Subgroups — C₉×#1855 #11766576

Order 1152.

417 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#108	C₂	1	2	yes	maximal, complemented
3	C₉×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₉×#341	C₂	1	2	yes	maximal, complemented
6	C₃×#1855	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	—	2	1	no	—
9	C₂×C₉×M₁₆	—	2	4	no	—
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₉	—	2	1	no	—
14	C₂²×C₈×C₉	C₂²	1	2	yes	—
15	C₉×#108	—	2	4	no	—
16	C₉×C₂²⋊C₈	—	2	1	no	—
17	C₉×C₂²⋊C₈	—	2	1	no	—
18	C₉×C₂²⋊C₈	C₄	1	2	yes	—
19	C₉×C₂²⋊C₈	C₄	1	2	yes	—
20	C₂²×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#108	C₂×C₃	1	2	yes	—
22	C₃×#407	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#341	C₂×C₃	1	2	yes	—
25	C₂⁴×C₉	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×M₁₆	—	2	4	no	—
35	C₉×M₁₆	—	2	2	no	—
36	C₉×M₁₆	—	2	2	no	—
37	C₉×M₁₆	—	2	2	no	—
38	C₉×M₁₆	D₄	1	4	yes	—
39	C₂×C₈×C₉	—	2	4	no	—
40	C₂×C₈×C₉	—	2	2	no	—
41	C₂×C₈×C₉	—	2	2	no	—
42	C₂×C₈×C₉	—	2	2	no	—
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
46	#1855	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃×M₁₆	—	2	1	no	—
49	C₂×C₃×M₁₆	—	2	4	no	—
50	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×C₈	—	2	1	no	—
54	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
55	C₃×#108	—	2	4	no	—
56	C₃×C₂²⋊C₈	—	2	1	no	—
57	C₃×C₂²⋊C₈	—	2	1	no	—
58	C₃×C₂²⋊C₈	C₃×C₄	1	2	yes	—
59	C₃×C₂²⋊C₈	C₃×C₄	1	2	yes	—
60	C₂³×C₉	—	2	2	no	—
61	C₂³×C₉	—	2	2	no	—
62	C₂³×C₉	—	4	1	no	—
63	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	#41	1	2	yes	—
79	C₈×C₉	—	2	4	no	—
80	C₈×C₉	—	2	4	no	—
81	C₈×C₉	—	4	1	no	—
82	C₈×C₉	—	4	1	no	—
83	C₂²×M₁₆	C₂×C₉	1	1	yes	characteristic
84	C₂×#108	C₂×C₉	1	2	yes	complemented
85	#407	C₂×C₉	1	1	yes	characteristic
86	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
87	#341	C₂×C₉	1	2	yes	complemented
88	C₂⁴×C₃	C₃×Q₈	1	1	yes	characteristic
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₃×Q₈	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₃×M₁₆	—	2	4	no	—
98	C₃×M₁₆	—	2	2	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	C₃×D₄	1	4	yes	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	—	2	1	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	1	no	—
115	C₂²×C₉	—	2	1	no	—
116	C₂²×C₉	—	4	4	no	—
117	C₂²×C₉	#108	1	2	yes	complemented
118	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
120	C₄×C₉	—	2	1	no	—
121	C₄×C₉	—	2	1	no	—
122	C₄×C₉	C₄×D₄	1	2	yes	—
123	C₄×C₉	#120	1	1	yes	characteristic
124	C₄×C₉	#123	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
126	C₂×M₁₆	—	2	1	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
129	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
130	C₂×M₁₆	C₂²×C₉	1	2	yes	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	C₂²×C₉	1	2	yes	—
133	#108	—	2	4	no	—
134	C₂²⋊C₈	—	2	1	no	—
135	C₂²⋊C₈	—	2	1	no	—
136	C₂²⋊C₈	C₄×C₉	1	2	yes	—
137	C₂²⋊C₈	C₄×C₉	1	2	yes	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	—	2	2	no	—
140	C₂³×C₃	—	4	1	no	—
141	C₂³×C₃	C₃×C₄⋊C₄	1	1	yes	characteristic
142	C₂³×C₃	C₃×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×#41	1	2	yes	—
157	C₃×C₈	—	2	4	no	—
158	C₃×C₈	—	2	4	no	—
159	C₃×C₈	—	4	1	no	—
160	C₃×C₈	—	4	1	no	—
161	C₂×C₉	—	2	2	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	4	1	no	—
164	C₂×C₉	#385	1	1	yes	characteristic
165	C₂×C₉	C₂×#108	1	1	yes	characteristic
166	C₂×C₉	#430	1	1	yes	characteristic
167	C₂⁴	Q₈×C₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
174	C₂²×C₄	Q₈×C₉	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
176	M₁₆	—	2	4	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	D₄×C₉	1	4	yes	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	C₂×C₄×C₉	1	4	yes	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	4	no	—
196	C₂²×C₃	C₃×#108	1	2	yes	—
197	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
202	C₃×C₄	C₃×#120	1	1	yes	characteristic
203	C₃×C₄	C₃×#123	1	1	yes	characteristic
204	C₉	#1855	1	1	yes	characteristic, direct product factor, Sylow
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	4	1	no	—
208	C₂³	C₉×C₄⋊C₄	1	1	yes	characteristic
209	C₂³	C₉×C₄⋊C₄	1	1	yes	characteristic
210	C₂³	C₂×Q₈×C₉	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	C₉×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₉×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
223	C₂×C₄	C₉×#41	1	2	yes	—
224	C₈	—	2	4	no	—
225	C₈	—	2	4	no	—
226	C₈	—	4	1	no	—
227	C₈	—	4	1	no	—
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	4	1	no	—
231	C₂×C₃	C₃×#385	1	1	yes	characteristic
232	C₂×C₃	C₂×C₃×#108	1	1	yes	characteristic
233	C₂×C₃	C₃×#430	1	1	yes	characteristic
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₉×#108	1	2	yes	complemented
243	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
244	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	C₄×D₄×C₉	1	2	yes	—
248	C₄	C₉×#120	1	1	yes	characteristic
249	C₄	C₉×#123	1	1	yes	characteristic
250	C₃	C₃×#1855	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	1	no	—
254	C₂	C₉×#385	1	1	yes	characteristic
255	C₂	C₂×C₉×#108	1	1	yes	characteristic
256	C₂	C₉×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₄⋊C₄
prime radical	C₂⁴×C₃	C₃×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#108	C₂	semidirect
5. C₉×#341	C₂	semidirect
46. #1855	C₉	direct (N × H)
84. C₂×#108	C₂×C₉	semidirect
87. #341	C₂×C₉	semidirect
117. C₂²×C₉	#108	semidirect
204. C₉	#1855	direct (N × H)
242. C₂²	C₉×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂×C₉×#108	2
C₉×#407	2
C₂×C₉×C₂²⋊C₈	2
C₉×#341	2
C₃×#1855	3

Subgroups — C9×#1855 #11766576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1855 #11766576