Subgroups — #5008

Order 192.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	#869	C₂	1	1	yes	maximal, characteristic, complemented
3	#832	C₂	1	2	yes	maximal
4	#808	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#476	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄	—	2	1	no	—
9	C₂×Dic₆	—	2	1	no	—
10	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₃×C₂²⋊C₄	—	2	1	no	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₃	—	2	1	no	—
14	C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	—	3	1	no	—
17	C₄×D₄	D₃	1	1	yes	characteristic
18	#106	—	3	2	no	—
19	C₄⋊C₈	—	3	1	no	—
20	#102	—	3	2	no	—
21	C₂³×C₃	—	2	1	no	—
22	C₃×D₄	—	2	1	no	—
23	C₃×D₄	Q₈	1	2	yes	complemented
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	—	4	2	no	—
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×Dic₃	—	2	2	no	—
30	Dic₆	—	4	2	no	—
31	C₃⋊C₈	—	2	2	no	—
32	C₂×Q₈	—	6	1	no	—
33	C₂×D₄	C₂×D₃	1	1	yes	characteristic
34	C₂²×C₄	—	2	1	no	—
35	C₂×C₈	—	3	2	no	—
36	C₄⋊C₄	—	3	2	no	—
37	C₄⋊C₄	—	6	1	no	—
38	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²⋊C₄	—	2	1	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	—	2	2	no	—
42	C₂²×C₃	—	4	1	no	—
43	C₂²×C₃	—	4	1	no	—
44	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	4	1	no	—
47	C₃×C₄	SD₁₆	1	2	yes	—
48	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₃×C₄	#41	1	1	yes	characteristic
50	Dic₃	—	4	2	no	—
51	C₂³	—	2	1	no	—
52	Q₈	—	12	2	no	—
53	D₄	—	2	1	no	—
54	D₄	Dic₆	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₂×C₄	#67	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	—	6	2	no	—
62	C₂×C₃	—	4	2	no	—
63	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	C₂×#67	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	12	2	no	—
73	C₄	#214	1	2	yes	—
74	C₄	C₂×Dic₆	1	1	yes	characteristic
75	C₄	#235	1	1	yes	characteristic
76	C₃	#476	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	4	2	no	—
78	C₂	#924	1	1	yes	characteristic
79	C₂	C₂×#214	1	1	yes	characteristic
80	C₂	#949	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×D₄	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×C₃×D₄	C₂²
hypercenter	C₄×D₄	D₃
hypocenter	C₃	#476
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #869	C₂	semidirect
4. #808	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
23. C₃×D₄	Q₈	semidirect
54. D₄	Dic₆	semidirect
76. C₃	#476	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄	2
#869	2
#832	2
#808	2
C₄⋊(C₃⋊C₈)	2
#476	3