Subgroups — C₂²×#3748 #10596252

Order 576.

894 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#3748	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂²×#208	—	3	4	no	maximal
5	C₂⁴×C₃²	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₄	C₄	1	6	yes	—
7	C₂²×C₃²×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₄	C₂²	1	6	yes	—
10	#3748	C₂²	1	16	yes	direct product factor
11	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	12	yes	complemented
12	C₂³×C₃×C₄	D₃	1	4	yes	—
13	C₂×#208	—	3	48	no	—
14	C₂²×C₃⋊C₈	—	3	8	no	—
15	C₂³×C₃²	—	2	4	no	—
16	C₂³×C₃²	C₂×C₄	1	6	yes	—
17	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₄	C₂×C₄	1	12	yes	—
19	C₂×C₃²×C₄	C₂×C₄	1	3	yes	—
20	C₂×C₃²×C₄	C₂×C₄	1	6	yes	—
21	C₂×C₃²×C₄	C₂³	1	3	yes	—
22	C₂×C₃²×C₄	C₂³	1	4	yes	—
23	C₃⋊(C₃⋊C₈)	C₂³	1	8	yes	complemented
24	C₂²×M₁₆	—	9	1	no	Sylow
25	C₂⁴×C₃	Dic₃	1	4	yes	—
26	C₂²×C₃×C₄	Dic₃	1	24	yes	—
27	C₂²×C₃×C₄	Dic₃	1	4	yes	—
28	C₂²×C₃×C₄	C₂×D₃	1	24	yes	—
29	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
30	#208	—	3	64	no	—
31	C₂×C₃⋊C₈	—	3	48	no	—
32	C₂²×C₃²	—	2	12	no	—
33	C₂²×C₃²	M₁₆	1	4	yes	complemented
34	C₂²×C₃²	C₂²×C₄	1	4	yes	—
35	C₂²×C₃²	C₂²×C₄	1	3	yes	—
36	C₃²×C₄	C₂²×C₄	1	3	yes	—
37	C₃²×C₄	C₂²×C₄	1	4	yes	—
38	C₃²×C₄	C₂⁴	1	1	yes	characteristic
39	C₂³×C₄	C₃⋊D₃	1	1	yes	characteristic
40	C₂×M₁₆	—	9	12	no	—
41	C₂²×C₈	—	9	2	no	—
42	C₂³×C₃	—	2	16	no	—
43	C₂³×C₃	C₂×Dic₃	1	24	yes	—
44	C₂³×C₃	C₂×Dic₃	1	4	yes	—
45	C₂×C₃×C₄	C₂×Dic₃	1	48	yes	—
46	C₂×C₃×C₄	C₂×Dic₃	1	24	yes	—
47	C₂×C₃×C₄	C₂×Dic₃	1	12	yes	—
48	C₂×C₃×C₄	C₂²×D₃	1	12	yes	—
49	C₂×C₃×C₄	C₂²×D₃	1	16	yes	—
50	C₃⋊C₈	—	3	32	no	—
51	C₂×C₃²	—	2	4	no	—
52	C₂×C₃²	C₂×M₁₆	1	6	yes	complemented
53	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
54	C₂⁴	C₃⋊Dic₃	1	1	yes	characteristic
55	C₂²×C₄	C₃⋊Dic₃	1	1	yes	characteristic
56	C₂²×C₄	C₃⋊Dic₃	1	6	yes	—
57	C₂²×C₄	C₂×C₃⋊D₃	1	6	yes	—
58	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
59	M₁₆	—	9	16	no	—
60	C₂×C₈	—	9	12	no	—
61	C₂²×C₃	—	2	48	no	—
62	C₂²×C₃	#208	1	16	yes	complemented
63	C₂²×C₃	C₂²×Dic₃	1	16	yes	—
64	C₂²×C₃	C₂²×Dic₃	1	12	yes	—
65	C₃×C₄	C₂²×Dic₃	1	12	yes	—
66	C₃×C₄	C₂²×Dic₃	1	16	yes	—
67	C₃×C₄	C₂³×D₃	1	4	yes	—
68	C₃²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂³	—	2	4	no	—
70	C₂³	C₂×C₃⋊Dic₃	1	6	yes	—
71	C₂³	C₂×C₃⋊Dic₃	1	1	yes	characteristic
72	C₂×C₄	C₂×C₃⋊Dic₃	1	12	yes	—
73	C₂×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
74	C₂×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
75	C₂×C₄	C₂²×C₃⋊D₃	1	3	yes	—
76	C₂×C₄	C₂²×C₃⋊D₃	1	4	yes	—
77	C₈	—	9	8	no	—
78	C₂×C₃	—	2	16	no	—
79	C₂×C₃	C₂×#208	1	24	yes	complemented
80	C₂×C₃	C₂³×Dic₃	1	4	yes	—
81	C₂²	—	2	12	no	—
82	C₂²	#3748	1	4	yes	direct product factor
83	C₂²	C₂²×C₃⋊Dic₃	1	4	yes	—
84	C₂²	C₂²×C₃⋊Dic₃	1	3	yes	—
85	C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
86	C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
87	C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
88	C₃	C₂²×#208	1	4	yes	complemented
89	C₂	—	2	4	no	—
90	C₂	C₂×#3748	1	6	yes	direct product factor
91	C₂	C₂³×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₄
Frattini	C₄	C₂³×C₃⋊D₃
Fitting	C₂³×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂×C₄
prime radical	C₂⁴×C₃²	C₄
cosocle	C₃²×C₄	C₂⁴
hypercenter	C₂³×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#3748	C₂	direct (N × H)
3. C₂²×C₃⋊(C₃⋊C₈)	C₂	semidirect
10. #3748	C₂²	direct (N × H)
11. C₂×C₃⋊(C₃⋊C₈)	C₂²	semidirect
23. C₃⋊(C₃⋊C₈)	C₂³	semidirect
33. C₂²×C₃²	M₁₆	semidirect
52. C₂×C₃²	C₂×M₁₆	semidirect
62. C₂²×C₃	#208	semidirect
68. C₃²	C₂²×M₁₆	semidirect
79. C₂×C₃	C₂×#208	semidirect
82. C₂²	#3748	direct (N × H)
88. C₃	C₂²×#208	semidirect
90. C₂	C₂×#3748	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×C₃²×C₄	2
C₂×#3748	2
C₂²×C₃⋊(C₃⋊C₈)	2
C₂²×#208	3

Subgroups — C22×#3748 #10596252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3748 #10596252