Subgroups — #4964

Order 192.

118 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	#926	C₂	1	1	yes	maximal, characteristic
3	C₂×#208	C₂	1	1	yes	maximal, characteristic
4	#808	C₂	1	2	yes	maximal, complemented
5	#807	C₂	1	2	yes	maximal, complemented
6	#428	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	—	2	1	no	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₄	—	2	1	no	—
10	C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	#217	—	2	1	no	—
12	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₃	—	2	1	no	—
15	#208	C₄	1	4	yes	complemented
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×M₁₆	—	3	1	no	—
18	#117	—	3	1	no	—
19	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
20	#107	—	3	2	no	—
21	#106	—	3	2	no	—
22	C₂³×C₃	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	4	2	no	—
25	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	Q₈	1	2	yes	—
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×Dic₃	—	2	1	no	—
30	C₂×Dic₃	—	2	1	no	—
31	C₃⋊C₈	C₂×C₄	1	4	yes	complemented
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
34	M₁₆	—	3	4	no	—
35	C₂×C₈	—	3	2	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	3	1	no	—
38	C₄⋊C₄	—	3	1	no	—
39	C₄⋊C₄	C₂×D₃	1	2	yes	—
40	C₂²⋊C₄	—	6	1	no	—
41	C₄²	—	6	1	no	—
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	—	2	1	no	—
44	C₂²×C₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄	—	4	2	no	—
47	C₃×C₄	C₄⋊C₄	1	2	yes	—
48	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
50	Dic₃	—	4	1	no	—
51	Dic₃	—	4	1	no	—
52	C₂³	#67	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	Dic₆	1	2	yes	—
58	C₂×C₄	C₄×D₃	1	2	yes	—
59	C₂×C₄	#67	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	—	3	4	no	—
62	C₂×C₃	—	2	2	no	—
63	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₃	#137	1	1	yes	characteristic
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Dic₃⋊C₄	1	2	yes	—
69	C₂²	C₂×#67	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	12	1	no	—
72	C₄	—	12	1	no	—
73	C₄	Dic₃⋊C₄	1	2	yes	—
74	C₄	C₂×Dic₆	1	1	yes	characteristic
75	C₄	C₂×C₄×D₃	1	1	yes	characteristic
76	C₃	#428	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	2	no	—
78	C₂	C₂×Dic₃⋊C₄	1	1	yes	characteristic
79	C₂	#932	1	1	yes	characteristic
80	C₂	#942	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#428
LCS	G ▷ 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. #808	C₂	semidirect
5. #807	C₂	semidirect
15. #208	C₄	semidirect
31. C₃⋊C₈	C₂×C₄	semidirect
76. C₃	#428	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
#926	2
C₂×#208	2
#808	2
#807	2
#428	3