Subgroups — #5570

Order 192.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃	C₂	1	2	yes	maximal
3	#890	C₂	1	6	yes	maximal
4	#869	C₂	1	3	yes	maximal
5	C₄×Dic₆	C₂	1	3	yes	maximal
6	#557	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×Dic₆	C₂²	1	3	yes	—
9	C₃×C₄⋊C₄	C₂²	1	3	yes	—
10	C₃×C₄²	C₂²	1	3	yes	—
11	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃	C₂²	1	6	yes	—
13	C₄⋊Dic₃	C₂²	1	6	yes	—
14	Dic₃⋊C₄	C₂²	1	6	yes	—
15	C₄×Dic₃	C₂²	1	6	yes	—
16	C₄⋊Q₈	—	3	3	no	—
17	#125	—	3	6	no	—
18	C₄×Q₈	—	3	2	no	—
19	C₄×Q₈	—	3	3	no	—
20	C₄×Q₈	D₃	1	1	yes	characteristic
21	C₃×Q₈	Q₈	1	4	yes	complemented
22	C₂×C₃×C₄	C₂³	1	3	yes	—
23	C₂×C₃×C₄	C₂³	1	3	yes	—
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	C₂³	1	2	yes	—
26	C₂×Dic₃	C₂³	1	6	yes	—
27	Dic₆	—	2	6	no	—
28	C₂×Q₈	—	3	3	no	—
29	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
30	C₄⋊C₄	—	3	1	no	—
31	C₄⋊C₄	—	3	6	no	—
32	C₄⋊C₄	—	3	6	no	—
33	C₄⋊C₄	—	3	6	no	—
34	C₄⋊C₄	C₂×D₃	1	3	yes	—
35	C₄²	—	3	6	no	—
36	C₄²	C₂×D₃	1	3	yes	—
37	C₂²×C₃	C₂⁴	1	1	yes	characteristic
38	C₃×C₄	—	2	3	no	—
39	C₃×C₄	C₂×Q₈	1	6	yes	—
40	C₃×C₄	#41	1	2	yes	—
41	Dic₃	—	2	6	no	—
42	Dic₃	—	2	2	no	—
43	Q₈	—	6	6	no	—
44	Q₈	Dic₆	1	4	yes	complemented
45	C₂×C₄	—	3	2	no	—
46	C₂×C₄	—	3	6	no	—
47	C₂×C₄	C₂²×D₃	1	3	yes	—
48	C₂×C₄	C₂²×D₃	1	3	yes	—
49	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
50	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₃	C₂×#41	1	1	yes	characteristic
52	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
53	C₂²	C₂³×D₃	1	1	yes	characteristic
54	C₄	—	2	3	no	—
55	C₄	—	6	6	no	—
56	C₄	—	6	2	no	—
57	C₄	C₂×Dic₆	1	6	yes	—
58	C₄	#239	1	2	yes	—
59	C₃	#557	1	1	yes	characteristic, complemented, Sylow
60	C₂	C₂²×Dic₆	1	1	yes	characteristic
61	C₂	C₂×#239	1	1	yes	characteristic
62	C₂	#1010	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄×Q₈	C₂
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₃×Q₈	Q₈	semidirect
44. Q₈	Dic₆	semidirect
59. C₃	#557	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈	2
Q₈×Dic₃	2
#890	2
#869	2
C₄×Dic₆	2
#557	3