Subgroups — #5009

Order 192.

102 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	#870	C₂	1	1	yes	maximal, characteristic, complemented
3	#832	C₂	1	1	yes	maximal, characteristic
4	#832	C₂	1	1	yes	maximal, characteristic
5	#808	C₂	1	1	yes	maximal, characteristic, complemented
6	#807	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#478	—	3	1	no	maximal, Sylow
9	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄	—	2	1	no	—
11	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	Dic₃⋊C₄	—	2	1	no	—
17	Dic₃⋊C₄	—	2	1	no	—
18	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#125	—	3	1	no	—
21	C₄×D₄	D₃	1	1	yes	characteristic
22	#107	—	3	1	no	—
23	#106	—	3	1	no	—
24	C₄⋊C₈	—	3	1	no	—
25	#102	—	3	1	no	—
26	#102	—	3	1	no	—
27	C₂³×C₃	—	2	1	no	—
28	C₃×D₄	—	2	1	no	—
29	C₃×D₄	Q₈	1	2	yes	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	—	4	2	no	—
32	C₂×C₃×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×Dic₃	—	2	1	no	—
36	C₂×Dic₃	—	2	1	no	—
37	C₃⋊C₈	—	2	1	no	—
38	C₃⋊C₈	—	2	1	no	—
39	C₂×D₄	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₄	—	2	1	no	—
41	C₂×C₈	—	3	1	no	—
42	C₂×C₈	—	3	1	no	—
43	C₄⋊C₄	—	3	1	no	—
44	C₄⋊C₄	—	3	1	no	—
45	C₄⋊C₄	—	6	1	no	—
46	C₄⋊C₄	—	6	1	no	—
47	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃	—	2	2	no	—
51	C₂²×C₃	—	4	1	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₃×C₄	#41	1	1	yes	characteristic
59	Dic₃	—	4	1	no	—
60	Dic₃	—	4	1	no	—
61	C₂³	—	2	1	no	—
62	D₄	—	2	1	no	—
63	D₄	Dic₆	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	6	1	no	—
67	C₂×C₄	—	6	1	no	—
68	C₂×C₄	#67	1	1	yes	characteristic
69	C₂×C₄	#67	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₈	—	6	1	no	—
72	C₈	—	6	1	no	—
73	C₂×C₃	—	4	2	no	—
74	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₃	#135	1	1	yes	characteristic
76	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²	—	2	2	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	1	no	—
80	C₂²	C₂×#67	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	12	1	no	—
85	C₄	—	12	1	no	—
86	C₄	C₂×Dic₆	1	1	yes	characteristic
87	C₄	#235	1	1	yes	characteristic
88	C₃	#478	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	4	2	no	—
90	C₂	#924	1	1	yes	characteristic
91	C₂	#932	1	1	yes	characteristic
92	C₂	#950	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	#478
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. #870	C₂	semidirect
5. #808	C₂	semidirect
6. #807	C₂	semidirect
7. C₄⋊(C₃⋊C₈)	C₂	semidirect
88. C₃	#478	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
#870	2
#832	2
#808	2
#807	2
C₄⋊(C₃⋊C₈)	2
#478	3