Subgroups — #11720708

Order 864.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1088	C₂	1	1	yes	maximal, characteristic
2	#88626	C₂	1	2	yes	maximal
3	C₂×#1088	C₄	1	1	yes	characteristic
4	C₂×#1088	C₄	1	1	yes	characteristic
5	C₄×#272	C₂²	1	1	yes	characteristic
6	#6188	—	2	2	no	—
7	C₂×#272	C₂×C₄	1	1	yes	characteristic
8	#1088	—	2	1	no	—
9	#1088	C₈	1	2	yes	complemented
10	C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
11	#1084	D₄	1	1	yes	characteristic
12	C₃×C₄⋊C₈	—	9	1	no	maximal
13	C₃×C₄×D₃	—	12	1	no	—
14	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
15	#272	C₂×C₈	1	1	yes	characteristic
16	#272	M₁₆	1	1	yes	characteristic
17	C₂×C₃×C₈	—	9	2	no	—
18	C₃×C₄²	—	9	1	no	—
19	C₂×C₃×D₃	—	12	1	no	—
20	C₃²×C₄	—	4	1	no	—
21	C₃×Dic₃	—	12	1	no	—
22	C₄⋊C₈	—	9	1	no	Sylow
23	H(3,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃×C₄	—	9	1	no	—
25	C₂×C₃×C₄	—	9	1	no	—
26	C₂×C₃×C₄	—	9	1	no	—
27	C₄×D₃	—	12	1	no	—
28	C₃×C₈	—	18	2	no	—
29	C₂×C₃²	—	4	1	no	—
30	C₃×D₃	—	12	1	no	—
31	C₃×D₃	—	12	1	no	—
32	C₂×C₈	—	9	2	no	—
33	C₄²	—	9	1	no	—
34	C₂²×C₃	—	9	1	no	—
35	C₂×D₃	—	12	1	no	—
36	C₃×C₄	—	9	2	no	—
37	C₃×C₄	—	9	1	no	—
38	C₃×C₄	—	12	1	no	—
39	C₃×C₄	—	18	1	no	—
40	C₃×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
41	Dic₃	—	12	1	no	—
42	C₃²	—	4	1	no	—
43	C₂×C₄	—	9	1	no	—
44	C₂×C₄	—	9	1	no	—
45	C₂×C₄	—	9	1	no	—
46	C₈	—	18	2	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	—	9	1	no	—
49	C₂×C₃	—	12	1	no	—
50	C₂×C₃	#3777	1	1	yes	characteristic
51	D₃	—	12	1	no	—
52	D₃	—	12	1	no	—
53	C₂²	—	9	1	no	—
54	C₄	—	9	2	no	—
55	C₄	—	9	1	no	—
56	C₄	—	18	1	no	—
57	C₄	SU(3,𝔽₂)	1	1	yes	characteristic
58	C₃	—	12	1	no	—
59	C₃	#63744	1	1	yes	characteristic
60	C₂	—	9	1	no	—
61	C₂	—	9	1	no	—
62	C₂	#88592	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃×C₄	PSU(3,𝔽₂)
commutator	#272	C₂×C₈
Fitting	C₄×H(3,𝔽₃)	Q₈
socle	C₂×C₃	#3777
prime radical	C₂×#272	C₂×C₄
cosocle	C₄×#272	C₂²
hypocenter	H(3,𝔽₃)	C₄⋊C₈
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
9. #1088	C₈	semidirect
23. H(3,𝔽₃)	C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1088	2
#88626	2
C₃×C₄⋊C₈	9