Subgroups — #11782571

Order 1152.

534 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592393	C₂	1	3	yes	maximal
2	#10592380	C₂	1	3	yes	maximal, complemented
3	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	#68061	—	3	4	no	maximal
5	#64083	C₂²	1	1	yes	characteristic
6	#64078	C₂²	1	3	yes	—
7	C₃²×C₄×C₈	C₂²	1	3	yes	—
8	#63648	—	2	6	no	—
9	#63643	—	2	6	no	—
10	#4694	—	3	12	no	—
11	#4681	—	3	12	no	—
12	C₃×C₈²	D₃	1	4	yes	complemented
13	C₂×C₄⋊(C₃⋊D₃)	—	2	3	no	—
14	C₂×#634	—	2	3	no	—
15	C₂×C₃²×C₈	D₄	1	6	yes	—
16	C₃²×C₄²	C₂³	1	1	yes	characteristic
17	C₄⋊(C₃⋊Dic₃)	—	2	3	no	—
18	C₄⋊(C₃⋊Dic₃)	—	2	3	no	—
19	#1711	—	9	1	no	Sylow
20	#874	—	3	4	no	—
21	#869	—	3	12	no	—
22	C₃×C₄×C₈	C₂×D₃	1	12	yes	—
23	#821	—	6	24	no	—
24	#816	—	6	24	no	—
25	C₂²×C₃⋊D₃	—	4	1	no	—
26	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
27	C₂×C₃⋊Dic₃	—	4	3	no	—
28	C₄⋊(C₃⋊D₃)	—	4	6	no	—
29	#634	—	4	6	no	—
30	C₃²×C₈	—	2	6	no	—
31	#487	—	9	3	no	—
32	#486	—	9	3	no	—
33	C₈²	C₃⋊D₃	1	1	yes	characteristic, complemented
34	C₂×D₁₂	—	6	12	no	—
35	C₂×Dic₆	—	6	12	no	—
36	C₂×C₃×C₈	D₁₂	1	24	yes	—
37	C₃×C₄²	C₂²×D₃	1	4	yes	—
38	C₄⋊Dic₃	—	6	12	no	—
39	C₄⋊Dic₃	—	6	12	no	—
40	C₂²×C₃²	#127	1	1	yes	characteristic
41	C₂×C₃⋊D₃	—	8	3	no	—
42	C₃²×C₄	#135	1	6	yes	—
43	C₃⋊Dic₃	—	8	3	no	—
44	C₄⋊Q₈	—	9	3	no	—
45	#127	—	9	1	no	—
46	#103	—	18	6	no	—
47	#102	—	18	6	no	—
48	C₄×C₈	C₂×C₃⋊D₃	1	3	yes	—
49	C₂²×D₃	—	12	4	no	—
50	C₂×C₃×C₄	C₂×D₁₂	1	12	yes	—
51	C₂×Dic₃	—	12	12	no	—
52	D₁₂	—	12	24	no	—
53	Dic₆	—	12	24	no	—
54	C₃×C₈	—	2	24	no	—
55	C₂×C₃²	#495	1	3	yes	—
56	C₃⋊D₃	—	16	1	no	—
57	C₂×Q₈	—	18	3	no	—
58	C₂×D₄	—	18	3	no	—
59	C₂×C₈	C₄⋊(C₃⋊D₃)	1	6	yes	—
60	C₄⋊C₄	—	18	3	no	—
61	C₄⋊C₄	—	18	3	no	—
62	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₂²×C₃	#874	1	4	yes	—
64	C₂×D₃	—	24	12	no	—
65	C₃×C₄	#904	1	24	yes	—
66	Dic₃	—	24	12	no	—
67	C₃²	#1711	1	1	yes	characteristic, complemented, Sylow
68	C₂³	—	36	1	no	—
69	Q₈	—	36	6	no	—
70	D₄	—	36	6	no	—
71	C₂×C₄	—	36	3	no	—
72	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₂×C₃	#4696	1	12	yes	—
75	D₃	—	48	4	no	—
76	C₂²	—	72	3	no	—
77	C₂²	#64083	1	1	yes	characteristic
78	C₄	—	72	3	no	—
79	C₄	#64113	1	6	yes	—
80	C₃	#68061	1	4	yes	complemented
81	C₂	—	144	1	no	—
82	C₂	#10592395	1	3	yes	—

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×C₈²	C₂
socle	C₂²×C₃²	#127
prime radical	#64083	C₂²
hypercenter	C₈²	C₃⋊D₃
hypocenter	C₃²	#1711
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #10592380	C₂	semidirect
3. C₃²×C₈²	C₂	semidirect
12. C₃×C₈²	D₃	semidirect
33. C₈²	C₃⋊D₃	semidirect
67. C₃²	#1711	semidirect
80. C₃	#68061	semidirect

Maximal subgroups

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

subgroup	index
#10592393	2
#10592380	2
C₃²×C₈²	2
#68061	3