Subgroups — #5516

Order 192.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#117	C₂	1	1	yes	maximal, characteristic
3	#926	C₂	1	1	yes	maximal, characteristic
4	#887	C₂	1	4	yes	maximal, complemented
5	#877	C₂	1	4	yes	maximal
6	C₄×Dic₆	C₂	1	4	yes	maximal, complemented
7	#519	—	3	1	no	maximal, Sylow
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃	C₂²	1	2	yes	—
10	C₂×Dic₆	C₄	1	8	yes	complemented
11	C₂×Dic₆	C₂²	1	2	yes	—
12	C₂×Dic₆	C₂²	1	2	yes	—
13	C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄²	C₂²	1	2	yes	—
16	#217	C₂²	1	2	yes	—
17	C₄⋊Dic₃	C₂²	1	2	yes	—
18	Dic₃⋊C₄	C₂²	1	8	yes	—
19	C₄×Dic₃	C₂²	1	8	yes	—
20	C₄×Dic₃	C₂²	1	2	yes	—
21	C₂²×Q₈	—	3	1	no	—
22	C₄×Q₈	—	3	4	no	—
23	C₄×Q₈	—	3	4	no	—
24	#117	—	3	1	no	—
25	#117	—	3	4	no	—
26	#117	D₃	1	1	yes	characteristic
27	C₂³×C₃	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄	C₂³	1	4	yes	—
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	C₂×C₄	1	8	yes	—
33	C₂×Dic₃	C₂³	1	4	yes	—
34	C₂×Dic₃	C₂³	1	4	yes	—
35	Dic₆	C₂×C₄	1	8	yes	—
36	Dic₆	C₂×C₄	1	8	yes	—
37	C₂×Q₈	—	3	8	no	—
38	C₂×Q₈	—	3	2	no	—
39	C₂×Q₈	—	3	2	no	—
40	C₂²×C₄	—	3	2	no	—
41	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₄⋊C₄	—	3	8	no	—
43	C₄⋊C₄	—	3	2	no	—
44	C₄⋊C₄	C₂×D₃	1	2	yes	—
45	C₂²⋊C₄	—	3	2	no	—
46	C₂²⋊C₄	C₂×D₃	1	2	yes	—
47	C₄²	—	3	8	no	—
48	C₄²	—	3	2	no	—
49	C₄²	C₂×D₃	1	2	yes	—
50	C₂²×C₃	—	2	2	no	—
51	C₂²×C₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₃	C₂⁴	1	1	yes	characteristic
53	C₃×C₄	—	2	4	no	—
54	C₃×C₄	C₂²×C₄	1	2	yes	—
55	C₃×C₄	C₂²×C₄	1	2	yes	—
56	Dic₃	—	2	4	no	—
57	Dic₃	C₂²×C₄	1	8	yes	—
58	C₂³	C₂²×D₃	1	1	yes	characteristic
59	Q₈	—	3	8	no	—
60	Q₈	—	3	8	no	—
61	C₂×C₄	—	3	8	no	—
62	C₂×C₄	—	3	4	no	—
63	C₂×C₄	—	3	4	no	—
64	C₂×C₄	C₄×D₃	1	4	yes	—
65	C₂×C₄	C₂²×D₃	1	4	yes	—
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₃	2₋¹⁺⁴	1	2	yes	—
71	C₂²	—	2	2	no	—
72	C₂²	C₂×C₄×D₃	1	2	yes	—
73	C₂²	C₂³×D₃	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	3	8	no	—
76	C₄	—	6	4	no	—
77	C₄	C₂×C₄×D₃	1	2	yes	—
78	C₄	C₂×C₄×D₃	1	2	yes	—
79	C₃	#519	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₃	1	1	yes	characteristic
82	C₂	#1010	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting	C₃×#117	C₂
socle, cosocle	C₂²×C₃	C₂⁴
prime radical	C₂³×C₃	C₂³
hypercenter	#117	D₃
hypocenter	C₃	#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
4. #887	C₂	semidirect
6. C₄×Dic₆	C₂	semidirect
10. C₂×Dic₆	C₄	semidirect
79. C₃	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆	2
C₃×#117	2
#926	2
#887	2
#877	2
C₄×Dic₆	2
#519	3