Subgroups — #5731

Order 192.

192 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, complemented
2	#946	C₂	1	2	yes	maximal
3	Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
4	#898	C₂	1	4	yes	maximal
5	#889	C₂	1	4	yes	maximal, complemented
6	#872	C₂	1	1	yes	maximal, characteristic
7	#870	C₂	1	1	yes	maximal, characteristic, complemented
8	#541	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	2	yes	—
10	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
14	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
15	C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₄⋊Dic₃	C₂²	1	4	yes	—
17	Dic₃⋊C₄	C₂²	1	4	yes	—
18	Dic₃⋊C₄	C₂²	1	2	yes	—
19	Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₃	C₂²	1	4	yes	—
22	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
23	#126	—	3	4	no	—
24	#125	—	3	4	no	—
25	#125	—	3	1	no	—
26	C₂²⋊Q₈	—	3	2	no	—
27	C₄×Q₈	—	3	2	no	—
28	#117	—	3	1	no	—
29	C₂²×D₃	C₂³	1	1	yes	characteristic
30	C₃×Q₈	—	2	4	no	—
31	C₂×C₃×C₄	C₂³	1	4	yes	—
32	C₂×C₃×C₄	C₂³	1	2	yes	—
33	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂×Dic₃	C₂³	1	2	yes	—
35	C₂×Dic₃	C₂³	1	4	yes	—
36	C₂×Dic₃	C₂³	1	1	yes	characteristic
37	C₄×D₃	—	2	2	no	—
38	C₂×Q₈	C₂×D₃	1	2	yes	—
39	C₂²×C₄	—	3	1	no	—
40	C₄⋊C₄	—	3	2	no	—
41	C₄⋊C₄	—	3	2	no	—
42	C₄⋊C₄	—	3	4	no	—
43	C₄⋊C₄	—	3	4	no	—
44	C₄⋊C₄	—	3	4	no	—
45	C₄⋊C₄	C₂×D₃	1	4	yes	—
46	C₂²⋊C₄	—	3	2	no	—
47	C₂²⋊C₄	—	3	4	no	—
48	C₄²	—	3	1	no	—
49	C₄²	—	3	4	no	—
50	C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₃	—	2	1	no	—
53	C₂×D₃	—	2	2	no	—
54	C₃×C₄	—	2	4	no	—
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	#41	1	2	yes	—
57	Dic₃	—	2	1	no	—
58	Dic₃	—	2	2	no	—
59	Dic₃	—	2	4	no	—
60	C₂³	—	3	1	no	—
61	Q₈	—	2	4	no	—
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	3	4	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	6	2	no	—
66	C₂×C₄	C₂²×D₃	1	4	yes	—
67	C₂×C₄	C₂²×D₃	1	2	yes	—
68	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₃	2₋¹⁺⁴	1	2	yes	—
71	D₃	—	4	1	no	—
72	C₂²	—	6	1	no	—
73	C₂²	—	6	2	no	—
74	C₂²	C₂³×D₃	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	2	2	no	—
77	C₄	—	6	1	no	—
78	C₄	—	6	2	no	—
79	C₄	—	6	4	no	—
80	C₄	#237	1	2	yes	—
81	C₃	#541	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	12	1	no	—
83	C₂	C₂×#237	1	1	yes	characteristic
84	C₂	#1007	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₃	C₂³
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#541
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
1. C₃×C₄⋊Q₈	C₂	semidirect
3. Q₈×Dic₃	C₂	semidirect
5. #889	C₂	semidirect
7. #870	C₂	semidirect
22. C₄⋊Q₈	D₃	semidirect
81. C₃	#541	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
#946	2
Q₈×Dic₃	2
#898	2
#889	2
#872	2
#870	2
#541	3