Subgroups — #10593565

Order 576.

129 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	C₂×#3792	C₂	1	1	yes	maximal, characteristic
3	#63748	C₂	1	4	yes	maximal, complemented
4	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
5	#3792	—	2	2	no	—
6	#3792	C₄	1	4	yes	—
7	#3792	C₂²	1	4	yes	—
8	C₂×#622	—	2	2	no	—
9	C₂×#622	C₂²	1	2	yes	—
10	C₂³×C₃²	Q₈	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
12	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
13	C₂×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
15	#622	—	2	4	no	—
16	#622	C₂×C₄	1	4	yes	—
17	#430	—	9	1	no	maximal, Sylow
18	C₂²×Dic₃	—	12	1	no	—
19	C₂²×C₃²	—	4	1	no	—
20	C₂²×C₃²	C₄⋊C₄	1	2	yes	—
21	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
22	C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
23	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
24	C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
25	C₂×M₁₆	—	9	1	no	—
26	C₂×M₁₆	—	9	2	no	—
27	#108	—	9	4	no	—
28	C₂³×C₃	—	4	1	no	—
29	C₂×Dic₃	—	12	2	no	—
30	C₂×Dic₃	—	12	2	no	—
31	C₂×Dic₃	—	12	1	no	—
32	C₂×Dic₃	—	12	1	no	—
33	C₂×C₃²	—	2	2	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄	—	9	1	no	—
37	M₁₆	—	9	4	no	—
38	M₁₆	—	9	4	no	—
39	M₁₆	—	18	2	no	—
40	C₂×C₈	—	9	2	no	—
41	C₂×C₈	—	18	2	no	—
42	C₂²×C₃	—	4	4	no	—
43	C₂²×C₃	—	4	1	no	—
44	C₂²×C₃	—	4	2	no	—
45	Dic₃	—	12	2	no	—
46	Dic₃	—	12	1	no	—
47	Dic₃	—	12	1	no	—
48	C₃²	#430	1	1	yes	characteristic, complemented, Sylow
49	C₂³	PSU(3,𝔽₂)	1	1	yes	characteristic
50	C₂×C₄	—	9	1	no	—
51	C₂×C₄	—	9	1	no	—
52	C₂×C₄	—	9	2	no	—
53	C₂×C₄	—	9	2	no	—
54	C₈	—	9	4	no	—
55	C₈	—	18	4	no	—
56	C₂×C₃	—	4	2	no	—
57	C₂×C₃	—	4	4	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂²	—	4	1	no	—
60	C₂²	#3777	1	2	yes	—
61	C₂²	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
62	C₄	—	9	1	no	—
63	C₄	—	9	1	no	—
64	C₄	—	9	2	no	—
65	C₃	—	4	1	no	—
66	C₂	—	2	1	no	—
67	C₂	—	2	2	no	—
68	C₂	C₂×#3777	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#3777
commutator	C₃⋊Dic₃	C₂²×C₄
Frattini	C₂²	C₂×PSU(3,𝔽₂)
Fitting, prime radical	C₂³×C₃²	Q₈
socle	C₂×C₃²	C₂×C₄⋊C₄
cosocle	C₂×C₃⋊Dic₃	C₂³
hypercenter	C₂³	PSU(3,𝔽₂)
hypocenter	C₃²	#430
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. #63748	C₂	semidirect
48. C₃²	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3792	2
#63748	2
#430	9