Subgroups — C₄²×#1088 #13147318

Order 1728.

603 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#1088	C₂	1	6	yes	maximal
2	C₄²×#272	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1088	C₂²	1	4	yes	—
4	C₂×C₄×#272	C₂²	1	3	yes	—
5	C₄×#1088	C₄	1	24	yes	direct product factor
6	C₄²×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
7	C₄×#1084	C₄	1	3	yes	complemented
8	C₃×C₄²×D₃	—	6	2	no	—
9	C₂²×#272	C₂³	1	1	yes	characteristic
10	C₂×#1088	C₂×C₄	1	24	yes	—
11	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
12	C₂×#1084	C₂×C₄	1	3	yes	—
13	C₄×#272	C₂×C₄	1	12	yes	—
14	C₃×C₄³	—	9	1	no	maximal
15	C₂×C₃×C₄×D₃	—	6	6	no	—
16	C₃²×C₄²	—	2	2	no	—
17	C₃×C₄×Dic₃	—	6	6	no	—
18	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
19	C₂×#272	C₂²×C₄	1	6	yes	—
20	#1088	C₄²	1	16	yes	direct product factor
21	C₄×H(3,𝔽₃)	C₄²	1	6	yes	complemented
22	#1084	C₄²	1	6	yes	complemented
23	C₂×C₃×C₄²	—	9	6	no	—
24	C₂×C₃×C₄²	—	9	1	no	—
25	C₄²×D₃	—	6	2	no	—
26	C₂²×C₃×D₃	—	6	2	no	—
27	C₂×C₃²×C₄	—	2	6	no	—
28	C₂×C₃×Dic₃	—	6	6	no	—
29	C₃×C₄×D₃	—	6	24	no	—
30	C₄³	—	9	1	no	Sylow
31	C₂×H(3,𝔽₃)	C₂×C₄²	1	3	yes	—
32	#272	C₂×C₄²	1	4	yes	—
33	C₂²×C₃×C₄	—	9	4	no	—
34	C₂²×C₃×C₄	—	9	3	no	—
35	C₂×C₄×D₃	—	6	6	no	—
36	C₃×C₄²	—	6	2	no	—
37	C₃×C₄²	—	9	24	no	—
38	C₃×C₄²	—	9	3	no	—
39	C₃×C₄²	#157	1	1	yes	characteristic
40	C₄×Dic₃	—	6	6	no	—
41	C₂²×C₃²	—	2	2	no	—
42	C₂×C₃×D₃	—	6	12	no	—
43	C₃²×C₄	—	2	12	no	—
44	C₃×Dic₃	—	6	12	no	—
45	C₂×C₄²	—	9	6	no	—
46	C₂×C₄²	—	9	1	no	—
47	H(3,𝔽₃)	C₄³	1	1	yes	characteristic, complemented, Sylow
48	C₂³×C₃	—	9	1	no	—
49	C₂²×D₃	—	6	2	no	—
50	C₂×C₃×C₄	—	6	6	no	—
51	C₂×C₃×C₄	—	9	24	no	—
52	C₂×C₃×C₄	—	9	3	no	—
53	C₂×C₃×C₄	—	9	12	no	—
54	C₂×C₃×C₄	C₂×#157	1	3	yes	—
55	C₂×Dic₃	—	6	6	no	—
56	C₄×D₃	—	6	24	no	—
57	C₂×C₃²	—	2	6	no	—
58	C₃×D₃	—	6	8	no	—
59	C₂²×C₄	—	9	4	no	—
60	C₂²×C₄	—	9	3	no	—
61	C₄²	—	9	24	no	—
62	C₄²	—	9	3	no	—
63	C₄²	#1088	1	1	yes	characteristic, direct product factor
64	C₂²×C₃	—	6	2	no	—
65	C₂²×C₃	—	9	6	no	—
66	C₂²×C₃	C₂²×#157	1	1	yes	characteristic
67	C₂×D₃	—	6	12	no	—
68	C₃×C₄	—	6	12	no	—
69	C₃×C₄	—	9	16	no	—
70	C₃×C₄	—	9	6	no	—
71	C₃×C₄	C₄×#157	1	6	yes	—
72	Dic₃	—	6	12	no	—
73	C₃²	—	2	2	no	—
74	C₂³	—	9	1	no	—
75	C₂×C₄	—	9	24	no	—
76	C₂×C₄	—	9	3	no	—
77	C₂×C₄	—	9	12	no	—
78	C₂×C₄	C₂×#1088	1	3	yes	—
79	C₂×C₃	—	6	6	no	—
80	C₂×C₃	—	9	4	no	—
81	C₂×C₃	C₂×C₄×#157	1	3	yes	—
82	D₃	—	6	8	no	—
83	C₂²	—	9	6	no	—
84	C₂²	C₂²×#1088	1	1	yes	characteristic
85	C₄	—	9	16	no	—
86	C₄	—	9	6	no	—
87	C₄	C₄×#1088	1	6	yes	direct product factor
88	C₃	—	6	2	no	—
89	C₃	C₄²×#157	1	1	yes	characteristic
90	C₂	—	9	4	no	—
91	C₂	C₂×C₄×#1088	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₃×C₄²	#157
commutator, hypocenter	H(3,𝔽₃)	C₄³
Frattini, socle	C₂²×C₃	C₂²×#157
Fitting	C₄²×H(3,𝔽₃)	C₄
prime radical, cosocle	C₂²×#272	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×#1088	C₄	direct (N × H)
6. C₄²×H(3,𝔽₃)	C₄	semidirect
7. C₄×#1084	C₄	semidirect
20. #1088	C₄²	direct (N × H)
21. C₄×H(3,𝔽₃)	C₄²	semidirect
22. #1084	C₄²	semidirect
47. H(3,𝔽₃)	C₄³	semidirect
63. C₄²	#1088	direct (N × H)
87. C₄	C₄×#1088	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#1088	2
C₄²×#272	2
C₃×C₄³	9

Subgroups — C42×#1088 #13147318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#1088 #13147318