Subgroups — #5517

Order 192.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#117	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	#890	C₂	1	2	yes	maximal, complemented
5	#888	C₂	1	2	yes	maximal, complemented
6	#878	C₂	1	4	yes	maximal
7	C₄×Dic₆	C₂	1	4	yes	maximal, complemented
8	#533	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	C₂²	1	2	yes	—
11	C₂×Dic₆	C₂²	1	4	yes	—
12	C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₃×C₄²	C₂²	1	2	yes	—
15	#217	C₂²	1	2	yes	—
16	C₄⋊Dic₃	C₂²	1	4	yes	—
17	C₄⋊Dic₃	C₂²	1	2	yes	—
18	Dic₃⋊C₄	C₂²	1	4	yes	—
19	Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₃	C₂²	1	2	yes	—
21	C₄×Dic₃	C₂²	1	2	yes	—
22	C₄×Dic₃	C₂²	1	1	yes	characteristic
23	C₄×Dic₃	C₂²	1	1	yes	characteristic
24	C₄⋊Q₈	—	3	2	no	—
25	#125	—	3	2	no	—
26	C₂²⋊Q₈	—	3	4	no	—
27	C₄×Q₈	—	3	4	no	—
28	#117	—	3	1	no	—
29	#117	D₃	1	1	yes	characteristic
30	C₂×C₄²	—	3	1	no	—
31	C₂³×C₃	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₄	Q₈	1	4	yes	complemented
33	C₂×C₃×C₄	C₂³	1	4	yes	—
34	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂×Dic₃	—	2	4	no	—
37	C₂×Dic₃	C₂³	1	2	yes	—
38	C₂×Dic₃	C₂³	1	4	yes	—
39	C₂×Dic₃	C₂³	1	2	yes	—
40	Dic₆	—	2	8	no	—
41	C₂×Q₈	—	3	4	no	—
42	C₂²×C₄	—	3	2	no	—
43	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₄⋊C₄	—	3	2	no	—
45	C₄⋊C₄	—	3	4	no	—
46	C₄⋊C₄	—	3	4	no	—
47	C₄⋊C₄	—	3	4	no	—
48	C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂²⋊C₄	—	3	2	no	—
50	C₂²⋊C₄	C₂×D₃	1	2	yes	—
51	C₄²	—	3	1	no	—
52	C₄²	—	3	1	no	—
53	C₄²	—	3	2	no	—
54	C₄²	—	3	2	no	—
55	C₄²	C₂×D₃	1	2	yes	—
56	C₂²×C₃	—	2	2	no	—
57	C₂²×C₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₃	C₂⁴	1	1	yes	characteristic
59	C₃×C₄	—	2	4	no	—
60	C₃×C₄	C₂×Q₈	1	2	yes	—
61	C₃×C₄	C₂×Q₈	1	2	yes	—
62	Dic₃	—	2	4	no	—
63	Dic₃	—	2	2	no	—
64	Dic₃	#41	1	4	yes	—
65	C₂³	C₂²×D₃	1	1	yes	characteristic
66	Q₈	—	6	8	no	—
67	C₂×C₄	—	3	2	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	3	4	no	—
70	C₂×C₄	—	6	4	no	—
71	C₂×C₄	Dic₆	1	4	yes	complemented
72	C₂×C₄	C₂²×D₃	1	4	yes	—
73	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃	—	2	2	no	—
76	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₃	C₂×#41	1	2	yes	—
78	C₂²	—	2	2	no	—
79	C₂²	C₂×Dic₆	1	2	yes	—
80	C₂²	C₂³×D₃	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	3	4	no	—
83	C₄	—	6	2	no	—
84	C₄	—	6	4	no	—
85	C₄	C₂×Dic₆	1	2	yes	—
86	C₄	C₂×Dic₆	1	2	yes	—
87	C₃	#533	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₆	1	1	yes	characteristic
90	C₂	D₃×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₃×#117	C₂
prime radical	C₂³×C₃	C₂³
hypercenter	#117	D₃
hypocenter	C₃	#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #890	C₂	semidirect
5. #888	C₂	semidirect
7. C₄×Dic₆	C₂	semidirect
32. C₂×C₃×C₄	Q₈	semidirect
71. C₂×C₄	Dic₆	semidirect
87. C₃	#533	semidirect

Maximal subgroups

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

subgroup	index
C₃×#117	2
#926	2
C₂×C₄×Dic₃	2
#890	2
#888	2
#878	2
C₄×Dic₆	2
#533	3