Subgroups — #5060

Order 192.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#216	C₂	1	2	yes	maximal, complemented
3	C₂×#214	C₂	1	2	yes	maximal
4	#869	C₂	1	1	yes	maximal, characteristic, complemented
5	#803	C₂	1	1	yes	maximal, characteristic, complemented
6	#497	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₆	C₂²	1	2	yes	—
10	C₃×C₂²⋊C₄	—	2	2	no	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	#216	—	2	4	no	—
13	#214	—	2	4	no	—
14	C₄⋊Dic₃	—	2	2	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×Q₁₆	—	3	2	no	—
17	C₂×SD₁₆	—	3	2	no	—
18	C₄⋊Q₈	—	3	1	no	—
19	#124	D₃	1	1	yes	characteristic
20	#97	—	3	1	no	—
21	C₂³×C₃	—	2	1	no	—
22	C₃×Q₈	—	2	2	no	—
23	C₃×D₄	—	2	2	no	—
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	D₄	1	2	yes	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	2	no	—
28	Dic₆	—	2	2	no	—
29	Dic₆	—	2	2	no	—
30	C₃⋊C₈	D₄	1	4	yes	—
31	C₂×Q₈	—	3	2	no	—
32	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
33	C₂×D₄	C₂×D₃	1	1	yes	characteristic
34	Q₁₆	—	6	4	no	—
35	SD₁₆	—	6	4	no	—
36	C₂×C₈	—	3	2	no	—
37	C₄⋊C₄	—	6	2	no	—
38	C₂²⋊C₄	—	2	2	no	—
39	C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	—	4	1	no	—
41	C₂²×C₃	—	4	2	no	—
42	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄	—	2	2	no	—
44	C₃×C₄	—	4	1	no	—
45	C₃×C₄	C₂×D₄	1	2	yes	—
46	Dic₃	—	4	2	no	—
47	C₂³	—	2	1	no	—
48	Q₈	—	2	2	no	—
49	Q₈	—	6	2	no	—
50	Q₈	—	6	2	no	—
51	D₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	#67	1	2	yes	—
55	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₈	—	3	4	no	—
57	C₂×C₃	—	8	1	no	—
58	C₂×C₃	#127	1	1	yes	characteristic
59	C₂×C₃	#137	1	2	yes	—
60	C₂²	—	4	1	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×#67	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	4	1	no	—
65	C₄	—	12	2	no	—
66	C₄	D₃×D₄	1	2	yes	—
67	C₃	#497	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	8	1	no	—
69	C₂	#940	1	1	yes	characteristic
70	C₂	#951	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#124	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×C₃×D₄	C₂²
hypercenter	#124	D₃
hypocenter	C₃	#497
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. C₂×#216	C₂	semidirect
4. #869	C₂	semidirect
5. #803	C₂	semidirect
67. C₃	#497	semidirect

Maximal subgroups

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

subgroup	index
C₃×#124	2
C₂×#216	2
C₂×#214	2
#869	2
#803	2
#497	3