Subgroups — #5569

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	1	yes	maximal, characteristic
3	Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#890	C₂	1	3	yes	maximal
5	#888	C₂	1	3	yes	maximal
6	#870	C₂	1	3	yes	maximal
7	C₄×Dic₆	C₂	1	3	yes	maximal
8	#557	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₆	C₂²	1	3	yes	—
11	C₃×C₄⋊C₄	C₂²	1	3	yes	—
12	C₃×C₄²	C₂²	1	3	yes	—
13	C₄⋊Dic₃	C₂²	1	3	yes	—
14	C₄⋊Dic₃	C₂²	1	3	yes	—
15	C₄⋊Dic₃	C₂²	1	3	yes	—
16	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
17	Dic₃⋊C₄	C₂²	1	3	yes	—
18	Dic₃⋊C₄	C₂²	1	6	yes	—
19	C₄×Dic₃	C₂²	1	3	yes	—
20	C₄×Dic₃	C₂²	1	3	yes	—
21	C₄⋊Q₈	—	3	3	no	—
22	#125	—	3	3	no	—
23	#125	—	3	3	no	—
24	C₄×Q₈	—	3	1	no	—
25	C₄×Q₈	—	3	1	no	—
26	C₄×Q₈	—	3	3	no	—
27	C₄×Q₈	D₃	1	1	yes	characteristic
28	C₃×Q₈	Q₈	1	4	yes	complemented
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	3	yes	—
31	C₂×C₃×C₄	C₂³	1	3	yes	—
32	C₂×Dic₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₃	C₂³	1	3	yes	—
34	C₂×Dic₃	C₂³	1	3	yes	—
35	C₂×Dic₃	C₂³	1	1	yes	characteristic
36	Dic₆	—	2	6	no	—
37	C₂×Q₈	—	3	3	no	—
38	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
39	C₄⋊C₄	—	3	1	no	—
40	C₄⋊C₄	—	3	3	no	—
41	C₄⋊C₄	—	3	3	no	—
42	C₄⋊C₄	—	3	3	no	—
43	C₄⋊C₄	—	3	3	no	—
44	C₄⋊C₄	—	3	6	no	—
45	C₄⋊C₄	C₂×D₃	1	3	yes	—
46	C₄²	—	3	3	no	—
47	C₄²	—	3	3	no	—
48	C₄²	C₂×D₃	1	3	yes	—
49	C₂²×C₃	C₂⁴	1	1	yes	characteristic
50	C₃×C₄	—	2	3	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	C₂×Q₈	1	6	yes	—
53	Dic₃	—	2	3	no	—
54	Dic₃	—	2	1	no	—
55	Dic₃	—	2	3	no	—
56	Dic₃	#41	1	2	yes	—
57	Q₈	—	6	6	no	—
58	Q₈	Dic₆	1	4	yes	complemented
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	3	3	no	—
62	C₂×C₄	—	3	3	no	—
63	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃	1	3	yes	—
65	C₂×C₄	C₂²×D₃	1	3	yes	—
66	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
69	C₂²	C₂³×D₃	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	3	no	—
72	C₄	—	3	2	no	—
73	C₄	—	6	1	no	—
74	C₄	—	6	3	no	—
75	C₄	—	6	3	no	—
76	C₄	C₂×Dic₆	1	6	yes	—
77	C₃	#557	1	1	yes	characteristic, complemented, Sylow
78	C₂	C₂²×Dic₆	1	1	yes	characteristic
79	C₂	#1007	1	1	yes	characteristic
80	C₂	D₃×#41	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
28. C₃×Q₈	Q₈	semidirect
58. Q₈	Dic₆	semidirect
77. 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
#888	2
#870	2
C₄×Dic₆	2
#557	3