Subgroups — #1581

Order 128.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#335	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈	C₄	1	1	yes	characteristic
9	C₂×M₁₆	—	2	1	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₄×D₄	C₂²	1	1	yes	characteristic
12	C₄×D₄	C₂²	1	1	yes	characteristic
13	#107	C₄	1	1	yes	characteristic
14	C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	1	yes	characteristic
16	#102	C₄	1	1	yes	characteristic
17	#102	C₄	1	1	yes	characteristic
18	C₂²⋊C₈	—	2	1	no	—
19	C₂²⋊C₈	—	2	1	no	—
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄	—	2	1	no	—
26	C₂²×C₄	—	2	1	no	—
27	D₈	—	2	2	no	—
28	M₁₆	—	4	2	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	4	2	no	—
33	C₂×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈	D₄	1	1	yes	characteristic
36	C₂×C₈	D₄	1	1	yes	characteristic
37	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄	—	2	1	no	—
40	C₂²⋊C₄	—	2	1	no	—
41	C₄²	C₂³	1	1	yes	characteristic
42	C₂³	—	2	1	no	—
43	C₂³	—	2	1	no	—
44	D₄	—	2	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	2	1	no	—
47	D₄	M₁₆	1	2	yes	complemented
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	#41	1	1	yes	characteristic
57	C₈	—	2	1	no	—
58	C₈	—	2	1	no	—
59	C₈	—	2	1	no	—
60	C₈	—	4	1	no	—
61	C₈	—	4	1	no	—
62	C₈	—	4	1	no	—
63	C₈	D₈	1	2	yes	complemented
64	C₂²	—	2	2	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₄×D₄	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	C₂×M₁₆	1	1	yes	characteristic
75	C₄	#131	1	1	yes	characteristic
76	C₄	C₂×D₈	1	1	yes	characteristic
77	C₄	#135	1	1	yes	characteristic
78	C₂	—	4	2	no	—
79	C₂	—	8	1	no	—
80	C₂	#435	1	1	yes	characteristic
81	C₂	C₄×D₈	1	1	yes	characteristic
82	C₂	#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #436	C₂	semidirect
3. C₈×D₄	C₂	semidirect
4. #335	C₂	semidirect
5. D₄⋊C₈	C₂	semidirect
6. D₄⋊C₈	C₂	semidirect
7. #322	C₂	semidirect
47. D₄	M₁₆	semidirect
63. C₈	D₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈	2
#436	2
C₈×D₄	2
#335	2
D₄⋊C₈	2
#322	2