Subgroups — #5945

Order 192.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴⋊C₃	C₂	1	3	yes	maximal, complemented
2	#994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#41⋊C₃	C₂	1	3	yes	maximal, complemented
4	#585	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
5	C₃×#41	—	4	1	no	maximal
6	#41⋊C₃	C₂²	1	1	yes	characteristic, complemented
7	#41⋊C₃	C₂²	1	3	yes	complemented
8	C₂×SL(2,𝔽₃)	C₂²	1	3	yes	complemented
9	2₋¹⁺⁴	—	3	1	no	—
10	2₋¹⁺⁴	C₂×C₃	1	3	yes	complemented
11	H(5,𝔽₂)	—	3	3	no	—
12	H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×#41	—	3	3	no	—
14	C₂×#41	—	3	1	no	—
15	C₂×#41	C₂×C₃	1	3	yes	complemented
16	C₃×Q₈	—	4	1	no	—
17	C₃×D₄	—	4	3	no	—
18	C₂×C₃×C₄	—	4	3	no	—
19	SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
20	#41	—	3	6	no	—
21	#41	—	3	1	no	—
22	#41	—	3	3	no	—
23	#41	—	3	6	no	—
24	#41	—	3	3	no	—
25	#41	—	3	3	no	—
26	#41	—	3	3	no	—
27	#41	A₄	1	1	yes	characteristic
28	#41	C₂²×C₃	1	1	yes	characteristic
29	#41	C₂²×C₃	1	3	yes	—
30	C₂×Q₈	—	3	3	no	—
31	C₂×Q₈	—	3	1	no	—
32	C₂×Q₈	C₂²×C₃	1	3	yes	—
33	C₂×D₄	—	3	3	no	—
34	C₂×D₄	—	3	6	no	—
35	C₂×D₄	—	3	3	no	—
36	C₂×D₄	—	3	3	no	—
37	C₂²×C₄	—	3	2	no	—
38	C₂²×C₄	—	3	3	no	—
39	C₂²×C₃	—	4	3	no	—
40	C₃×C₄	—	4	3	no	—
41	C₃×C₄	—	4	1	no	—
42	C₂³	—	3	2	no	—
43	C₂³	—	3	3	no	—
44	Q₈	—	3	3	no	—
45	Q₈	—	3	3	no	—
46	Q₈	C₂×A₄	1	1	yes	characteristic
47	Q₈	C₂³×C₃	1	1	yes	characteristic
48	D₄	—	3	1	no	—
49	D₄	—	3	3	no	—
50	D₄	—	3	6	no	—
51	D₄	—	3	6	no	—
52	D₄	—	3	3	no	—
53	D₄	C₂×A₄	1	3	yes	—
54	C₂×C₄	—	3	6	no	—
55	C₂×C₄	—	3	3	no	—
56	C₂×C₄	—	3	3	no	—
57	C₂×C₄	—	3	3	no	—
58	C₂×C₄	—	3	3	no	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	C₂×A₄	1	3	yes	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	8	3	no	—
63	C₂²	—	3	3	no	—
64	C₂²	—	3	1	no	—
65	C₂²	—	12	3	no	—
66	C₂²	—	12	2	no	—
67	C₂²	C₂²×A₄	1	3	yes	—
68	C₄	—	3	3	no	—
69	C₄	—	3	1	no	—
70	C₄	C₂²×A₄	1	3	yes	—
71	C₄	C₂²×A₄	1	1	yes	characteristic
72	C₃	—	4	1	no	Sylow
73	C₂	—	2	3	no	—
74	C₂	—	6	1	no	—
75	C₂	—	6	3	no	—
76	C₂	C₂³×A₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂³×C₃
Frattini, socle	C₂	C₂³×A₄
Fitting	#585	C₃
hypercenter	#41	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴⋊C₃	C₂	semidirect
2. #994	C₂	semidirect
3. C₂×#41⋊C₃	C₂	semidirect
4. #585	C₃	semidirect
6. #41⋊C₃	C₂²	semidirect
7. #41⋊C₃	C₂²	semidirect
8. C₂×SL(2,𝔽₃)	C₂²	semidirect
10. 2₋¹⁺⁴	C₂×C₃	semidirect
12. H(5,𝔽₂)	C₂×C₃	semidirect
15. C₂×#41	C₂×C₃	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴⋊C₃	2
#994	2
C₂×#41⋊C₃	2
#585	3
C₃×#41	4