Subgroups — #4970

Order 192.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#924	C₂	1	1	yes	maximal, characteristic
3	#830	C₂	1	1	yes	maximal, characteristic
4	#810	C₂	1	2	yes	maximal, complemented
5	#807	C₂	1	2	yes	maximal, complemented
6	#484	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	—	2	1	no	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₆	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊C₄	—	2	1	no	—
11	C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	#217	—	2	1	no	—
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	Dic₃⋊C₄	—	2	1	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂²⋊Q₈	—	3	1	no	—
17	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
18	#107	—	3	2	no	—
19	#103	—	3	2	no	—
20	C₂²⋊C₈	—	3	1	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	4	2	no	—
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	C₂×Dic₃	—	2	1	no	—
29	Dic₆	—	2	2	no	—
30	C₃⋊C₈	—	2	2	no	—
31	C₂×Q₈	—	3	1	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₈	—	3	2	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	3	1	no	—
37	C₄⋊C₄	—	6	1	no	—
38	C₄⋊C₄	C₂×D₃	1	2	yes	—
39	C₂²⋊C₄	—	6	1	no	—
40	C₂²×C₃	—	2	1	no	—
41	C₂²×C₃	—	2	1	no	—
42	C₂²×C₃	Q₁₆	1	2	yes	—
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	4	2	no	—
46	C₃×C₄	#41	1	2	yes	—
47	Dic₃	—	4	1	no	—
48	Dic₃	—	4	1	no	—
49	C₂³	#67	1	1	yes	characteristic
50	Q₈	—	6	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	#67	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	6	2	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	#123	1	1	yes	characteristic
61	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#216	1	2	yes	—
66	C₂²	C₂×#67	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	2	no	—
69	C₄	—	12	1	no	—
70	C₄	—	12	1	no	—
71	C₄	#235	1	2	yes	—
72	C₃	#484	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	2	2	no	—
74	C₂	#929	1	1	yes	characteristic
75	C₂	#932	1	1	yes	characteristic
76	C₂	C₂×#216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#484
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #810	C₂	semidirect
5. #807	C₂	semidirect
72. C₃	#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
#924	2
#830	2
#810	2
#807	2
#484	3