Subgroups — #10596485

Order 576.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64287	C₂	1	1	yes	maximal, characteristic
2	#64218	C₂	1	2	yes	maximal, complemented
3	C₂×#3771	C₂	1	2	yes	maximal, complemented
4	#64214	C₂	1	1	yes	maximal, characteristic, complemented
5	#64213	C₂	1	1	yes	maximal, characteristic, complemented
6	#3830	C₄	1	1	yes	characteristic
7	#3792	C₄	1	2	yes	—
8	C₂×#622	C₂²	1	2	yes	—
9	#3788	C₄	1	1	yes	characteristic
10	#3787	C₂²	1	1	yes	characteristic
11	#3771	C₂²	1	1	yes	characteristic
12	#3771	C₂²	1	1	yes	characteristic
13	#3771	C₂²	1	2	yes	—
14	C₃²×D₄	C₈	1	1	yes	characteristic
15	#638	C₈	1	2	yes	—
16	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
17	C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
18	#634	C₈	1	1	yes	characteristic
19	#622	C₂×C₄	1	1	yes	characteristic
20	#622	C₂×C₄	1	2	yes	—
21	#622	C₂³	1	1	yes	characteristic
22	#504	—	9	1	no	maximal, Sylow
23	#237	—	12	1	no	—
24	C₂²×C₃²	C₂×C₈	1	2	yes	—
25	C₂×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
26	C₃²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₃⋊Dic₃	C₂×C₈	1	2	yes	—
28	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
29	#131	—	9	1	no	—
30	M₃₂	—	9	1	no	—
31	M₃₂	—	9	2	no	—
32	C₂×C₁₆	—	9	1	no	—
33	C₂×C₁₆	—	9	2	no	—
34	C₃×D₄	—	4	1	no	—
35	#67	—	12	2	no	—
36	C₂×Dic₃	—	12	2	no	—
37	C₄×D₃	—	12	1	no	—
38	Dic₆	—	12	1	no	—
39	C₂×C₃²	—	2	2	no	—
40	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
41	C₃⋊D₃	—	2	1	no	—
42	#41	—	9	1	no	—
43	M₁₆	—	9	1	no	—
44	M₁₆	—	9	2	no	—
45	C₂×C₈	—	9	1	no	—
46	C₂×C₈	—	9	2	no	—
47	C₁₆	—	9	1	no	—
48	C₁₆	—	9	1	no	—
49	C₁₆	—	9	2	no	—
50	C₂²×C₃	—	4	2	no	—
51	C₂×D₃	—	12	1	no	—
52	C₃×C₄	—	4	1	no	—
53	Dic₃	—	12	2	no	—
54	Dic₃	—	12	1	no	—
55	C₃²	#504	1	1	yes	characteristic, complemented, Sylow
56	Q₈	—	9	1	no	—
57	D₄	—	9	2	no	—
58	D₄	AGL(1,𝔽₉)	1	1	yes	characteristic
59	C₂×C₄	—	9	2	no	—
60	C₂×C₄	—	9	1	no	—
61	C₈	—	9	1	no	—
62	C₈	—	9	1	no	—
63	C₈	—	9	2	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	—	8	2	no	—
66	D₃	—	24	1	no	—
67	C₂²	—	9	1	no	—
68	C₂²	C₂×AGL(1,𝔽₉)	1	2	yes	—
69	C₄	—	9	1	no	—
70	C₄	—	9	2	no	—
71	C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
72	C₃	—	4	1	no	—
73	C₂	—	2	2	no	—
74	C₂	—	18	1	no	—
75	C₂	C₂²×AGL(1,𝔽₉)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×AGL(1,𝔽₉)
commutator, socle	C₂×C₃²	C₂²×C₈
Fitting	C₃²×D₄	C₈
prime radical	#3830	C₄
cosocle	#622	C₂³
hypercenter	D₄	AGL(1,𝔽₉)
hypocenter	C₃²	#504
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #64218	C₂	semidirect
3. C₂×#3771	C₂	semidirect
4. #64214	C₂	semidirect
5. #64213	C₂	semidirect
55. C₃²	#504	semidirect

Maximal subgroups

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

subgroup	index
#64287	2
#64218	2
C₂×#3771	2
#64214	2
#64213	2
#504	9