Subgroups — C₉×#2982 #11767703

Order 1152.

594 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₉	C₂	1	8	yes	maximal, complemented
3	C₉×#431	C₂	1	6	yes	maximal
4	C₃×#2982	C₃	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₉	C₄	1	4	yes	—
7	C₄×Q₈×C₉	C₂²	1	4	yes	—
8	C₂×C₉×C₄⋊C₄	C₄	1	3	yes	—
9	C₂×C₄²×C₉	C₂²	1	3	yes	—
10	C₉×C₄⋊C₈	C₂²	1	12	yes	—
11	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₄×C₈×C₉	C₂²	1	12	yes	—
13	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
14	C₃×C₈×Q₈	C₂×C₃	1	8	yes	—
15	C₃×#431	C₂×C₃	1	6	yes	—
16	C₂×Q₈×C₉	C₈	1	8	yes	complemented
17	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
19	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
20	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₉	C₂³	1	8	yes	—
22	C₉×C₄⋊C₄	C₂×C₄	1	12	yes	—
23	C₄²×C₉	C₂×C₄	1	6	yes	—
24	C₄²×C₉	C₂³	1	6	yes	—
25	#2982	C₉	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
27	C₃×C₄×Q₈	C₃×C₄	1	4	yes	—
28	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
29	C₂×C₃×C₄⋊C₄	C₃×C₄	1	3	yes	—
30	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
31	C₃×C₄⋊C₈	C₂²×C₃	1	12	yes	—
32	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
33	C₃×C₄×C₈	C₂²×C₃	1	12	yes	—
34	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₉	C₂×C₈	1	16	yes	—
36	C₂×C₄×C₉	—	2	2	no	—
37	C₂×C₄×C₉	—	4	3	no	—
38	C₂×C₄×C₉	C₂×C₈	1	12	yes	—
39	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
41	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
42	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
43	C₈×C₉	—	2	8	no	—
44	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
45	C₈×Q₈	C₂×C₉	1	8	yes	complemented
46	#431	C₂×C₉	1	6	yes	—
47	C₂×C₃×Q₈	C₃×C₈	1	8	yes	—
48	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
49	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
50	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
51	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
52	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
53	C₃×C₄⋊C₄	C₂×C₃×C₄	1	12	yes	—
54	C₃×C₄²	C₂×C₃×C₄	1	6	yes	—
55	C₃×C₄²	C₂³×C₃	1	6	yes	—
56	C₂²×C₉	—	2	1	no	—
57	C₂²×C₉	—	2	1	no	—
58	C₂²×C₉	C₂²×C₈	1	2	yes	—
59	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₉	—	2	1	no	—
61	C₄×C₉	—	2	6	no	—
62	C₄×C₉	C₂²×C₈	1	12	yes	—
63	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
64	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
65	C₄×Q₈	C₄×C₉	1	4	yes	—
66	C₄×Q₈	C₂²×C₉	1	4	yes	—
67	C₂×C₄⋊C₄	C₄×C₉	1	3	yes	—
68	C₂×C₄²	C₂²×C₉	1	3	yes	—
69	C₄⋊C₈	C₂²×C₉	1	12	yes	—
70	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
71	C₄×C₈	C₂²×C₉	1	12	yes	—
72	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₃×Q₈	C₂×C₃×C₈	1	16	yes	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	4	3	no	—
76	C₂×C₃×C₄	C₂×C₃×C₈	1	12	yes	—
77	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
79	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
80	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
81	C₃×C₈	—	2	8	no	—
82	C₂×C₉	—	2	2	no	—
83	C₂×C₉	#519	1	1	yes	characteristic
84	C₂×C₉	C₂³×C₈	1	1	yes	characteristic
85	C₂×C₉	#568	1	1	yes	characteristic
86	C₂×Q₈	C₈×C₉	1	8	yes	complemented
87	C₂×Q₈	C₂×C₄×C₉	1	4	yes	—
88	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
89	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
90	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
91	C₂×C₈	C₂³×C₉	1	8	yes	—
92	C₄⋊C₄	C₂×C₄×C₉	1	12	yes	—
93	C₄²	C₂×C₄×C₉	1	6	yes	—
94	C₄²	C₂³×C₉	1	6	yes	—
95	C₂²×C₃	—	2	1	no	—
96	C₂²×C₃	—	2	1	no	—
97	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
98	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
99	C₃×C₄	—	2	6	no	—
100	C₃×C₄	—	2	1	no	—
101	C₃×C₄	C₂²×C₃×C₈	1	12	yes	—
102	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
103	C₉	#2982	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
105	Q₈	C₂×C₈×C₉	1	16	yes	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	C₂×C₈×C₉	1	12	yes	—
109	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
110	C₂×C₄	C₂²×C₄×C₉	1	6	yes	—
111	C₂×C₄	C₂²×C₄×C₉	1	6	yes	—
112	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
113	C₈	—	2	8	no	—
114	C₂×C₃	—	2	2	no	—
115	C₂×C₃	C₃×#519	1	1	yes	characteristic
116	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
117	C₂×C₃	C₃×#568	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×C₈×C₉	1	2	yes	—
121	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
122	C₄	—	2	6	no	—
123	C₄	—	2	1	no	—
124	C₄	C₂²×C₈×C₉	1	12	yes	—
125	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
126	C₃	C₃×#2982	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	C₉×#519	1	1	yes	characteristic
129	C₂	C₂³×C₈×C₉	1	1	yes	characteristic
130	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂	C₂³×C₈×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₉	C₂	semidirect
16. C₂×Q₈×C₉	C₈	semidirect
25. #2982	C₉	direct (N × H)
45. C₈×Q₈	C₂×C₉	semidirect
86. C₂×Q₈	C₈×C₉	semidirect
103. C₉	#2982	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₈×Q₈×C₉	2
C₉×#431	2
C₃×#2982	3

Subgroups — C9×#2982 #11767703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2982 #11767703