Subgroups — #4555

Order 192.

110 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	#926	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#339	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃	—	2	1	no	—
7	C₃×M₁₆	—	2	1	no	—
8	C₃×M₁₆	C₄	1	2	yes	complemented
9	C₂×C₃×C₈	—	2	1	no	—
10	#217	—	2	1	no	—
11	C₄⋊Dic₃	C₄	1	2	yes	complemented
12	C₄×Dic₃	—	2	1	no	—
13	C₄×Dic₃	—	2	1	no	—
14	C₄×Dic₃	C₄	1	2	yes	complemented
15	C₂×M₁₆	D₃	1	1	yes	characteristic
16	#117	—	3	1	no	—
17	C₂×C₄²	—	3	1	no	—
18	C₂³×C₃	D₄	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	Q₈	1	1	yes	characteristic
25	C₂×Dic₃	—	2	4	no	—
26	C₂×Dic₃	—	2	2	no	—
27	C₂×Dic₃	—	2	2	no	—
28	C₃×C₈	—	2	2	no	—
29	C₂²×C₄	—	6	1	no	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	M₁₆	—	2	1	no	—
32	M₁₆	Dic₃	1	2	yes	complemented
33	C₂×C₈	—	2	1	no	—
34	C₄⋊C₄	—	3	2	no	—
35	C₂²⋊C₄	—	6	1	no	—
36	C₄²	—	3	2	no	—
37	C₄²	—	6	1	no	—
38	C₄²	—	6	1	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
43	C₃×C₄	C₄²	1	1	yes	characteristic
44	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
47	Dic₃	—	2	4	no	—
48	Dic₃	—	4	2	no	—
49	C₂³	D₁₂	1	1	yes	characteristic
50	C₂×C₄	—	6	4	no	—
51	C₂×C₄	—	6	2	no	—
52	C₂×C₄	—	6	2	no	—
53	C₂×C₄	Dic₆	1	1	yes	characteristic
54	C₂×C₄	C₄×D₃	1	1	yes	characteristic
55	C₂×C₄	C₄×D₃	1	1	yes	characteristic
56	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₄	#67	1	1	yes	characteristic
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₈	—	2	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	#104	1	2	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
66	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
68	C₄	—	6	4	no	—
69	C₄	—	12	2	no	—
70	C₄	C₄×Dic₃	1	1	yes	characteristic
71	C₄	Dic₃⋊C₄	1	1	yes	characteristic
72	C₄	Dic₃⋊C₄	1	1	yes	characteristic
73	C₄	#217	1	1	yes	characteristic
74	C₃	#339	1	1	yes	characteristic, complemented, Sylow
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	#825	1	2	yes	—
78	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	C₂²⋊C₄
prime radical	C₂³×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#339
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
8. C₃×M₁₆	C₄	semidirect
11. C₄⋊Dic₃	C₄	semidirect
14. C₄×Dic₃	C₄	semidirect
32. M₁₆	Dic₃	semidirect
74. C₃	#339	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
#926	2
C₂×C₄×Dic₃	2
#339	3