Subgroups — #13136736

Order 1728.

183 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88631	C₂	1	3	yes	maximal
2	C₂²×#1084	C₂²	1	1	yes	characteristic
3	#88631	—	2	6	no	—
4	C₂×#6188	C₄	1	6	yes	—
5	C₂³×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
6	C₂×#1084	C₂×C₄	1	3	yes	—
7	C₂×#1084	D₄	1	3	yes	—
8	#6188	—	2	6	no	—
9	C₃×#338	—	9	1	no	maximal
10	C₂²×C₃×Dic₃	—	12	1	no	—
11	C₂²×H(3,𝔽₃)	—	4	1	no	—
12	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	3	yes	—
13	#1084	C₄²	1	1	yes	characteristic
14	#1084	C₂²⋊C₄	1	3	yes	—
15	C₂×C₃×M₁₆	—	9	3	no	—
16	C₂³×C₃²	—	4	1	no	—
17	C₂×C₃×Dic₃	—	12	3	no	—
18	C₂×C₃×Dic₃	—	12	3	no	—
19	#338	—	9	1	no	Sylow
20	C₂×H(3,𝔽₃)	—	2	3	no	—
21	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	9	1	no	—
23	C₂²×Dic₃	—	12	1	no	—
24	C₃×M₁₆	—	18	6	no	—
25	C₂×C₃×C₈	—	9	6	no	—
26	C₂²×C₃²	—	4	4	no	—
27	C₂²×C₃²	—	4	3	no	—
28	C₃×Dic₃	—	12	1	no	—
29	C₃×Dic₃	—	12	3	no	—
30	C₂×M₁₆	—	9	3	no	—
31	H(3,𝔽₃)	#338	1	1	yes	characteristic, complemented, Sylow
32	C₂³×C₃	—	12	1	no	—
33	C₂³×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic
34	C₂×C₃×C₄	—	9	3	no	—
35	C₂×C₃×C₄	—	9	3	no	—
36	C₂×Dic₃	—	12	3	no	—
37	C₂×Dic₃	—	12	3	no	—
38	C₃×C₈	—	18	6	no	—
39	C₂×C₃²	—	4	6	no	—
40	C₂×C₃²	—	4	1	no	—
41	C₂²×C₄	—	9	1	no	—
42	M₁₆	—	18	6	no	—
43	C₂×C₈	—	9	6	no	—
44	C₂²×C₃	—	4	1	no	—
45	C₂²×C₃	—	12	4	no	—
46	C₂²×C₃	—	12	3	no	—
47	C₂²×C₃	#3777	1	3	yes	—
48	C₃×C₄	—	9	3	no	—
49	C₃×C₄	—	9	1	no	—
50	Dic₃	—	12	3	no	—
51	Dic₃	—	12	1	no	—
52	C₃²	—	4	1	no	—
53	C₂³	SU(3,𝔽₂)	1	1	yes	characteristic
54	C₂×C₄	—	9	3	no	—
55	C₂×C₄	—	9	3	no	—
56	C₈	—	18	6	no	—
57	C₂×C₃	—	2	3	no	—
58	C₂×C₃	—	12	6	no	—
59	C₂×C₃	—	12	1	no	—
60	C₂×C₃	#63747	1	1	yes	characteristic
61	C₂²	—	4	1	no	—
62	C₂²	#88592	1	3	yes	—
63	C₄	—	9	3	no	—
64	C₄	—	9	1	no	—
65	C₃	—	12	1	no	—
66	C₃	#10590096	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#11720711	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#63747
commutator	#1084	C₄²
Frattini, hypercenter	C₂³×C₃	PSU(3,𝔽₂)
Fitting, prime radical	C₂³×H(3,𝔽₃)	Q₈
cosocle	C₂²×#1084	C₂²
hypocenter	H(3,𝔽₃)	#338
LCS	G ▷ #1084 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
31. H(3,𝔽₃)	#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88631	2
C₃×#338	9