Subgroups — #5231

Order 192.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₃)⋊C₄	C₂	1	3	yes	maximal
3	C₃⋊#95	C₂	1	3	yes	maximal, complemented
4	#393	—	3	1	no	maximal, Sylow
5	C₂³×D₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×Q₈	—	2	6	no	—
7	C₂²×C₃×C₄	C₂²	1	3	yes	—
8	C₂²×Dic₃	C₂²	1	3	yes	—
9	(C₂×D₃)⋊C₄	—	2	6	no	—
10	C₂²×Q₈	D₃	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊C₄	—	3	3	no	—
12	#95	—	3	3	no	—
13	C₂³×C₃	C₂³	1	1	yes	characteristic
14	C₂²×D₃	—	2	3	no	—
15	C₂²×D₃	—	2	3	no	—
16	C₂²×D₃	Q₈	1	2	yes	complemented
17	C₃×Q₈	—	2	8	no	—
18	C₂×C₃×C₄	—	2	6	no	—
19	C₂×C₃×C₄	D₄	1	6	yes	—
20	C₂×Dic₃	—	2	3	no	—
21	C₂×Dic₃	—	2	6	no	—
22	C₂⁴	—	3	1	no	—
23	C₂×Q₈	—	2	6	no	—
24	C₂²×C₄	—	3	3	no	—
25	C₂²×C₄	C₂×D₃	1	3	yes	—
26	C₂²⋊C₄	—	6	6	no	—
27	C₂²×C₃	C₂×D₄	1	3	yes	—
28	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₃	#41	1	3	yes	—
30	C₂×D₃	—	2	6	no	—
31	C₂×D₃	—	4	1	no	—
32	C₂×D₃	—	4	3	no	—
33	C₃×C₄	—	2	6	no	—
34	Dic₃	—	4	3	no	—
35	C₂³	—	3	2	no	—
36	C₂³	—	6	3	no	—
37	C₂³	—	6	3	no	—
38	C₂³	C₂²×D₃	1	1	yes	characteristic
39	Q₈	—	2	8	no	—
40	C₂×C₄	—	2	6	no	—
41	C₂×C₄	—	6	6	no	—
42	C₂×C₄	—	6	3	no	—
43	C₂×C₄	#67	1	6	yes	—
44	C₂×C₃	#120	1	1	yes	characteristic
45	C₂×C₃	C₂²⋊Q₈	1	3	yes	—
46	C₂×C₃	#124	1	3	yes	—
47	D₃	—	4	2	no	—
48	C₂²	—	6	6	no	—
49	C₂²	—	12	3	no	—
50	C₂²	—	12	1	no	—
51	C₂²	D₃×Q₈	1	1	yes	characteristic
52	C₂²	#239	1	3	yes	—
53	C₂²	C₂×#67	1	3	yes	—
54	C₄	—	2	6	no	—
55	C₄	—	12	3	no	—
56	C₃	#393	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	12	2	no	—
58	C₂	#946	1	3	yes	—
59	C₂	#947	1	3	yes	—
60	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator, socle, cosocle	C₂³×C₃	C₂³
Fitting	C₂²×C₃×Q₈	C₂
prime radical	C₂³×D₃	C₂²
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	#393
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
3. C₃⋊#95	C₂	semidirect
10. C₂²×Q₈	D₃	semidirect
16. C₂²×D₃	Q₈	semidirect
56. C₃	#393	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂×(C₂×D₃)⋊C₄	2
C₃⋊#95	2
#393	3