Subgroups — #5116

Order 192.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	#924	C₂	1	2	yes	maximal
3	C₂×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#818	C₂	1	1	yes	maximal, characteristic, complemented
5	#816	C₂	1	2	yes	maximal, complemented
6	#467	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₈	—	2	2	no	—
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	#217	—	2	2	no	—
13	C₄⋊Dic₃	C₂²	1	2	yes	—
14	Dic₃⋊C₄	—	2	2	no	—
15	Dic₁₂	—	2	2	no	—
16	C₂×Q₁₆	—	3	1	no	—
17	C₂²×C₈	D₃	1	1	yes	characteristic
18	C₂²⋊Q₈	—	3	2	no	—
19	#107	—	3	1	no	—
20	#103	—	3	2	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	—	2	1	no	—
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	—	2	2	no	—
27	C₂×Dic₃	—	2	2	no	—
28	Dic₆	—	2	2	no	—
29	Dic₆	—	2	2	no	—
30	C₃×C₈	—	2	1	no	—
31	C₃×C₈	D₄	1	2	yes	—
32	C₂×Q₈	—	3	2	no	—
33	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
34	Q₁₆	—	6	2	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	C₂×D₃	1	1	yes	characteristic
37	C₂×C₈	C₂×D₃	1	1	yes	characteristic
38	C₄⋊C₄	—	3	2	no	—
39	C₄⋊C₄	—	6	2	no	—
40	C₂²⋊C₄	—	6	2	no	—
41	C₂²×C₃	—	2	1	no	—
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	Q₁₆	1	2	yes	complemented
44	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄	#41	1	1	yes	characteristic
48	Dic₃	—	4	2	no	—
49	Dic₃	—	4	2	no	—
50	C₂³	D₁₂	1	1	yes	characteristic
51	Q₈	—	6	2	no	—
52	Q₈	—	6	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	6	2	no	—
56	C₂×C₄	—	6	2	no	—
57	C₂×C₄	D₁₂	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	#67	1	2	yes	—
61	C₂×C₃	—	2	2	no	—
62	C₂×C₃	#121	1	1	yes	characteristic
63	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₃	#135	1	1	yes	characteristic
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	Dic₁₂	1	2	yes	complemented
68	C₂²	C₂×D₁₂	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	12	2	no	—
71	C₄	—	12	2	no	—
72	C₄	#235	1	1	yes	characteristic
73	C₄	C₂×#67	1	1	yes	characteristic
74	C₃	#467	1	1	yes	characteristic, complemented, Sylow
75	C₂	—	2	2	no	—
76	C₂	#904	1	1	yes	characteristic
77	C₂	C₂×Dic₁₂	1	1	yes	characteristic
78	C₂	#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#467
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₁₂	C₂	semidirect
4. #818	C₂	semidirect
5. #816	C₂	semidirect
43. C₂²×C₃	Q₁₆	semidirect
67. C₂²	Dic₁₂	semidirect
74. C₃	#467	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈	2
#924	2
C₂×Dic₁₂	2
#818	2
#816	2
#467	3