Subgroups — #5095

Order 192.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1007	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#942	C₂	1	2	yes	maximal, complemented
4	#836	C₂	1	1	yes	maximal, characteristic, complemented
5	#805	C₂	1	2	yes	maximal, complemented
6	#456	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	#239	—	2	2	no	—
10	D₃×Q₈	—	2	2	no	—
11	#235	—	2	2	no	—
12	#235	C₂²	1	2	yes	—
13	C₃×C₄⋊C₄	—	2	2	no	—
14	C₃×C₄²	C₂²	1	1	yes	characteristic
15	#216	—	2	2	no	—
16	#215	—	2	2	no	—
17	#208	C₂²	1	2	yes	—
18	2₋¹⁺⁴	—	3	1	no	—
19	#137	—	3	2	no	—
20	C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
21	#104	—	3	2	no	—
22	#101	—	3	1	no	—
23	C₃×Q₈	—	2	2	no	—
24	C₃×Q₈	—	2	2	no	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	D₄	1	2	yes	—
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	#67	—	2	2	no	—
30	D₁₂	—	2	2	no	—
31	D₁₂	D₄	1	2	yes	—
32	C₄×D₃	—	2	4	no	—
33	C₄×D₃	—	2	2	no	—
34	Dic₆	—	2	2	no	—
35	Dic₆	D₄	1	2	yes	—
36	C₃⋊C₈	—	2	2	no	—
37	#41	—	3	2	no	—
38	#41	—	6	2	no	—
39	#41	—	6	2	no	—
40	C₂×Q₈	—	6	2	no	—
41	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
43	Q₁₆	—	6	2	no	—
44	SD₁₆	—	6	2	no	—
45	M₁₆	—	3	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₃	—	2	2	no	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	C₂×D₄	1	2	yes	—
54	Dic₃	—	2	2	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	3	2	no	—
58	Q₈	—	6	2	no	—
59	D₄	—	3	2	no	—
60	D₄	—	6	2	no	—
61	D₄	—	6	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	6	4	no	—
65	C₂×C₄	—	6	2	no	—
66	C₂×C₄	#67	1	2	yes	—
67	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₈	—	6	2	no	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	#120	1	1	yes	characteristic
71	D₃	—	4	2	no	—
72	C₂²	—	6	2	no	—
73	C₂²	C₂×#67	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	4	1	no	—
77	C₄	—	6	2	no	—
78	C₄	D₃×D₄	1	2	yes	—
79	C₃	#456	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	2	1	no	—
81	C₂	—	12	2	no	—
82	C₂	#937	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#937
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄⋊Q₈	C₂
socle	C₂×C₃	#120
prime radical	#1007	C₂
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#456
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊Q₈	C₂	semidirect
3. #942	C₂	semidirect
4. #836	C₂	semidirect
5. #805	C₂	semidirect
20. C₄⋊Q₈	D₃	semidirect
79. C₃	#456	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#1007	2
C₃×C₄⋊Q₈	2
#942	2
#836	2
#805	2
#456	3