Subgroups — #5229

Order 192.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃⋊#95	C₂	1	4	yes	maximal
5	#391	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₈	—	2	2	no	—
7	C₂×C₃×Q₈	—	2	2	no	—
8	C₂×C₃×Q₈	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃	C₂²	1	2	yes	—
12	C₂²×Dic₃	C₂²	1	2	yes	—
13	C₄⋊Dic₃	—	2	2	no	—
14	C₄×Dic₃	—	2	2	no	—
15	C₂²×Q₈	D₃	1	1	yes	characteristic
16	C₂×C₄⋊C₄	—	3	1	no	—
17	C₂×C₄²	—	3	1	no	—
18	#95	—	3	4	no	—
19	C₂³×C₃	C₂³	1	1	yes	characteristic
20	C₃×Q₈	—	2	8	no	—
21	C₂×C₃×C₄	—	2	2	no	—
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	D₄	1	2	yes	—
26	C₂×C₃×C₄	D₄	1	2	yes	—
27	C₂×Dic₃	—	2	2	no	—
28	C₂×Dic₃	—	2	2	no	—
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	—	2	4	no	—
31	C₂×Dic₃	Q₈	1	4	yes	complemented
32	C₂×Q₈	—	2	2	no	—
33	C₂×Q₈	—	2	2	no	—
34	C₂×Q₈	Dic₃	1	4	yes	complemented
35	C₂²×C₄	—	3	2	no	—
36	C₂²×C₄	—	3	2	no	—
37	C₂²×C₄	C₂×D₃	1	2	yes	—
38	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₄⋊C₄	—	6	2	no	—
40	C₄²	—	6	2	no	—
41	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₃	#41	1	2	yes	—
46	C₃×C₄	—	2	4	no	—
47	C₃×C₄	C₂²⋊C₄	1	4	yes	—
48	Dic₃	—	2	4	no	—
49	Dic₃	—	4	2	no	—
50	C₂³	C₂²×D₃	1	1	yes	characteristic
51	Q₈	—	2	8	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	3	4	no	—
55	C₂×C₄	—	6	2	no	—
56	C₂×C₄	—	6	2	no	—
57	C₂×C₄	—	6	4	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	C₂×Dic₃	1	4	yes	—
60	C₂×C₄	C₂×Dic₃	1	2	yes	—
61	C₂×C₄	#67	1	2	yes	—
62	C₂×C₄	#67	1	2	yes	—
63	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃	C₄×Q₈	1	2	yes	—
65	C₂×C₃	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₃	#124	1	1	yes	characteristic
67	C₂×C₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂²	D₃×Q₈	1	2	yes	—
69	C₂²	#239	1	2	yes	—
70	C₂²	C₂²×Dic₃	1	1	yes	characteristic
71	C₂²	C₂×#67	1	1	yes	characteristic
72	C₂²	C₂×#67	1	1	yes	characteristic
73	C₄	—	2	4	no	—
74	C₄	—	6	4	no	—
75	C₄	—	12	2	no	—
76	C₄	#217	1	4	yes	—
77	C₃	#391	1	1	yes	characteristic, complemented, Sylow
78	C₂	#944	1	1	yes	characteristic
79	C₂	Q₈×Dic₃	1	2	yes	—
80	C₂	#946	1	2	yes	—
81	C₂	#947	1	1	yes	characteristic
82	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄
Fitting	C₂²×C₃×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂³
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	#391
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
8. C₂×C₃×Q₈	C₄	semidirect
31. C₂×Dic₃	Q₈	semidirect
34. C₂×Q₈	Dic₃	semidirect
77. C₃	#391	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂×C₄⋊Dic₃	2
C₂×C₄×Dic₃	2
C₃⋊#95	2
#391	3