Subgroups — C₂³×#6188 #13164683

Order 1728.

979 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1084	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6188	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂²×#1084	C₄	1	7	yes	—
5	C₂²×#1084	C₂²	1	7	yes	—
6	C₂×#6188	C₂²	1	28	yes	direct product factor
7	C₂³×C₃×Dic₃	—	6	2	no	—
8	C₂³×H(3,𝔽₃)	C₈	1	8	yes	complemented
9	C₂³×H(3,𝔽₃)	C₂×C₄	1	7	yes	—
10	C₂×#1084	C₂×C₄	1	21	yes	—
11	C₂×#1084	C₂³	1	7	yes	—
12	#6188	C₂³	1	8	yes	direct product factor
13	C₂³×C₃×C₈	—	9	1	no	maximal
14	C₂⁴×C₃²	—	2	2	no	—
15	C₂²×C₃×Dic₃	—	6	14	no	—
16	C₂²×C₃×Dic₃	—	6	14	no	—
17	C₂²×H(3,𝔽₃)	C₂×C₈	1	28	yes	complemented
18	C₂²×H(3,𝔽₃)	C₂²×C₄	1	7	yes	—
19	#1084	C₂²×C₄	1	7	yes	—
20	#1084	C₂⁴	1	1	yes	characteristic
21	C₂³×C₃×C₄	—	9	1	no	—
22	C₂³×Dic₃	—	6	2	no	—
23	C₂²×C₃×C₈	—	9	14	no	—
24	C₂³×C₃²	—	2	16	no	—
25	C₂³×C₃²	—	2	14	no	—
26	C₂×C₃×Dic₃	—	6	42	no	—
27	C₂×C₃×Dic₃	—	6	14	no	—
28	C₂³×C₈	—	9	1	no	Sylow
29	C₂×H(3,𝔽₃)	C₂²×C₈	1	14	yes	complemented
30	C₂×H(3,𝔽₃)	C₂³×C₄	1	1	yes	characteristic
31	C₂⁴×C₃	—	6	2	no	—
32	C₂⁴×C₃	#157	1	1	yes	characteristic
33	C₂²×C₃×C₄	—	9	7	no	—
34	C₂²×C₃×C₄	—	9	7	no	—
35	C₂²×Dic₃	—	6	14	no	—
36	C₂²×Dic₃	—	6	14	no	—
37	C₂×C₃×C₈	—	9	28	no	—
38	C₂²×C₃²	—	2	14	no	—
39	C₂²×C₃²	—	2	56	no	—
40	C₃×Dic₃	—	6	14	no	—
41	C₃×Dic₃	—	6	2	no	—
42	C₂³×C₄	—	9	1	no	—
43	C₂²×C₈	—	9	14	no	—
44	H(3,𝔽₃)	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₃	—	6	16	no	—
46	C₂³×C₃	—	6	14	no	—
47	C₂³×C₃	#622	1	8	yes	—
48	C₂³×C₃	C₂×#157	1	7	yes	—
49	C₂×C₃×C₄	—	9	7	no	—
50	C₂×C₃×C₄	—	9	21	no	—
51	C₂×Dic₃	—	6	14	no	—
52	C₂×Dic₃	—	6	42	no	—
53	C₃×C₈	—	9	8	no	—
54	C₂×C₃²	—	2	28	no	—
55	C₂×C₃²	—	2	2	no	—
56	C₂⁴	#1088	1	1	yes	characteristic
57	C₂²×C₄	—	9	7	no	—
58	C₂²×C₄	—	9	7	no	—
59	C₂×C₈	—	9	28	no	—
60	C₂²×C₃	—	6	56	no	—
61	C₂²×C₃	—	6	14	no	—
62	C₂²×C₃	C₂×#622	1	28	yes	—
63	C₂²×C₃	C₂²×#157	1	7	yes	—
64	C₃×C₄	—	9	1	no	—
65	C₃×C₄	—	9	7	no	—
66	Dic₃	—	6	2	no	—
67	Dic₃	—	6	14	no	—
68	C₃²	—	2	2	no	—
69	C₂³	#6188	1	8	yes	direct product factor
70	C₂³	C₂×#1088	1	7	yes	—
71	C₂×C₄	—	9	7	no	—
72	C₂×C₄	—	9	21	no	—
73	C₈	—	9	8	no	—
74	C₂×C₃	—	6	2	no	—
75	C₂×C₃	—	6	28	no	—
76	C₂×C₃	C₂²×#622	1	14	yes	—
77	C₂×C₃	C₂³×#157	1	1	yes	characteristic
78	C₂²	C₂×#6188	1	28	yes	direct product factor
79	C₂²	C₂²×#1088	1	7	yes	—
80	C₄	—	9	1	no	—
81	C₄	—	9	7	no	—
82	C₃	—	6	2	no	—
83	C₃	C₂³×#622	1	1	yes	characteristic
84	C₂	C₂²×#6188	1	14	yes	direct product factor
85	C₂	C₂³×#1088	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁴×C₃	#157
commutator, hypocenter	H(3,𝔽₃)	C₂³×C₈
Frattini	C₂×C₃	C₂³×#157
Fitting, prime radical	C₂⁴×H(3,𝔽₃)	C₄
cosocle	#1084	C₂⁴
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×#6188	C₂	direct (N × H)
6. C₂×#6188	C₂²	direct (N × H)
8. C₂³×H(3,𝔽₃)	C₈	semidirect
12. #6188	C₂³	direct (N × H)
17. C₂²×H(3,𝔽₃)	C₂×C₈	semidirect
29. C₂×H(3,𝔽₃)	C₂²×C₈	semidirect
44. H(3,𝔽₃)	C₂³×C₈	semidirect
69. C₂³	#6188	direct (N × H)
78. C₂²	C₂×#6188	direct (N × H)
84. C₂	C₂²×#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
C₂³×#1084	2
C₂²×#6188	2
C₂³×C₃×C₈	9

Subgroups — C23×#6188 #13164683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6188 #13164683