Subgroups — #5121

Order 192.

118 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	#818	C₂	1	2	yes	maximal, complemented
5	#817	C₂	1	2	yes	maximal, complemented
6	#428	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	—	2	1	no	—
9	C₃×M₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₈	C₂²	1	2	yes	—
11	#217	—	2	1	no	—
12	C₄⋊Dic₃	—	2	1	no	—
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₃	—	2	1	no	—
17	C₂×M₁₆	D₃	1	1	yes	characteristic
18	#117	—	3	1	no	—
19	C₂×C₄⋊C₄	—	3	1	no	—
20	#107	—	3	2	no	—
21	#106	—	3	2	no	—
22	C₂³×C₃	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	Q₈	1	2	yes	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	C₂×Dic₃	—	2	1	no	—
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	—	4	2	no	—
31	C₃×C₈	C₂×C₄	1	4	yes	complemented
32	C₂²×C₄	—	6	1	no	—
33	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
34	M₁₆	Dic₃	1	4	yes	complemented
35	C₂×C₈	C₂×D₃	1	2	yes	—
36	C₄⋊C₄	—	3	1	no	—
37	C₄⋊C₄	—	3	2	no	—
38	C₄⋊C₄	—	3	1	no	—
39	C₄⋊C₄	—	6	1	no	—
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	C₄⋊C₄	1	2	yes	—
47	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
49	Dic₃	—	4	1	no	—
50	Dic₃	—	4	2	no	—
51	Dic₃	—	4	1	no	—
52	C₂³	D₁₂	1	1	yes	characteristic
53	C₂×C₄	—	6	1	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	—	6	2	no	—
56	C₂×C₄	—	12	2	no	—
57	C₂×C₄	Dic₆	1	2	yes	—
58	C₂×C₄	D₁₂	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₃	1	2	yes	—
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	C₂×Dic₃	1	4	yes	complemented
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	C₄⋊Dic₃	1	2	yes	—
69	C₂²	C₂×D₁₂	1	1	yes	characteristic
70	C₄	—	12	2	no	—
71	C₄	—	12	1	no	—
72	C₄	—	12	1	no	—
73	C₄	C₄⋊Dic₃	1	2	yes	—
74	C₄	C₂×Dic₆	1	1	yes	characteristic
75	C₄	C₂²×Dic₃	1	1	yes	characteristic
76	C₃	#428	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	2	no	—
78	C₂	#908	1	1	yes	characteristic
79	C₂	#909	1	1	yes	characteristic
80	C₂	C₂×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#428
LCS	G ▷ C₃×C₄ ▷ 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
4. #818	C₂	semidirect
5. #817	C₂	semidirect
9. C₃×M₁₆	C₄	semidirect
31. C₃×C₈	C₂×C₄	semidirect
34. M₁₆	Dic₃	semidirect
61. C₈	C₂×Dic₃	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×M₁₆	2
#926	2
C₂×C₄⋊Dic₃	2
#818	2
#817	2
#428	3