Subgroups — C₉×#2016 #11766737

Order 1152.

555 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#103	C₂	1	2	yes	maximal
4	C₉×#391	C₂	1	1	yes	maximal, characteristic
5	C₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₃×#2016	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Q₁₆	—	2	4	no	—
10	C₂×C₉×Q₁₆	—	2	4	no	—
11	C₂×C₉×Q₁₆	—	2	4	no	—
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₉×C₄⋊Q₈	—	2	4	no	—
14	C₂×C₉×C₄⋊C₄	—	2	2	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×#103	—	2	4	no	—
18	C₉×#95	—	2	2	no	—
19	C₂²×C₃×Q₁₆	C₂×C₃	1	2	yes	—
20	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#103	C₂×C₃	1	2	yes	—
22	C₃×#391	C₂×C₃	1	1	yes	characteristic
23	C₃×#336	C₂×C₃	1	1	yes	characteristic
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	1	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×Q₈×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	D₄	1	4	yes	—
33	C₂×Q₈×C₉	D₄	1	2	yes	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	D₄	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₉×Q₁₆	—	2	16	no	—
40	C₂×C₈×C₉	—	2	2	no	—
41	C₂×C₈×C₉	—	2	2	no	—
42	C₂×C₈×C₉	D₄	1	4	yes	—
43	C₉×C₄⋊C₄	—	2	8	no	—
44	C₉×C₄⋊C₄	—	2	2	no	—
45	C₄²×C₉	—	2	2	no	—
46	#2016	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
49	C₂×C₃×Q₁₆	—	2	4	no	—
50	C₂×C₃×Q₁₆	—	2	4	no	—
51	C₂×C₃×Q₁₆	—	2	4	no	—
52	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
53	C₃×C₄⋊Q₈	—	2	4	no	—
54	C₂×C₃×C₄⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₃×#103	—	2	4	no	—
58	C₃×#95	—	2	2	no	—
59	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
60	Q₈×C₉	—	2	4	no	—
61	Q₈×C₉	—	2	4	no	—
62	Q₈×C₉	—	2	4	no	—
63	Q₈×C₉	—	4	2	no	—
64	Q₈×C₉	—	4	2	no	—
65	Q₈×C₉	—	4	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	4	1	no	—
71	C₂×C₄×C₉	—	4	2	no	—
72	C₂×C₄×C₉	—	4	2	no	—
73	C₂×C₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₉	#41	1	1	yes	characteristic
81	C₈×C₉	—	2	4	no	—
82	C₂²×Q₁₆	C₂×C₉	1	2	yes	—
83	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
84	C₂×#103	C₂×C₉	1	2	yes	—
85	#391	C₂×C₉	1	1	yes	characteristic
86	#336	C₂×C₉	1	1	yes	characteristic
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×Q₈	—	2	1	no	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×Q₈	—	2	2	no	—
91	C₂×C₃×Q₈	—	2	2	no	—
92	C₂×C₃×Q₈	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	2	no	—
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
96	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×Q₁₆	—	2	16	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
106	C₃×C₄⋊C₄	—	2	8	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄²	—	2	2	no	—
109	C₂²×C₉	#120	1	1	yes	characteristic
110	C₂²×C₉	#121	1	1	yes	characteristic
111	C₂²×C₉	#127	1	1	yes	characteristic
112	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
113	C₂²×C₉	#137	1	2	yes	—
114	C₄×C₉	—	2	4	no	—
115	C₄×C₉	—	4	4	no	—
116	C₄×C₉	—	4	2	no	—
117	C₄×C₉	—	8	1	no	—
118	C₄×C₉	#120	1	2	yes	—
119	C₄×C₉	#121	1	2	yes	—
120	C₂²×Q₈	C₂²×C₉	1	2	yes	—
121	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	4	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂²×C₈	C₂²×C₉	1	2	yes	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
130	#103	—	2	4	no	—
131	#95	—	2	2	no	—
132	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	2	4	no	—
134	C₃×Q₈	—	2	4	no	—
135	C₃×Q₈	—	2	4	no	—
136	C₃×Q₈	—	4	2	no	—
137	C₃×Q₈	—	4	2	no	—
138	C₃×Q₈	—	4	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	1	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	4	1	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
154	C₃×C₈	—	2	4	no	—
155	C₂×C₉	#392	1	1	yes	characteristic
156	C₂×C₉	#451	1	2	yes	—
157	C₂×C₉	#462	1	2	yes	—
158	C₂×C₉	#494	1	1	yes	characteristic
159	C₂×C₉	#497	1	1	yes	characteristic
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	D₄×C₉	1	4	yes	—
169	C₂×Q₈	D₄×C₉	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	D₄×C₉	1	2	yes	—
174	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
175	Q₁₆	—	2	16	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	D₄×C₉	1	4	yes	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₂²×C₃	C₃×#120	1	1	yes	characteristic
183	C₂²×C₃	C₃×#121	1	1	yes	characteristic
184	C₂²×C₃	C₃×#127	1	1	yes	characteristic
185	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
186	C₂²×C₃	C₃×#137	1	2	yes	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	4	4	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	8	1	no	—
191	C₃×C₄	C₃×#120	1	2	yes	—
192	C₃×C₄	C₃×#121	1	2	yes	—
193	C₉	#2016	1	1	yes	characteristic, direct product factor, Sylow
194	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	2	4	no	—
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	4	2	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₉×Q₁₆	1	4	yes	complemented
212	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
215	C₂×C₄	C₉×#41	1	1	yes	characteristic
216	C₈	—	2	4	no	—
217	C₂×C₃	C₃×#392	1	1	yes	characteristic
218	C₂×C₃	C₃×#451	1	2	yes	—
219	C₂×C₃	C₃×#462	1	2	yes	—
220	C₂×C₃	C₃×#494	1	1	yes	characteristic
221	C₂×C₃	C₃×#497	1	1	yes	characteristic
222	C₂²	C₉×#120	1	1	yes	characteristic
223	C₂²	C₉×#121	1	1	yes	characteristic
224	C₂²	C₉×#127	1	1	yes	characteristic
225	C₂²	C₂×C₉×Q₁₆	1	2	yes	—
226	C₂²	C₉×#137	1	2	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	1	no	—
231	C₄	C₉×#120	1	2	yes	—
232	C₄	C₉×#121	1	2	yes	—
233	C₃	C₃×#2016	1	1	yes	characteristic
234	C₂	C₉×#392	1	1	yes	characteristic
235	C₂	C₉×#451	1	2	yes	—
236	C₂	C₉×#462	1	2	yes	—
237	C₂	C₉×#494	1	1	yes	characteristic
238	C₂	C₉×#497	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
46. #2016	C₉	direct (N × H)
76. C₂×C₄×C₉	Q₁₆	semidirect
193. C₉	#2016	direct (N × H)
211. C₂×C₄	C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#103	2
C₉×#391	2
C₉×#336	2
C₃×#2016	3

Subgroups — C9×#2016 #11766737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2016 #11766737