Subgroups — C₂×#64218 #10596490

Order 576.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#622	C₂	1	1	yes	maximal, characteristic
2	#64218	C₂	1	4	yes	maximal, direct product factor
3	C₂×#3771	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃⋊Dic₃	C₄	1	1	yes	characteristic
5	C₂×#622	C₄	1	2	yes	—
6	C₂×#622	C₄	1	1	yes	characteristic
7	C₂×#622	C₂²	1	1	yes	characteristic
8	C₂×#622	C₂²	1	2	yes	—
9	#3771	C₂²	1	4	yes	complemented
10	C₂³×C₃²	C₈	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₈	1	2	yes	—
12	C₂×C₃⋊Dic₃	C₈	1	1	yes	characteristic
13	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
14	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
15	#622	C₂×C₄	1	1	yes	characteristic
16	#622	C₂×C₄	1	2	yes	—
17	#622	C₂³	1	1	yes	characteristic
18	C₂×M₃₂	—	9	1	no	maximal, Sylow
19	C₂²×Dic₃	—	12	1	no	—
20	C₂²×C₃²	—	2	2	no	—
21	C₂²×C₃²	C₂×C₈	1	2	yes	—
22	C₂²×C₃²	C₂×C₈	1	1	yes	characteristic
23	C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
24	C₃⋊Dic₃	C₂×C₈	1	2	yes	—
25	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₈	—	9	1	no	—
27	M₃₂	—	9	4	no	—
28	C₂×C₁₆	—	9	2	no	—
29	C₂³×C₃	—	4	1	no	—
30	C₂×Dic₃	—	12	2	no	—
31	C₂×Dic₃	—	12	1	no	—
32	C₂×Dic₃	—	12	2	no	—
33	C₂×Dic₃	—	12	1	no	—
34	C₂×C₃²	—	2	2	no	—
35	C₂×C₃²	M₃₂	1	2	yes	complemented
36	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
37	C₂²×C₄	—	9	1	no	—
38	C₂×C₈	—	9	1	no	—
39	C₂×C₈	—	9	2	no	—
40	C₂×C₈	—	9	2	no	—
41	C₂×C₈	—	9	1	no	—
42	C₁₆	—	9	4	no	—
43	C₂²×C₃	—	4	4	no	—
44	C₂²×C₃	—	4	2	no	—
45	C₂²×C₃	—	4	1	no	—
46	Dic₃	—	12	2	no	—
47	Dic₃	—	12	1	no	—
48	Dic₃	—	12	1	no	—
49	C₃²	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
50	C₂³	AGL(1,𝔽₉)	1	1	yes	characteristic
51	C₂×C₄	—	9	2	no	—
52	C₂×C₄	—	9	2	no	—
53	C₂×C₄	—	9	1	no	—
54	C₂×C₄	—	9	1	no	—
55	C₈	—	9	1	no	—
56	C₈	—	9	1	no	—
57	C₈	—	9	2	no	—
58	C₂×C₃	—	4	4	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	4	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	C₂×AGL(1,𝔽₉)	1	2	yes	—
63	C₂²	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
64	C₄	—	9	1	no	—
65	C₄	—	9	2	no	—
66	C₄	—	9	1	no	—
67	C₃	—	4	1	no	—
68	C₂	—	2	2	no	—
69	C₂	#64218	1	2	yes	direct product factor
70	C₂	C₂²×AGL(1,𝔽₉)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×AGL(1,𝔽₉)
commutator	C₂×C₃²	C₂²×C₈
Frattini	C₂	C₂²×AGL(1,𝔽₉)
Fitting, prime radical	C₂³×C₃²	C₈
socle	C₂²×C₃²	C₂×C₈
cosocle	#622	C₂³
hypercenter	C₂³	AGL(1,𝔽₉)
hypocenter	C₃²	C₂×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. #64218	C₂	direct (N × H)
3. C₂×#3771	C₂	semidirect
9. #3771	C₂²	semidirect
35. C₂×C₃²	M₃₂	semidirect
49. C₃²	C₂×M₃₂	semidirect
69. C₂	#64218	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#622	2
#64218	2
C₂×#3771	2
C₂×M₃₂	9

Subgroups — C2×#64218 #10596490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#64218 #10596490