Subgroups — #5090

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
2	C₂×#216	C₂	1	2	yes	maximal
3	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
4	#835	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#462	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×Dic₆	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₄	—	2	1	no	—
10	C₃×C₄⋊C₄	—	2	1	no	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	#216	—	2	4	no	—
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	Dic₃⋊C₄	—	2	1	no	—
15	C₄×Dic₃	—	2	1	no	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×Q₁₆	—	3	2	no	—
18	C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₄×Q₈	—	3	1	no	—
20	C₄⋊C₈	—	3	1	no	—
21	#103	—	3	2	no	—
22	C₃×Q₈	—	2	2	no	—
23	C₃×Q₈	—	2	2	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×Dic₃	—	2	1	no	—
29	C₂×Dic₃	—	2	1	no	—
30	Dic₆	—	2	1	no	—
31	Dic₆	D₄	1	2	yes	—
32	C₃⋊C₈	—	2	2	no	—
33	C₂×Q₈	—	3	1	no	—
34	C₂×Q₈	C₂×D₃	1	2	yes	—
35	Q₁₆	—	6	4	no	—
36	C₂×C₈	—	3	2	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	3	1	no	—
40	C₄⋊C₄	—	6	1	no	—
41	C₄²	—	6	1	no	—
42	C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	4	2	no	—
46	C₃×C₄	Q₁₆	1	2	yes	complemented
47	C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄	#41	1	1	yes	characteristic
49	Dic₃	—	2	2	no	—
50	Dic₃	—	4	1	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	3	2	no	—
54	Q₈	—	6	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	—	6	1	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	#121	1	1	yes	characteristic
63	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₃	#137	1	1	yes	characteristic
65	C₂²	C₂×#67	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	4	2	no	—
68	C₄	—	6	2	no	—
69	C₄	—	12	1	no	—
70	C₄	#216	1	2	yes	complemented
71	C₄	D₃×D₄	1	1	yes	characteristic
72	C₄	#237	1	1	yes	characteristic
73	C₃	#462	1	1	yes	characteristic, complemented, Sylow
74	C₂	#938	1	1	yes	characteristic
75	C₂	#942	1	1	yes	characteristic
76	C₂	C₂×#216	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	#462
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
46. C₃×C₄	Q₁₆	semidirect
70. C₄	#216	semidirect
73. C₃	#462	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×#216	2
C₄×Dic₆	2
#835	2
C₄⋊(C₃⋊C₈)	2
#462	3