Subgroups — #4554

Order 192.

98 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×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#208	C₂	1	1	yes	maximal, characteristic
4	#343	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃	C₄	1	2	yes	—
7	C₃×M₁₆	—	2	1	no	—
8	C₃×M₁₆	C₄	1	2	yes	complemented
9	C₂×C₃×C₈	—	2	1	no	—
10	C₄⋊Dic₃	—	2	1	no	—
11	C₄⋊Dic₃	—	2	1	no	—
12	#208	—	2	1	no	—
13	#208	C₄	1	2	yes	complemented
14	C₂×C₃⋊C₈	—	2	1	no	—
15	C₂×M₁₆	—	3	1	no	—
16	C₂×M₁₆	D₃	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	3	1	no	—
18	C₂³×C₃	C₂×C₄	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	D₄	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	Q₈	1	1	yes	characteristic
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	—	4	2	no	—
27	C₃×C₈	—	2	2	no	—
28	C₃⋊C₈	—	2	2	no	—
29	C₂²×C₄	—	3	2	no	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	M₁₆	—	2	1	no	—
32	M₁₆	—	3	2	no	—
33	M₁₆	—	6	1	no	—
34	M₁₆	Dic₃	1	2	yes	complemented
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	6	1	no	—
37	C₄⋊C₄	—	6	1	no	—
38	C₄⋊C₄	—	6	1	no	—
39	C₂²×C₃	—	2	1	no	—
40	C₂²×C₃	—	2	1	no	—
41	C₂²×C₃	C₄²	1	1	yes	characteristic
42	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
44	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
48	Dic₃	—	4	2	no	—
49	C₂³	C₄×D₃	1	1	yes	characteristic
50	C₂×C₄	—	6	2	no	—
51	C₂×C₄	—	12	2	no	—
52	C₂×C₄	Dic₆	1	1	yes	characteristic
53	C₂×C₄	C₄×D₃	1	1	yes	characteristic
54	C₂×C₄	D₁₂	1	1	yes	characteristic
55	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₄	#67	1	1	yes	characteristic
57	C₂×C₄	#67	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	6	2	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	#95	1	1	yes	characteristic
63	C₂×C₃	#100	1	1	yes	characteristic
64	C₂×C₃	#101	1	1	yes	characteristic
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	C₄×Dic₃	1	1	yes	characteristic
68	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
69	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
70	C₄	—	12	2	no	—
71	C₄	Dic₃⋊C₄	1	1	yes	characteristic
72	C₄	C₄⋊Dic₃	1	1	yes	characteristic
73	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
74	C₄	#217	1	1	yes	characteristic
75	C₃	#343	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	#823	1	1	yes	characteristic
79	C₂	#824	1	1	yes	characteristic
80	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#343
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
8. C₃×M₁₆	C₄	semidirect
13. #208	C₄	semidirect
34. M₁₆	Dic₃	semidirect
75. C₃	#343	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×C₄⋊Dic₃	2
C₂×#208	2
#343	3