Subgroups — C₂²×#6188 #11722453

Order 864.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1084	C₂	1	1	yes	maximal, characteristic
2	C₂×#6188	C₂	1	6	yes	maximal, direct product factor
3	C₂³×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂×#1084	C₄	1	3	yes	—
5	C₂×#1084	C₂²	1	3	yes	—
6	#6188	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×Dic₃	—	6	2	no	—
8	C₂²×H(3,𝔽₃)	C₈	1	4	yes	complemented
9	C₂²×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
10	#1084	C₂×C₄	1	3	yes	—
11	#1084	C₂³	1	1	yes	characteristic
12	C₂²×C₃×C₈	—	9	1	no	maximal
13	C₂³×C₃²	—	2	2	no	—
14	C₂×C₃×Dic₃	—	6	6	no	—
15	C₂×C₃×Dic₃	—	6	6	no	—
16	C₂×H(3,𝔽₃)	C₂×C₈	1	6	yes	complemented
17	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄	—	9	1	no	—
19	C₂²×Dic₃	—	6	2	no	—
20	C₂×C₃×C₈	—	9	6	no	—
21	C₂²×C₃²	—	2	8	no	—
22	C₂²×C₃²	—	2	6	no	—
23	C₃×Dic₃	—	6	6	no	—
24	C₃×Dic₃	—	6	2	no	—
25	C₂²×C₈	—	9	1	no	Sylow
26	H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₃	—	6	2	no	—
28	C₂³×C₃	#157	1	1	yes	characteristic
29	C₂×C₃×C₄	—	9	3	no	—
30	C₂×C₃×C₄	—	9	3	no	—
31	C₂×Dic₃	—	6	6	no	—
32	C₂×Dic₃	—	6	6	no	—
33	C₃×C₈	—	9	4	no	—
34	C₂×C₃²	—	2	2	no	—
35	C₂×C₃²	—	2	12	no	—
36	C₂²×C₄	—	9	1	no	—
37	C₂×C₈	—	9	6	no	—
38	C₂²×C₃	—	6	8	no	—
39	C₂²×C₃	—	6	6	no	—
40	C₂²×C₃	#622	1	4	yes	—
41	C₂²×C₃	C₂×#157	1	3	yes	—
42	C₃×C₄	—	9	1	no	—
43	C₃×C₄	—	9	3	no	—
44	Dic₃	—	6	2	no	—
45	Dic₃	—	6	6	no	—
46	C₃²	—	2	2	no	—
47	C₂³	#1088	1	1	yes	characteristic
48	C₂×C₄	—	9	3	no	—
49	C₂×C₄	—	9	3	no	—
50	C₈	—	9	4	no	—
51	C₂×C₃	—	6	12	no	—
52	C₂×C₃	—	6	2	no	—
53	C₂×C₃	C₂×#622	1	6	yes	—
54	C₂×C₃	C₂²×#157	1	1	yes	characteristic
55	C₂²	#6188	1	4	yes	direct product factor
56	C₂²	C₂×#1088	1	3	yes	—
57	C₄	—	9	1	no	—
58	C₄	—	9	3	no	—
59	C₃	—	6	2	no	—
60	C₃	C₂²×#622	1	1	yes	characteristic
61	C₂	C₂×#6188	1	6	yes	direct product factor
62	C₂	C₂²×#1088	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	#157
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₈
Frattini	C₂×C₃	C₂²×#157
Fitting, prime radical	C₂³×H(3,𝔽₃)	C₄
cosocle	#1084	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂×#6188	C₂	direct (N × H)
6. #6188	C₂²	direct (N × H)
8. C₂²×H(3,𝔽₃)	C₈	semidirect
16. C₂×H(3,𝔽₃)	C₂×C₈	semidirect
26. H(3,𝔽₃)	C₂²×C₈	semidirect
55. C₂²	#6188	direct (N × H)
61. C₂	C₂×#6188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#1084	2
C₂×#6188	2
C₂²×C₃×C₈	9

Subgroups — C22×#6188 #11722453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#6188 #11722453