Subgroups — #5085

Order 192.

142 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	C₂×#215	C₂	1	4	yes	maximal, complemented
3	#874	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#492	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₈	C₂²	1	2	yes	—
7	C₂×D₁₂	—	2	2	no	—
8	C₂×D₁₂	C₂²	1	2	yes	—
9	C₃×C₄⋊C₄	—	2	2	no	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	#215	—	2	8	no	—
12	C₂×C₃⋊C₈	C₂²	1	2	yes	—
13	C₂×SD₁₆	—	3	4	no	—
14	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
15	#127	—	3	1	no	—
16	C₄×C₈	—	3	1	no	—
17	C₂²×D₃	—	2	2	no	—
18	C₃×Q₈	—	2	4	no	—
19	C₂×C₃×C₄	—	2	2	no	—
20	C₂×C₃×C₄	D₄	1	2	yes	—
21	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
22	D₁₂	—	2	4	no	—
23	D₁₂	—	4	4	no	—
24	C₃⋊C₈	D₄	1	4	yes	complemented
25	C₂×Q₈	C₂×D₃	1	2	yes	—
26	C₂×D₄	—	3	2	no	—
27	C₂×D₄	—	6	2	no	—
28	SD₁₆	—	6	8	no	—
29	C₂×C₈	—	3	2	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄²	C₂×D₃	1	1	yes	characteristic
32	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₃	—	4	2	no	—
34	C₂×D₃	—	4	4	no	—
35	C₃×C₄	—	4	2	no	—
36	C₃×C₄	SD₁₆	1	4	yes	complemented
37	C₃×C₄	C₂×D₄	1	2	yes	—
38	C₂³	—	6	2	no	—
39	Q₈	—	2	4	no	—
40	D₄	—	6	4	no	—
41	D₄	—	12	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	#67	1	2	yes	—
44	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
45	C₈	—	3	4	no	—
46	C₂×C₃	#127	1	1	yes	characteristic
47	C₂×C₃	C₂×SD₁₆	1	2	yes	—
48	D₃	—	8	2	no	—
49	C₂²	—	12	2	no	—
50	C₂²	—	12	4	no	—
51	C₂²	C₂×#67	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	#215	1	4	yes	complemented
54	C₄	D₃×D₄	1	2	yes	—
55	C₃	#492	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	24	2	no	—
57	C₂	#940	1	1	yes	characteristic
58	C₂	C₂×#215	1	2	yes	—

	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	#874	C₂
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#492
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
2. C₂×#215	C₂	semidirect
4. C₄×C₃⋊C₈	C₂	semidirect
14. C₄⋊Q₈	D₃	semidirect
24. C₃⋊C₈	D₄	semidirect
36. C₃×C₄	SD₁₆	semidirect
53. C₄	#215	semidirect
55. C₃	#492	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
C₂×#215	2
#874	2
C₄×C₃⋊C₈	2
#492	3