Subgroups — #10596404

Order 576.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#64222	C₂	1	1	yes	maximal, characteristic
3	C₄×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic
4	PSU(3,𝔽₂)⋊C₄	C₂	1	4	yes	maximal
5	C₂×PSU(3,𝔽₂)	C₂²	1	2	yes	—
6	C₂×AGL(1,𝔽₉)	C₂²	1	2	yes	—
7	#3797	—	2	1	no	—
8	C₄⋊#157	—	2	1	no	—
9	C₄×#157	C₂²	1	1	yes	characteristic
10	#3777	—	2	1	no	—
11	#3773	—	2	1	no	—
12	#3773	C₂²	1	2	yes	—
13	C₂×#157	—	2	2	no	—
14	C₂×#157	D₄	1	1	yes	characteristic
15	C₂×#157	C₂³	1	1	yes	characteristic
16	PSU(3,𝔽₂)	—	2	4	no	—
17	AGL(1,𝔽₉)	—	2	2	no	—
18	C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
19	C₃×Dic₆	—	4	1	no	—
20	#627	—	4	1	no	—
21	#624	—	2	2	no	—
22	#487	—	9	1	no	maximal, Sylow
23	D₃×Q₈	—	12	1	no	—
24	C₂×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
25	#157	—	4	2	no	—
26	#157	Q₁₆	1	2	yes	—
27	#157	#41	1	2	yes	—
28	C₃²×C₄	SD₁₆	1	1	yes	characteristic
29	C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
30	C₃×Dic₃	—	4	2	no	—
31	C₄⋊Q₈	—	9	1	no	—
32	C₄⋊Q₈	—	9	1	no	—
33	#103	—	9	4	no	—
34	C₄×C₈	—	9	1	no	—
35	C₃×Q₈	—	12	1	no	—
36	C₄×D₃	—	12	1	no	—
37	C₄×D₃	—	12	2	no	—
38	Dic₆	—	4	1	no	—
39	Dic₆	—	12	2	no	—
40	C₂×C₃²	C₂×SD₁₆	1	1	yes	characteristic
41	C₃⋊D₃	#124	1	1	yes	characteristic
42	C₃⋊D₃	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×Q₈	—	9	2	no	—
44	C₂×Q₈	—	18	1	no	—
45	C₂×C₈	—	9	2	no	—
46	C₄⋊C₄	—	9	2	no	—
47	C₄⋊C₄	—	18	1	no	—
48	C₄⋊C₄	—	18	1	no	—
49	C₄⋊C₄	—	18	1	no	—
50	C₄²	—	9	1	no	—
51	C₂×D₃	—	12	1	no	—
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	—	12	2	no	—
54	Dic₃	—	4	2	no	—
55	Dic₃	—	12	1	no	—
56	C₃²	#487	1	1	yes	characteristic, complemented, Sylow
57	Q₈	—	12	1	no	—
58	Q₈	—	18	4	no	—
59	Q₈	—	36	1	no	—
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	9	1	no	—
62	C₂×C₄	—	9	1	no	—
63	C₂×C₄	—	18	2	no	—
64	C₂×C₄	—	18	2	no	—
65	C₈	—	18	2	no	—
66	C₂×C₃	—	4	1	no	—
67	D₃	—	12	1	no	—
68	D₃	—	12	1	no	—
69	C₂²	—	9	1	no	—
70	C₄	—	9	2	no	—
71	C₄	—	9	2	no	—
72	C₄	—	9	1	no	—
73	C₄	—	12	2	no	—
74	C₄	—	36	2	no	—
75	C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
76	C₃	—	4	1	no	—
77	C₂	—	9	1	no	—
78	C₂	—	9	1	no	—
79	C₂	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×AΓL(1,𝔽₉)
commutator, cosocle	C₂×#157	C₂³
Fitting	C₃²×C₄	SD₁₆
socle	C₂×C₃²	C₂×SD₁₆
prime radical	C₂×C₃⋊D₃	C₂×D₄
hypercenter	C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#487
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
56. C₃²	#487	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊PSU(3,𝔽₂)	2
#64222	2
C₄×AGL(1,𝔽₉)	2
PSU(3,𝔽₂)⋊C₄	2
#487	9