Subgroups — #11782556

Order 1152.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592381	C₂	1	2	yes	maximal
2	#10592380	C₂	1	4	yes	maximal
3	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic
4	#68064	—	3	4	no	maximal
5	#64078	C₂²	1	4	yes	—
6	C₃²×C₄×C₈	C₂²	1	2	yes	—
7	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#63645	—	2	4	no	—
9	#63643	—	2	8	no	—
10	#4682	—	3	8	no	—
11	#4681	—	3	16	no	—
12	C₃×C₈²	D₃	1	4	yes	—
13	C₂×#634	—	2	4	no	—
14	C₂×C₃²×C₈	D₄	1	4	yes	—
15	C₂×C₃²×C₈	D₄	1	2	yes	—
16	C₃²×C₄²	C₂³	1	1	yes	characteristic
17	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
18	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
19	#1710	—	9	1	no	Sylow
20	#869	—	3	16	no	—
21	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
22	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
23	#818	—	6	16	no	—
24	#816	—	6	32	no	—
25	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
26	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
27	C₂×C₃⋊Dic₃	—	4	4	no	—
28	#634	—	4	8	no	—
29	C₃²×C₈	—	2	4	no	—
30	C₃²×C₈	Q₁₆	1	4	yes	—
31	#500	—	9	2	no	—
32	#487	—	9	4	no	—
33	C₈²	C₃⋊D₃	1	1	yes	characteristic
34	C₂×Dic₆	—	6	16	no	—
35	C₂×C₃×C₈	D₁₂	1	16	yes	—
36	C₂×C₃×C₈	D₁₂	1	8	yes	—
37	C₃×C₄²	C₂²×D₃	1	4	yes	—
38	C₄⋊Dic₃	—	6	16	no	—
39	C₄⋊Dic₃	—	6	16	no	—
40	C₂²×C₃²	#127	1	1	yes	characteristic
41	C₃²×C₄	C₂×Q₁₆	1	2	yes	—
42	C₃²×C₄	#135	1	4	yes	—
43	C₃⋊Dic₃	—	8	4	no	—
44	C₄⋊Q₈	—	9	4	no	—
45	#107	—	18	4	no	—
46	#103	—	18	8	no	—
47	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
48	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
50	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
51	C₂×Dic₃	—	12	16	no	—
52	Dic₆	—	12	32	no	—
53	C₃×C₈	—	2	16	no	—
54	C₃×C₈	Dic₁₂	1	16	yes	—
55	C₂×C₃²	#494	1	1	yes	characteristic
56	C₂×C₃²	#495	1	2	yes	—
57	C₂×Q₈	—	18	4	no	—
58	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
59	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
60	C₄⋊C₄	—	18	4	no	—
61	C₄⋊C₄	—	18	4	no	—
62	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₂²×C₃	#874	1	4	yes	—
64	C₃×C₄	#904	1	16	yes	—
65	C₃×C₄	C₂×Dic₁₂	1	8	yes	—
66	Dic₃	—	24	16	no	—
67	C₃²	#1710	1	1	yes	characteristic, complemented, Sylow
68	Q₈	—	36	8	no	—
69	C₂×C₄	—	36	4	no	—
70	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
71	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₈	—	2	4	no	—
73	C₈	#3746	1	4	yes	—
74	C₂×C₃	#4696	1	8	yes	—
75	C₂×C₃	#4699	1	4	yes	—
76	C₂²	#64083	1	1	yes	characteristic
77	C₄	—	72	4	no	—
78	C₄	#64113	1	4	yes	—
79	C₄	C₂×#3746	1	2	yes	—
80	C₃	#68064	1	4	yes	complemented
81	C₂	#10592395	1	2	yes	—
82	C₂	#10592398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×C₈²	C₂
socle, prime radical	C₂²×C₃²	#127
hypercenter	C₈²	C₃⋊D₃
hypocenter	C₃²	#1710
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
67. C₃²	#1710	semidirect
80. C₃	#68064	semidirect

Maximal subgroups

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

subgroup	index
#10592381	2
#10592380	2
C₃²×C₈²	2
#68064	3