Subgroups — #5572

Order 192.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#892	C₂	1	3	yes	maximal
5	#887	C₂	1	3	yes	maximal, complemented
6	#872	C₂	1	3	yes	maximal
7	C₄×Dic₆	C₂	1	3	yes	maximal, complemented
8	#519	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	D₃×Q₈	C₄	1	8	yes	complemented
11	C₂×C₄×D₃	C₂²	1	3	yes	—
12	C₂×Dic₆	C₂²	1	3	yes	—
13	C₃×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²	C₂²	1	3	yes	—
15	(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
16	(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
17	C₄⋊Dic₃	C₂²	1	3	yes	—
18	Dic₃⋊C₄	C₂²	1	3	yes	—
19	Dic₃⋊C₄	C₂²	1	3	yes	—
20	C₄×Dic₃	C₂²	1	6	yes	—
21	C₄×Dic₃	C₂²	1	3	yes	—
22	C₂²×Q₈	—	3	1	no	—
23	C₄×Q₈	—	3	1	no	—
24	C₄×Q₈	—	3	3	no	—
25	C₄×Q₈	—	3	3	no	—
26	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
27	#117	—	3	3	no	—
28	#117	—	3	3	no	—
29	C₂²×D₃	C₂³	1	1	yes	characteristic
30	C₃×Q₈	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₄	C₂³	1	3	yes	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂³	1	3	yes	—
34	C₂×Dic₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₃	C₂³	1	3	yes	—
36	C₂×Dic₃	C₂³	1	3	yes	—
37	C₄×D₃	C₂×C₄	1	12	yes	—
38	Dic₆	C₂×C₄	1	12	yes	—
39	C₂×Q₈	—	3	8	no	—
40	C₂×Q₈	—	3	3	no	—
41	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₄	—	3	3	no	—
43	C₄⋊C₄	—	3	3	no	—
44	C₄⋊C₄	—	3	3	no	—
45	C₄⋊C₄	—	3	3	no	—
46	C₄⋊C₄	C₂×D₃	1	3	yes	—
47	C₂²⋊C₄	—	3	3	no	—
48	C₂²⋊C₄	—	3	1	no	—
49	C₄²	—	3	6	no	—
50	C₄²	—	3	3	no	—
51	C₄²	C₂×D₃	1	3	yes	—
52	C₂²×C₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₃	—	2	1	no	—
54	C₂×D₃	—	2	1	no	—
55	C₂×D₃	C₂²×C₄	1	2	yes	—
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	—	2	3	no	—
58	C₃×C₄	C₂²×C₄	1	6	yes	—
59	Dic₃	—	2	3	no	—
60	Dic₃	—	2	1	no	—
61	Dic₃	C₂²×C₄	1	6	yes	—
62	C₂³	—	3	1	no	—
63	Q₈	—	3	12	no	—
64	Q₈	C₄×D₃	1	4	yes	—
65	C₂×C₄	—	3	3	no	—
66	C₂×C₄	—	3	12	no	—
67	C₂×C₄	—	3	3	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	C₂²×D₃	1	3	yes	—
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃	1	3	yes	—
72	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
74	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
75	D₃	—	2	2	no	—
76	C₂²	—	3	2	no	—
77	C₂²	—	6	1	no	—
78	C₂²	—	6	1	no	—
79	C₂²	C₂³×D₃	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	3	no	—
82	C₄	—	3	6	no	—
83	C₄	—	6	3	no	—
84	C₄	—	6	1	no	—
85	C₄	C₂×C₄×D₃	1	6	yes	—
86	C₃	#519	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	6	2	no	—
88	C₂	C₂²×C₄×D₃	1	1	yes	characteristic
89	C₂	#1007	1	1	yes	characteristic
90	C₂	#1010	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting	C₃×C₄×Q₈	C₂
socle, cosocle	C₂²×C₃	C₂⁴
prime radical	C₂²×D₃	C₂³
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₈	C₂	semidirect
3. Q₈×Dic₃	C₂	semidirect
5. #887	C₂	semidirect
7. C₄×Dic₆	C₂	semidirect
10. D₃×Q₈	C₄	semidirect
26. C₄×Q₈	D₃	semidirect
86. C₃	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₈	2
C₃×C₄×Q₈	2
Q₈×Dic₃	2
#892	2
#887	2
#872	2
C₄×Dic₆	2
#519	3