Subgroups — C₉×#2986 #11767707

Order 1152.

495 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₉×#446	C₂	1	4	yes	maximal, complemented
3	C₈×Q₈×C₉	C₂	1	4	yes	maximal, complemented
4	C₉×#433	C₂	1	4	yes	maximal
5	C₉×#431	C₂	1	2	yes	maximal
6	C₃×#2986	C₃	1	1	yes	maximal, characteristic
7	C₉×C₄⋊Q₈	C₄	1	2	yes	—
8	C₉×#125	C₄	1	2	yes	—
9	C₉×C₂²⋊Q₈	C₄	1	4	yes	—
10	C₄×Q₈×C₉	C₂²	1	4	yes	—
11	C₉×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊C₈	C₂²	1	2	yes	—
14	C₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₉×C₄⋊C₈	C₂²	1	8	yes	—
16	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
17	C₉×#97	C₂²	1	4	yes	—
18	C₄×C₈×C₉	C₂²	1	4	yes	—
19	C₄×C₈×C₉	C₂²	1	4	yes	—
20	C₃×#533	C₂×C₃	1	1	yes	characteristic
21	C₃×#446	C₂×C₃	1	4	yes	—
22	C₃×C₈×Q₈	C₂×C₃	1	4	yes	—
23	C₃×#433	C₂×C₃	1	4	yes	—
24	C₃×#431	C₂×C₃	1	2	yes	—
25	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉	C₂³	1	4	yes	—
29	C₂×C₈×C₉	C₂³	1	4	yes	—
30	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₉×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₉	C₂³	1	2	yes	—
37	C₄²×C₉	C₂³	1	4	yes	—
38	#2986	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₄⋊Q₈	C₃×C₄	1	2	yes	—
40	C₃×#125	C₃×C₄	1	2	yes	—
41	C₃×C₂²⋊Q₈	C₃×C₄	1	4	yes	—
42	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
43	C₃×#117	C₂²×C₃	1	2	yes	—
44	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
45	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
46	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
47	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
48	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
49	C₃×#97	C₂²×C₃	1	4	yes	—
50	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
51	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
52	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₉	—	2	8	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	2	2	no	—
56	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
59	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
61	C₈×C₉	—	2	4	no	—
62	C₈×C₉	—	2	4	no	—
63	#533	C₂×C₉	1	1	yes	characteristic
64	#446	C₂×C₉	1	4	yes	complemented
65	C₈×Q₈	C₂×C₉	1	4	yes	complemented
66	#433	C₂×C₉	1	4	yes	—
67	#431	C₂×C₉	1	2	yes	—
68	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
69	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
70	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
72	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
73	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
74	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
75	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
76	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
77	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
78	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
79	C₃×C₄²	C₂³×C₃	1	2	yes	—
80	C₃×C₄²	C₂³×C₃	1	4	yes	—
81	C₂²×C₉	—	2	1	no	—
82	C₂²×C₉	—	2	2	no	—
83	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₉	—	2	1	no	—
85	C₄×C₉	—	2	2	no	—
86	C₄×C₉	—	2	8	no	—
87	C₄×C₉	#131	1	4	yes	—
88	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
89	C₄⋊Q₈	C₄×C₉	1	2	yes	—
90	#125	C₄×C₉	1	2	yes	—
91	C₂²⋊Q₈	C₄×C₉	1	4	yes	—
92	C₄×Q₈	C₂²×C₉	1	4	yes	—
93	#117	C₂²×C₉	1	2	yes	—
94	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
95	C₄⋊C₈	C₂²×C₉	1	2	yes	—
96	C₄⋊C₈	C₂²×C₉	1	2	yes	—
97	C₄⋊C₈	C₂²×C₉	1	8	yes	—
98	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
99	#97	C₂²×C₉	1	4	yes	—
100	C₄×C₈	C₂²×C₉	1	4	yes	—
101	C₄×C₈	C₂²×C₉	1	4	yes	—
102	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
103	C₃×Q₈	—	2	8	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
108	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
109	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
110	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
111	C₃×C₈	—	2	4	no	—
112	C₃×C₈	—	2	4	no	—
113	C₂×C₉	—	4	1	no	—
114	C₂×C₉	#519	1	1	yes	characteristic
115	C₂×C₉	C₂×#131	1	2	yes	—
116	C₂×Q₈	C₂×C₄×C₉	1	4	yes	—
117	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
118	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
119	C₂×C₈	C₂³×C₉	1	4	yes	—
120	C₂×C₈	C₂³×C₉	1	4	yes	—
121	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
122	C₄⋊C₄	C₂×C₄×C₉	1	8	yes	—
123	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
124	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
125	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
126	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
127	C₄²	C₂³×C₉	1	2	yes	—
128	C₄²	C₂³×C₉	1	4	yes	—
129	C₂²×C₃	—	2	2	no	—
130	C₂²×C₃	—	2	1	no	—
131	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	8	no	—
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	C₃×#131	1	4	yes	—
136	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
137	C₉	#2986	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
139	Q₈	—	2	8	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
144	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
145	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
146	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	4	no	—
149	C₂×C₃	—	4	1	no	—
150	C₂×C₃	C₃×#519	1	1	yes	characteristic
151	C₂×C₃	C₂×C₃×#131	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	1	no	—
154	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	8	no	—
157	C₄	—	2	1	no	—
158	C₄	C₉×#131	1	4	yes	—
159	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
160	C₃	C₃×#2986	1	1	yes	characteristic
161	C₂	—	4	1	no	—
162	C₂	C₉×#519	1	1	yes	characteristic
163	C₂	C₂×C₉×#131	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
3. C₈×Q₈×C₉	C₂	semidirect
38. #2986	C₉	direct (N × H)
64. #446	C₂×C₉	semidirect
65. C₈×Q₈	C₂×C₉	semidirect
137. C₉	#2986	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₉×#446	2
C₈×Q₈×C₉	2
C₉×#433	2
C₉×#431	2
C₃×#2986	3

Subgroups — C9×#2986 #11767707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2986 #11767707