Subgroups — #11772201

Order 1152.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10590150	C₂	1	2	yes	maximal
2	#10590148	C₂	1	1	yes	maximal, characteristic
3	#63773	—	2	2	no	—
4	C₂×#3708	C₄	1	1	yes	characteristic
5	C₂×#3708	C₄	1	1	yes	characteristic
6	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
7	C₃²×C₄²	Q₈	1	1	yes	characteristic
8	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
10	#3708	—	2	1	no	—
11	#3708	—	4	2	no	—
12	#3708	C₈	1	2	yes	—
13	#1369	—	9	1	no	maximal, Sylow
14	C₄×C₃⋊C₈	—	12	1	no	—
15	C₂×C₃²×C₄	—	2	1	no	—
16	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
17	C₃⋊(C₃⋊C₈)	C₂×C₈	1	1	yes	characteristic
18	C₃⋊(C₃⋊C₈)	M₁₆	1	1	yes	characteristic
19	C₃⋊(C₃⋊C₈)	SD₁₆	1	2	yes	—
20	#364	—	9	2	no	—
21	#346	—	9	1	no	—
22	C₃×C₄²	—	4	1	no	—
23	C₂×C₃⋊C₈	—	12	1	no	—
24	C₂×C₃⋊C₈	—	12	1	no	—
25	C₂²×C₃²	#108	1	1	yes	characteristic
26	C₃²×C₄	—	4	1	no	—
27	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
28	C₃²×C₄	#106	1	1	yes	characteristic
29	M₃₂	—	18	2	no	—
30	C₂×C₁₆	—	9	1	no	—
31	C₂×C₁₆	—	9	1	no	—
32	C₄×C₈	—	9	1	no	—
33	C₂×C₃×C₄	—	4	2	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	C₃⋊C₈	—	12	1	no	—
36	C₃⋊C₈	—	12	1	no	—
37	C₃⋊C₈	—	12	2	no	—
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	#334	1	1	yes	characteristic
40	C₂×C₈	—	9	1	no	—
41	C₂×C₈	—	9	1	no	—
42	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
43	C₁₆	—	9	2	no	—
44	C₁₆	—	18	1	no	—
45	C₁₆	—	36	2	no	—
46	C₂²×C₃	—	4	1	no	—
47	C₃×C₄	—	4	4	no	—
48	C₃×C₄	—	4	1	no	—
49	C₃×C₄	—	4	1	no	—
50	C₃²	#1369	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	#3777	1	1	yes	characteristic
53	C₈	—	9	1	no	—
54	C₈	—	9	1	no	—
55	C₈	—	9	2	no	—
56	C₂×C₃	—	4	2	no	—
57	C₂×C₃	—	4	1	no	—
58	C₂²	#63748	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	#63744	1	1	yes	characteristic
61	C₄	#63745	1	1	yes	characteristic
62	C₃	—	4	1	no	—
63	C₂	—	2	1	no	—
64	C₂	#10590088	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63745
commutator	C₃⋊(C₃⋊C₈)	C₂×C₈
Frattini, hypercenter	C₄²	PSU(3,𝔽₂)
Fitting	C₃²×C₄²	Q₈
socle	C₂×C₃²	#334
prime radical	C₂²×C₃²	#108
cosocle	C₄×C₃⋊(C₃⋊C₈)	C₂²
hypocenter	C₃²	#1369
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
50. C₃²	#1369	semidirect

Maximal subgroups

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

subgroup	index
#10590150	2
#10590148	2
#1369	9