Subgroups — C₂²×#216 #5809

Order 192.

258 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	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#216	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₈	—	2	3	no	—
7	C₂×C₃×Q₈	C₂²	1	6	yes	—
8	C₂²×C₃×C₄	—	2	1	no	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	—	2	1	no	—
11	C₂×Dic₆	—	2	3	no	—
12	C₂×Dic₆	C₂²	1	6	yes	—
13	#216	C₂²	1	16	yes	direct product factor
14	C₂×C₃⋊C₈	C₂²	1	6	yes	—
15	C₂²×Q₈	—	3	1	no	—
16	C₂²×Q₈	D₃	1	1	yes	characteristic
17	C₂×Q₁₆	—	3	12	no	—
18	C₂²×C₈	—	3	1	no	—
19	C₂³×C₃	D₄	1	1	yes	characteristic
20	C₃×Q₈	—	2	6	no	—
21	C₃×Q₈	C₂³	1	4	yes	—
22	C₂×C₃×C₄	—	2	6	no	—
23	C₂×C₃×C₄	D₄	1	3	yes	—
24	C₂×C₃×C₄	C₂³	1	3	yes	—
25	C₂×Dic₃	—	2	6	no	—
26	Dic₆	—	2	6	no	—
27	Dic₆	C₂³	1	4	yes	—
28	C₃⋊C₈	C₂³	1	4	yes	—
29	C₂×Q₈	—	2	3	no	—
30	C₂×Q₈	—	3	6	no	—
31	C₂×Q₈	—	6	3	no	—
32	C₂×Q₈	C₂×D₃	1	6	yes	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	—	6	1	no	—
35	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
36	Q₁₆	—	3	16	no	—
37	C₂×C₈	—	3	6	no	—
38	C₂²×C₃	Q₁₆	1	4	yes	complemented
39	C₂²×C₃	C₂×D₄	1	3	yes	—
40	C₃×C₄	—	2	4	no	—
41	C₃×C₄	C₂×D₄	1	3	yes	—
42	C₃×C₄	C₂⁴	1	1	yes	characteristic
43	Dic₃	—	2	4	no	—
44	C₂³	#67	1	1	yes	characteristic
45	Q₈	—	2	6	no	—
46	Q₈	—	3	4	no	—
47	Q₈	—	6	6	no	—
48	Q₈	C₂²×D₃	1	4	yes	—
49	C₂×C₄	—	2	6	no	—
50	C₂×C₄	—	6	6	no	—
51	C₂×C₄	#67	1	3	yes	—
52	C₂×C₄	C₂²×D₃	1	3	yes	—
53	C₈	—	3	4	no	—
54	C₂×C₃	C₂×Q₁₆	1	6	yes	complemented
55	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
56	C₂²	#216	1	4	yes	direct product factor
57	C₂²	C₂×#67	1	3	yes	—
58	C₄	—	2	4	no	—
59	C₄	—	6	4	no	—
60	C₄	C₂×#67	1	3	yes	—
61	C₄	C₂³×D₃	1	1	yes	characteristic
62	C₃	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂	C₂×#216	1	6	yes	direct product factor
64	C₂	C₂²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#67
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle, prime radical	C₂³×C₃	D₄
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂²×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#216	C₂	direct (N × H)
13. #216	C₂²	direct (N × H)
38. C₂²×C₃	Q₁₆	semidirect
54. C₂×C₃	C₂×Q₁₆	semidirect
56. C₂²	#216	direct (N × H)
62. C₃	C₂²×Q₁₆	semidirect
63. C₂	C₂×#216	direct (N × H)

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
C₂²×Dic₆	2
C₂×#216	2
C₂²×C₃⋊C₈	2
C₂²×Q₁₆	3

Subgroups — C22×#216 #5809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#216 #5809