes	—
33	C₂×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄	C₂³	1	1	yes	characteristic
37	C₂²×C₄	C₂³	1	4	yes	—
38	M₁₆	—	4	4	no	—
39	C₂×C₈	—	4	4	no	—
40	C₂×C₈	C₂³	1	2	yes	—
41	C₂×C₈	C₂³	1	2	yes	—
42	C₂×C₈	C₂³	1	4	yes	—
43	C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄	C₂×C₄	1	4	yes	—
47	C₄²	C₂×C₄	1	1	yes	characteristic
48	C₄²	C₂×C₄	1	1	yes	characteristic
49	C₄²	C₂³	1	1	yes	characteristic
50	C₄²	C₂³	1	1	yes	characteristic
51	C₂³	C₂²×C₄	1	4	yes	—
52	C₂³	C₂²×C₄	1	1	yes	characteristic
53	Q₈	—	4	2	no	—
54	D₄	—	2	4	no	—
55	D₄	—	4	2	no	—
56	D₄	—	4	2	no	—
57	D₄	—	4	4	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×C₄	1	4	yes	—
68	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄	C₂⁴	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	4	no	—
72	C₈	—	2	2	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	1	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂³×C₄	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	4	no	—
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	#131	1	2	yes	—
86	C₄	H(5,𝔽₂)	1	2	yes	—
87	C₂	—	4	1	no	—
88	C₂	—	4	4	no	—
89	C₂	#518	1	1	yes	characteristic
90	C₂	C₂×#131	1	1	yes	characteristic
91	C₂	#568	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	#532	C₂
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. #436	C₂	semidirect
3. #435	C₂	semidirect
4. C₈×D₄	C₂	semidirect
5. #432	C₂	semidirect
6. #431	C₂	semidirect
7. #408	C₂	semidirect

Maximal subgroups

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

subgroup	index
#532	2
#436	2
#435	2
C₈×D₄	2
#432	2
#431	2
#408	2