Subgroups — #11773056

Order 1152.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#368	C₂	1	1	yes	maximal, characteristic
2	#10589372	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#67581	—	3	4	no	maximal
5	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	C₃²×#108	—	2	1	no	—
7	#63646	—	2	1	no	—
8	C₃⋊(C₃⋊C₃₂)	C₄	1	2	yes	—
9	C₃×#368	D₃	1	4	yes	—
10	#4506	—	3	4	no	—
11	C₂×C₃⋊C₃₂	—	3	4	no	—
12	C₃²×M₁₆	—	4	1	no	—
13	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
14	#3748	—	4	1	no	—
15	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₁₆	Q₈	1	1	yes	characteristic
17	#1425	—	9	1	no	Sylow
18	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
19	C₃×#108	—	2	4	no	—
20	#819	—	6	4	no	—
21	C₃⋊C₃₂	—	3	8	no	—
22	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
23	C₃²×C₈	—	8	1	no	—
24	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₃²×C₈	Q₁₆	1	1	yes	characteristic
26	C₃⋊(C₃⋊C₈)	—	8	1	no	—
27	C₂×C₃₂	—	9	1	no	—
28	#368	—	9	1	no	—
29	#368	C₃⋊D₃	1	1	yes	characteristic
30	C₃×M₁₆	—	4	4	no	—
31	C₂×C₃×C₈	#67	1	4	yes	—
32	#208	—	12	4	no	—
33	C₃×C₁₆	Dic₆	1	4	yes	—
34	C₃×C₁₆	C₄×D₃	1	4	yes	—
35	C₂²×C₃²	Q₃₂	1	1	yes	characteristic
36	C₃²×C₄	#107	1	1	yes	characteristic
37	C₃²×C₄	D₁₆	1	1	yes	characteristic
38	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
39	#108	—	2	1	no	—
40	#108	—	18	1	no	—
41	C₃₂	—	9	2	no	—
42	C₂×C₃×C₄	#213	1	4	yes	—
43	C₃×C₈	—	8	4	no	—
44	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
45	C₃×C₈	#216	1	4	yes	—
46	C₃⋊C₈	—	24	4	no	—
47	C₂×C₃²	—	2	1	no	—
48	C₂×C₃²	#366	1	1	yes	characteristic
49	M₁₆	—	4	1	no	—
50	M₁₆	—	36	1	no	—
51	C₂×C₈	#638	1	1	yes	characteristic
52	C₁₆	#634	1	1	yes	characteristic
53	C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
54	C₂²×C₃	#829	1	4	yes	—
55	C₃×C₄	#807	1	4	yes	—
56	C₃×C₄	#826	1	4	yes	—
57	C₃²	#1425	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄	#3753	1	1	yes	characteristic
59	C₈	—	8	1	no	—
60	C₈	—	72	1	no	—
61	C₈	#3750	1	1	yes	characteristic
62	C₈	#3756	1	1	yes	characteristic
63	C₂×C₃	—	2	4	no	—
64	C₂×C₃	#4489	1	4	yes	—
65	C₂²	#63656	1	1	yes	characteristic
66	C₄	#63634	1	1	yes	characteristic
67	C₄	#63653	1	1	yes	characteristic
68	C₃	#67581	1	4	yes	complemented
69	C₂	—	2	1	no	—
70	C₂	#10589355	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63653
commutator	C₃²×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#368	C₂
socle	C₂×C₃²	#366
prime radical	C₂²×C₃²	Q₃₂
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	#368	C₃⋊D₃
hypocenter	C₃²	#1425
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
57. C₃²	#1425	semidirect
68. C₃	#67581	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#368	2
#10589372	2
C₂×C₃⋊(C₃⋊C₃₂)	2
#67581	3