Subgroups — C₂²×#3792 #10596652

Order 576.

534 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#3792	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#622	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₃²	C₄	1	1	yes	characteristic
5	C₂²×C₃⋊Dic₃	C₄	1	6	yes	—
6	C₂²×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₃	C₂²	1	6	yes	—
9	#3792	C₂²	1	16	yes	direct product factor
10	C₂×#622	C₂²	1	12	yes	complemented
11	C₂³×Dic₃	—	6	2	no	—
12	C₂³×C₃²	—	2	4	no	—
13	C₂³×C₃²	C₂×C₄	1	6	yes	—
14	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	C₂×C₄	1	12	yes	—
16	C₂×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
17	C₂×C₃⋊Dic₃	C₂×C₄	1	6	yes	—
18	C₂×C₃⋊Dic₃	C₂³	1	3	yes	—
19	C₂×C₃⋊Dic₃	C₂³	1	4	yes	—
20	#622	C₂³	1	8	yes	complemented
21	C₂²×M₁₆	—	9	1	no	maximal, Sylow
22	C₂⁴×C₃	—	2	2	no	—
23	C₂²×Dic₃	—	6	12	no	—
24	C₂²×Dic₃	—	6	2	no	—
25	C₂²×Dic₃	—	6	12	no	—
26	C₂²×Dic₃	—	6	2	no	—
27	C₂²×C₃²	—	2	12	no	—
28	C₂²×C₃²	M₁₆	1	4	yes	complemented
29	C₂²×C₃²	C₂²×C₄	1	4	yes	—
30	C₂²×C₃²	C₂²×C₄	1	3	yes	—
31	C₃⋊Dic₃	C₂²×C₄	1	3	yes	—
32	C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
33	C₃⋊Dic₃	C₂⁴	1	1	yes	characteristic
34	C₂³×C₄	—	9	1	no	—
35	C₂×M₁₆	—	9	12	no	—
36	C₂²×C₈	—	9	2	no	—
37	C₂³×C₃	—	2	16	no	—
38	C₂³×C₃	—	2	12	no	—
39	C₂³×C₃	—	2	2	no	—
40	C₂×Dic₃	—	6	24	no	—
41	C₂×Dic₃	—	6	12	no	—
42	C₂×Dic₃	—	6	6	no	—
43	C₂×Dic₃	—	6	6	no	—
44	C₂×Dic₃	—	6	8	no	—
45	C₂×C₃²	—	2	4	no	—
46	C₂×C₃²	C₂×M₁₆	1	6	yes	complemented
47	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
48	C₂⁴	#157	1	1	yes	characteristic
49	C₂²×C₄	—	9	6	no	—
50	C₂²×C₄	—	9	6	no	—
51	C₂²×C₄	—	9	1	no	—
52	C₂²×C₄	—	9	1	no	—
53	M₁₆	—	9	16	no	—
54	C₂×C₈	—	9	12	no	—
55	C₂²×C₃	—	2	48	no	—
56	C₂²×C₃	—	2	8	no	—
57	C₂²×C₃	—	2	6	no	—
58	C₂²×C₃	—	2	8	no	—
59	Dic₃	—	6	6	no	—
60	Dic₃	—	6	8	no	—
61	Dic₃	—	6	2	no	—
62	C₃²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	2	4	no	—
64	C₂³	C₂×#157	1	6	yes	—
65	C₂³	C₂×#157	1	1	yes	characteristic
66	C₂×C₄	—	9	3	no	—
67	C₂×C₄	—	9	4	no	—
68	C₂×C₄	—	9	12	no	—
69	C₂×C₄	—	9	6	no	—
70	C₂×C₄	—	9	3	no	—
71	C₈	—	9	8	no	—
72	C₂×C₃	—	2	16	no	—
73	C₂×C₃	—	2	12	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂²	—	2	12	no	—
76	C₂²	#3792	1	4	yes	direct product factor
77	C₂²	C₂²×#157	1	4	yes	—
78	C₂²	C₂²×#157	1	3	yes	—
79	C₄	—	9	1	no	—
80	C₄	—	9	3	no	—
81	C₄	—	9	4	no	—
82	C₃	—	2	2	no	—
83	C₂	—	2	4	no	—
84	C₂	C₂×#3792	1	6	yes	direct product factor
85	C₂	C₂³×#157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#157
commutator	C₂×C₃²	C₂³×C₄
Frattini	C₂	C₂³×#157
Fitting, prime radical	C₂⁴×C₃²	C₄
socle	C₂³×C₃²	C₂×C₄
cosocle	C₃⋊Dic₃	C₂⁴
hypercenter	C₂⁴	#157
hypocenter	C₃²	C₂²×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂×#3792	C₂	direct (N × H)
3. C₂²×#622	C₂	semidirect
9. #3792	C₂²	direct (N × H)
10. C₂×#622	C₂²	semidirect
20. #622	C₂³	semidirect
28. C₂²×C₃²	M₁₆	semidirect
46. C₂×C₃²	C₂×M₁₆	semidirect
62. C₃²	C₂²×M₁₆	semidirect
76. C₂²	#3792	direct (N × H)
84. C₂	C₂×#3792	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×C₃⋊Dic₃	2
C₂×#3792	2
C₂²×#622	2
C₂²×M₁₆	9

Subgroups — C22×#3792 #10596652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3792 #10596652