Subgroups — #4558

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#341	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₃×M₁₆	—	2	1	no	—
7	C₃×M₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₈	—	2	1	no	—
9	#208	—	2	1	no	—
10	#208	C₄	1	2	yes	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₂×C₃⋊C₈	—	2	1	no	—
13	C₂×C₃⋊C₈	—	2	1	no	—
14	C₂×C₃⋊C₈	C₄	1	2	yes	—
15	C₂×M₁₆	—	3	1	no	—
16	C₂×M₁₆	D₃	1	1	yes	characteristic
17	C₂²×C₈	—	3	1	no	—
18	C₂³×C₃	Q₈	1	1	yes	characteristic
19	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₃×C₈	—	2	2	no	—
26	C₃⋊C₈	—	2	2	no	—
27	C₃⋊C₈	—	2	2	no	—
28	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
29	M₁₆	—	2	1	no	—
30	M₁₆	—	3	2	no	—
31	M₁₆	—	6	1	no	—
32	M₁₆	Dic₃	1	2	yes	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	3	2	no	—
35	C₂×C₈	—	6	1	no	—
36	C₂×C₈	—	6	1	no	—
37	C₂×C₈	—	6	1	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
42	C₃×C₄	C₄²	1	1	yes	characteristic
43	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂³	Dic₆	1	1	yes	characteristic
47	C₂×C₄	C₄×D₃	1	1	yes	characteristic
48	C₂×C₄	C₄×D₃	1	1	yes	characteristic
49	C₂×C₄	D₁₂	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
51	C₂×C₄	#67	1	1	yes	characteristic
52	C₂×C₄	#67	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₈	—	6	2	no	—
55	C₈	—	6	2	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	#95	1	1	yes	characteristic
59	C₂×C₃	#108	1	2	yes	—
60	C₂²	—	2	2	no	—
61	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
62	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
63	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
64	C₄	C₄×Dic₃	1	1	yes	characteristic
65	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
66	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
67	C₄	#217	1	1	yes	characteristic
68	C₃	#341	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#822	1	2	yes	—
72	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂²×C₃	C₄⋊C₄
prime radical	C₂³×C₃	Q₈
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#341
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
68. C₃	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₁₆	2
C₂×#208	2
C₂²×C₃⋊C₈	2
#341	3