Subgroups — #5089

Order 192.

100 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	#875	C₂	1	1	yes	maximal, characteristic
3	#810	C₂	1	2	yes	maximal, complemented
4	#809	C₂	1	2	yes	maximal
5	#803	C₂	1	1	yes	maximal, characteristic, complemented
6	#489	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	—	2	1	no	—
8	C₂×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×Dic₆	C₂²	1	1	yes	characteristic
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	(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
17	#124	—	3	1	no	—
18	#103	—	3	2	no	—
19	#102	—	3	2	no	—
20	#97	—	3	1	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	2	no	—
29	Dic₆	—	2	2	no	—
30	C₃⋊C₈	—	2	2	no	—
31	C₂×Q₈	—	2	1	no	—
32	C₂×Q₈	—	3	1	no	—
33	C₂×D₄	—	3	1	no	—
34	C₂×C₈	—	3	2	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	C₂×D₃	1	2	yes	—
37	C₂²⋊C₄	—	6	1	no	—
38	C₂²⋊C₄	—	6	1	no	—
39	C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
41	C₂×D₃	—	4	1	no	—
42	C₂×D₃	—	4	1	no	—
43	C₂×D₃	—	4	1	no	—
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	4	2	no	—
47	C₃×C₄	#41	1	2	yes	—
48	Dic₃	—	4	1	no	—
49	C₂³	—	6	1	no	—
50	Q₈	—	4	2	no	—
51	Q₈	—	6	2	no	—
52	D₄	—	6	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	#67	1	1	yes	characteristic
56	C₂×C₄	#67	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	6	2	no	—
59	C₂×C₃	#124	1	1	yes	characteristic
60	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₃	#137	1	1	yes	characteristic
62	D₃	—	8	1	no	—
63	C₂²	—	12	1	no	—
64	C₂²	—	12	1	no	—
65	C₂²	—	12	1	no	—
66	C₂²	C₂×#67	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	12	1	no	—
71	C₄	#239	1	2	yes	—
72	C₃	#489	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	24	1	no	—
74	C₂	#932	1	1	yes	characteristic
75	C₂	#942	1	1	yes	characteristic
76	C₂	#947	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	#489
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
3. #810	C₂	semidirect
5. #803	C₂	semidirect
16. C₄⋊Q₈	D₃	semidirect
72. C₃	#489	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
#875	2
#810	2
#809	2
#803	2
#489	3