Subgroups — C₂×#63748 #10593564

Order 576.

133 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3792	C₂	1	2	yes	maximal
2	C₂²×#622	C₂	1	1	yes	maximal, characteristic
3	#63748	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
5	#3792	—	2	2	no	—
6	#3792	C₂²	1	4	yes	—
7	C₂×#622	—	2	2	no	—
8	C₂×#622	C₄	1	2	yes	—
9	C₂×#622	C₄	1	2	yes	—
10	C₂×#622	C₂²	1	2	yes	—
11	C₂³×C₃²	Q₈	1	1	yes	characteristic
12	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
17	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
18	#622	—	2	4	no	—
19	#622	C₂×C₄	1	4	yes	—
20	C₂×#108	—	9	1	no	maximal, Sylow
21	C₂²×Dic₃	—	12	1	no	—
22	C₂²×C₃²	—	2	2	no	—
23	C₂²×C₃²	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₃²	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
26	C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
27	C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
28	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
29	C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
30	C₂×M₁₆	—	9	2	no	—
31	C₂²×C₈	—	9	1	no	—
32	#108	—	9	4	no	—
33	C₂³×C₃	—	4	1	no	—
34	C₂×Dic₃	—	12	1	no	—
35	C₂×Dic₃	—	12	1	no	—
36	C₂×Dic₃	—	12	1	no	—
37	C₂×Dic₃	—	12	1	no	—
38	C₂×Dic₃	—	12	1	no	—
39	C₂×Dic₃	—	12	1	no	—
40	C₂×C₃²	—	2	1	no	—
41	C₂×C₃²	—	2	1	no	—
42	C₂×C₃²	#108	1	2	yes	complemented
43	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄	—	9	1	no	—
45	M₁₆	—	9	4	no	—
46	M₁₆	—	18	2	no	—
47	C₂×C₈	—	9	2	no	—
48	C₂×C₈	—	9	2	no	—
49	C₂×C₈	—	9	2	no	—
50	C₂×C₈	—	18	2	no	—
51	C₂²×C₃	—	4	4	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂²×C₃	—	4	1	no	—
54	C₂²×C₃	—	4	1	no	—
55	Dic₃	—	12	1	no	—
56	Dic₃	—	12	1	no	—
57	Dic₃	—	12	1	no	—
58	Dic₃	—	12	1	no	—
59	C₃²	C₂×#108	1	1	yes	characteristic, complemented, Sylow
60	C₂³	PSU(3,𝔽₂)	1	1	yes	characteristic
61	C₂×C₄	—	9	1	no	—
62	C₂×C₄	—	9	1	no	—
63	C₂×C₄	—	9	1	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	9	1	no	—
66	C₂×C₄	—	9	1	no	—
67	C₈	—	9	4	no	—
68	C₈	—	18	4	no	—
69	C₂×C₃	—	4	2	no	—
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	—	4	2	no	—
72	C₂×C₃	—	4	1	no	—
73	C₂²	—	2	2	no	—
74	C₂²	#3777	1	1	yes	characteristic
75	C₂²	#3777	1	1	yes	characteristic
76	C₂²	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
77	C₄	—	9	1	no	—
78	C₄	—	9	1	no	—
79	C₄	—	9	1	no	—
80	C₄	—	9	1	no	—
81	C₃	—	4	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#63748	1	2	yes	direct product factor
85	C₂	C₂×#3777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#3777
commutator	C₃⋊Dic₃	C₂²×C₄
Frattini	C₂²	C₂×PSU(3,𝔽₂)
Fitting, prime radical	C₂³×C₃²	Q₈
socle	C₂²×C₃²	C₄⋊C₄
cosocle	C₂×C₃⋊Dic₃	C₂³
hypercenter	C₂³	PSU(3,𝔽₂)
hypocenter	C₃²	C₂×#108
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. #63748	C₂	direct (N × H)
42. C₂×C₃²	#108	semidirect
59. C₃²	C₂×#108	semidirect
84. C₂	#63748	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×#3792	2
C₂²×#622	2
#63748	2
C₂×#108	9

Subgroups — C2×#63748 #10593564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63748 #10593564