Subgroups — #5084

Order 192.

108 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	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	#809	C₂	1	2	yes	maximal
4	#808	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#476	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	—	2	1	no	—
8	C₂×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃	—	2	1	no	—
10	C₃×C₄⋊C₄	—	2	1	no	—
11	C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
17	C₄×D₄	—	3	1	no	—
18	#106	—	3	2	no	—
19	C₄⋊C₈	—	3	1	no	—
20	#102	—	3	2	no	—
21	C₂²×D₃	—	2	1	no	—
22	C₃×Q₈	—	4	2	no	—
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	D₁₂	—	2	1	no	—
29	D₁₂	Q₈	1	2	yes	complemented
30	C₄×D₃	—	4	2	no	—
31	C₃⋊C₈	—	2	2	no	—
32	C₂×Q₈	—	2	1	no	—
33	C₂×D₄	—	3	1	no	—
34	C₂²×C₄	—	6	1	no	—
35	C₂×C₈	—	3	2	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	3	1	no	—
38	C₄⋊C₄	C₂×D₃	1	2	yes	—
39	C₂²⋊C₄	—	6	1	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
42	C₂×D₃	—	2	2	no	—
43	C₂×D₃	—	4	1	no	—
44	C₂×D₃	—	4	1	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	4	2	no	—
47	C₃×C₄	SD₁₆	1	2	yes	—
48	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₃×C₄	#41	1	1	yes	characteristic
50	Dic₃	—	4	1	no	—
51	C₂³	—	6	1	no	—
52	Q₈	—	4	2	no	—
53	D₄	—	3	2	no	—
54	D₄	—	6	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	—	12	2	no	—
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₂×C₄	#67	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	—	6	2	no	—
62	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	D₃	—	4	2	no	—
66	C₂²	—	6	2	no	—
67	C₂²	—	12	1	no	—
68	C₂²	—	12	1	no	—
69	C₂²	C₂×#67	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	2	no	—
72	C₄	—	12	1	no	—
73	C₄	#215	1	2	yes	—
74	C₄	D₃×Q₈	1	1	yes	characteristic
75	C₄	#239	1	1	yes	characteristic
76	C₃	#476	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	12	2	no	—
78	C₂	#932	1	1	yes	characteristic
79	C₂	C₂×#215	1	1	yes	characteristic
80	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	C₂²×C₃	C₂×D₄
prime radical	C₂×D₁₂	C₂²
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#476
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
1. C₃×C₄⋊Q₈	C₂	semidirect
4. #808	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
16. C₄⋊Q₈	D₃	semidirect
29. D₁₂	Q₈	semidirect
76. C₃	#476	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×D₁₂	2
#809	2
#808	2
C₄⋊(C₃⋊C₈)	2
#476	3