Subgroups — #5091

Order 192.

96 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	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	#810	C₂	1	2	yes	maximal
4	#807	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#477	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	—	2	1	no	—
8	C₂×Dic₆	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₄	—	2	1	no	—
10	C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	Dic₃⋊C₄	—	2	1	no	—
14	C₄×Dic₃	—	2	1	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	D₃	1	1	yes	characteristic
17	C₄×Q₈	—	3	1	no	—
18	#107	—	3	2	no	—
19	C₄⋊C₈	—	3	1	no	—
20	#103	—	3	2	no	—
21	C₃×Q₈	—	4	2	no	—
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	—	2	1	no	—
27	C₂×Dic₃	—	2	1	no	—
28	Dic₆	—	2	1	no	—
29	Dic₆	Q₈	1	2	yes	complemented
30	C₃⋊C₈	—	2	2	no	—
31	C₂×Q₈	—	2	1	no	—
32	C₂×Q₈	—	3	1	no	—
33	C₂×C₈	—	3	2	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	3	1	no	—
36	C₄⋊C₄	—	6	1	no	—
37	C₄⋊C₄	C₂×D₃	1	2	yes	—
38	C₄²	—	6	1	no	—
39	C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	4	2	no	—
43	C₃×C₄	Q₁₆	1	2	yes	—
44	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₃×C₄	#41	1	1	yes	characteristic
46	Dic₃	—	2	2	no	—
47	Dic₃	—	4	1	no	—
48	Q₈	—	3	2	no	—
49	Q₈	—	4	2	no	—
50	Q₈	—	6	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	6	1	no	—
53	C₂×C₄	—	6	1	no	—
54	C₂×C₄	#67	1	1	yes	characteristic
55	C₂×C₄	#67	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₈	—	6	2	no	—
58	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²	C₂×#67	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	4	2	no	—
64	C₄	—	6	2	no	—
65	C₄	—	12	1	no	—
66	C₄	#216	1	2	yes	—
67	C₄	D₃×Q₈	1	1	yes	characteristic
68	C₄	#239	1	1	yes	characteristic
69	C₃	#477	1	1	yes	characteristic, complemented, Sylow
70	C₂	#932	1	1	yes	characteristic
71	C₂	C₂×#216	1	1	yes	characteristic
72	C₂	#946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#477
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
29. Dic₆	Q₈	semidirect
69. C₃	#477	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
C₄×Dic₆	2
#810	2
#807	2
C₄⋊(C₃⋊C₈)	2
#477	3