Subgroups — C₉×#1809 #11766530

Order 1152.

474 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#380	C₂	1	2	yes	maximal
5	C₉×#340	C₂	1	2	yes	maximal, complemented
6	C₃×#1809	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	—	2	2	no	—
9	C₂×C₉×M₁₆	—	2	2	no	—
10	C₂×C₉×M₁₆	C₄	1	4	yes	complemented
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
16	C₉×#107	—	2	2	no	—
17	C₉×#106	—	2	2	no	—
18	C₉×#95	—	2	2	no	—
19	C₂²×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
22	C₃×#380	C₂×C₃	1	2	yes	—
23	C₃×#340	C₂×C₃	1	2	yes	—
24	C₂⁴×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	1	no	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	1	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	—	2	2	no	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₉×M₁₆	—	2	8	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₉	D₄	1	2	yes	—
41	C₂×C₈×C₉	D₄	1	2	yes	—
42	C₉×C₄⋊C₄	—	2	2	no	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	—	2	2	no	—
45	C₉×C₄⋊C₄	—	2	2	no	—
46	C₉×C₂²⋊C₄	—	4	4	no	—
47	#1809	C₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×M₁₆	—	2	2	no	—
50	C₂×C₃×M₁₆	—	2	2	no	—
51	C₂×C₃×M₁₆	C₃×C₄	1	4	yes	—
52	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
57	C₃×#107	—	2	2	no	—
58	C₃×#106	—	2	2	no	—
59	C₃×#95	—	2	2	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	—	2	1	no	—
62	C₂³×C₉	—	2	2	no	—
63	C₂³×C₉	—	2	2	no	—
64	C₂³×C₉	C₄⋊C₄	1	2	yes	—
65	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	4	2	no	—
73	C₂×C₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	—	4	2	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	4	2	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₉	#41	1	1	yes	characteristic
84	C₂×C₄×C₉	#41	1	1	yes	characteristic
85	C₈×C₉	—	2	2	no	—
86	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
87	C₂²×M₁₆	C₂×C₉	1	1	yes	characteristic
88	C₂×#107	C₂×C₉	1	1	yes	characteristic, complemented
89	C₂×#106	C₂×C₉	1	1	yes	characteristic, complemented
90	#380	C₂×C₉	1	2	yes	—
91	#340	C₂×C₉	1	2	yes	complemented
92	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
100	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₃×M₁₆	—	2	8	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
108	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
109	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₂²⋊C₄	—	4	4	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	—	2	4	no	—
117	C₂²×C₉	—	4	2	no	—
118	C₂²×C₉	—	4	1	no	—
119	C₂²×C₉	—	4	1	no	—
120	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₉	#121	1	1	yes	characteristic
122	C₂²×C₉	#121	1	1	yes	characteristic
123	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
124	C₂²×C₉	#137	1	2	yes	—
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	—	8	2	no	—
127	C₄×C₉	—	8	2	no	—
128	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₉	#117	1	1	yes	characteristic
130	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	C₄×C₉	1	4	yes	complemented
136	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
137	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
140	C₂×C₂²⋊C₄	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	2	no	—
143	#95	—	2	2	no	—
144	C₂³×C₃	—	2	1	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
149	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
163	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
171	C₂×C₉	—	4	2	no	—
172	C₂×C₉	#380	1	1	yes	characteristic
173	C₂×C₉	#428	1	2	yes	—
174	C₂×C₉	#468	1	2	yes	—
175	C₂×C₉	#469	1	1	yes	characteristic
176	C₂×C₉	#470	1	1	yes	characteristic
177	C₂⁴	D₄×C₉	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
185	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
186	C₂²×C₄	Q₈×C₉	1	2	yes	—
187	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
192	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
193	C₂×C₈	D₄×C₉	1	2	yes	—
194	C₂×C₈	D₄×C₉	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
206	C₂²×C₃	C₃×#121	1	1	yes	characteristic
207	C₂²×C₃	C₃×#121	1	1	yes	characteristic
208	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
209	C₂²×C₃	C₃×#137	1	2	yes	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	8	2	no	—
212	C₃×C₄	—	8	2	no	—
213	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₃×C₄	C₃×#117	1	1	yes	characteristic
215	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
216	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
217	C₉	#1809	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₉×C₄⋊C₄	1	2	yes	—
223	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
240	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
241	C₂×C₄	C₉×#41	1	1	yes	characteristic
242	C₂×C₄	C₉×#41	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	C₉×C₂²⋊C₄	1	4	yes	complemented
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	C₃×#380	1	1	yes	characteristic
247	C₂×C₃	C₃×#428	1	2	yes	—
248	C₂×C₃	C₃×#468	1	2	yes	—
249	C₂×C₃	C₃×#469	1	1	yes	characteristic
250	C₂×C₃	C₃×#470	1	1	yes	characteristic
251	C₂²	—	2	2	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
257	C₂²	C₉×#121	1	1	yes	characteristic
258	C₂²	C₉×#121	1	1	yes	characteristic
259	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
260	C₂²	C₉×#137	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	2	no	—
264	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
265	C₄	C₉×#117	1	1	yes	characteristic
266	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
267	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
268	C₃	C₃×#1809	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	C₉×#380	1	1	yes	characteristic
271	C₂	C₉×#428	1	2	yes	—
272	C₂	C₉×#468	1	2	yes	—
273	C₂	C₉×#469	1	1	yes	characteristic
274	C₂	C₉×#470	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#107	C₂	semidirect
3. C₂×C₉×#106	C₂	semidirect
5. C₉×#340	C₂	semidirect
10. C₂×C₉×M₁₆	C₄	semidirect
47. #1809	C₉	direct (N × H)
86. C₈×C₉	C₂²⋊C₄	semidirect
88. C₂×#107	C₂×C₉	semidirect
89. C₂×#106	C₂×C₉	semidirect
91. #340	C₂×C₉	semidirect
135. C₂×M₁₆	C₄×C₉	semidirect
217. C₉	#1809	direct (N × H)
244. C₈	C₉×C₂²⋊C₄	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×#107	2
C₂×C₉×#106	2
C₉×#380	2
C₉×#340	2
C₃×#1809	3

Subgroups — C9×#1809 #11766530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1809 #11766530