Subgroups — C₉×#2041 #11766762

Order 1152.

603 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
4	C₉×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2041	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	—	2	1	no	—
8	C₂²×D₄×C₉	—	2	1	no	—
9	C₂²×D₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₉×#127	—	2	4	no	—
12	C₂×C₉×C₄⋊C₄	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#102	—	2	4	no	—
17	C₉×#102	—	2	4	no	—
18	C₉×#95	—	2	1	no	—
19	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#102	C₂×C₃	1	2	yes	—
21	C₂×C₃×#102	C₂×C₃	1	2	yes	—
22	C₃×#389	C₂×C₃	1	1	yes	characteristic
23	C₃×#336	C₂×C₃	1	1	yes	characteristic
24	C₂⁴×C₉	—	2	2	no	—
25	C₂×D₄×C₉	—	2	4	no	—
26	C₂×D₄×C₉	—	2	4	no	—
27	C₂×D₄×C₉	—	2	2	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	4	2	no	—
33	C₂×D₄×C₉	—	4	2	no	—
34	C₂×D₄×C₉	—	4	2	no	—
35	C₂×D₄×C₉	—	4	2	no	—
36	C₂×D₄×C₉	D₄	1	4	yes	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₉	—	2	2	no	—
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₉×C₄⋊C₄	—	2	1	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	4	2	no	—
48	C₉×C₄⋊C₄	D₄	1	2	yes	complemented
49	C₄²×C₉	—	2	2	no	—
50	#2041	C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₃×D₄	—	2	1	no	—
52	C₂²×C₃×D₄	—	2	1	no	—
53	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
54	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
55	C₃×#127	—	2	4	no	—
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₃×#102	—	2	4	no	—
61	C₃×#102	—	2	4	no	—
62	C₃×#95	—	2	1	no	—
63	C₂³×C₉	—	2	4	no	—
64	C₂³×C₉	—	4	2	no	—
65	C₂³×C₉	—	4	2	no	—
66	C₂³×C₉	—	4	2	no	—
67	C₂³×C₉	—	4	2	no	—
68	C₂³×C₉	—	4	2	no	—
69	C₂³×C₉	—	4	2	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	D₄×C₉	—	2	4	no	—
72	D₄×C₉	—	2	4	no	—
73	D₄×C₉	—	4	8	no	—
74	D₄×C₉	—	4	8	no	—
75	D₄×C₉	—	4	2	no	—
76	D₄×C₉	—	4	2	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	—	4	1	no	—
85	C₂×C₄×C₉	—	4	1	no	—
86	C₂×C₄×C₉	—	4	1	no	—
87	C₂×C₄×C₉	D₈	1	2	yes	—
88	C₂×C₄×C₉	SD₁₆	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₉	#41	1	2	yes	—
92	C₂×C₄×C₉	#41	1	1	yes	characteristic
93	C₈×C₉	—	4	2	no	—
94	C₂×#127	C₂×C₉	1	1	yes	characteristic
95	C₂×#102	C₂×C₉	1	2	yes	complemented
96	C₂×#102	C₂×C₉	1	2	yes	complemented
97	#389	C₂×C₉	1	1	yes	characteristic, complemented
98	#336	C₂×C₉	1	1	yes	characteristic, complemented
99	C₂⁴×C₃	—	2	2	no	—
100	C₂×C₃×D₄	—	2	4	no	—
101	C₂×C₃×D₄	—	2	4	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	2	2	no	—
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	—	2	2	no	—
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	—	4	2	no	—
108	C₂×C₃×D₄	—	4	2	no	—
109	C₂×C₃×D₄	—	4	2	no	—
110	C₂×C₃×D₄	—	4	2	no	—
111	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	1	no	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	4	2	no	—
123	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
124	C₃×C₄²	—	2	2	no	—
125	C₂²×C₉	—	4	4	no	—
126	C₂²×C₉	—	4	4	no	—
127	C₂²×C₉	—	4	4	no	—
128	C₂²×C₉	—	8	2	no	—
129	C₂²×C₉	—	8	2	no	—
130	C₂²×C₉	—	8	2	no	—
131	C₂²×C₉	—	8	2	no	—
132	C₂²×C₉	#120	1	1	yes	characteristic
133	C₂²×C₉	#121	1	1	yes	characteristic
134	C₂²×C₉	#124	1	1	yes	characteristic
135	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
136	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
137	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	4	2	no	—
142	C₄×C₉	—	8	1	no	—
143	C₄×C₉	#121	1	2	yes	—
144	C₄×C₉	#123	1	2	yes	—
145	C₂²×D₄	—	2	1	no	—
146	C₂²×D₄	—	2	1	no	—
147	C₂²×D₄	C₂²×C₉	1	2	yes	—
148	C₂²×C₈	C₂²×C₉	1	2	yes	—
149	#127	—	2	4	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
153	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
154	#102	—	2	4	no	—
155	#102	—	2	4	no	—
156	#95	—	2	1	no	—
157	C₂³×C₃	—	2	4	no	—
158	C₂³×C₃	—	4	2	no	—
159	C₂³×C₃	—	4	2	no	—
160	C₂³×C₃	—	4	2	no	—
161	C₂³×C₃	—	4	2	no	—
162	C₂³×C₃	—	4	2	no	—
163	C₂³×C₃	—	4	2	no	—
164	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	4	8	no	—
168	C₃×D₄	—	4	8	no	—
169	C₃×D₄	—	4	2	no	—
170	C₃×D₄	—	4	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
182	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
185	C₂×C₃×C₄	C₃×#41	1	2	yes	—
186	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
187	C₃×C₈	—	4	2	no	—
188	C₂×C₉	—	8	4	no	—
189	C₂×C₉	#394	1	1	yes	characteristic
190	C₂×C₉	#447	1	1	yes	characteristic
191	C₂×C₉	#450	1	1	yes	characteristic
192	C₂×C₉	#459	1	1	yes	characteristic
193	C₂×C₉	#460	1	1	yes	characteristic
194	C₂×C₉	#486	1	1	yes	characteristic
195	C₂×C₉	#490	1	1	yes	characteristic
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	D₄×C₉	1	4	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
212	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	D₄×C₉	1	2	yes	complemented
221	C₄²	—	2	2	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	—	4	4	no	—
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	—	8	2	no	—
226	C₂²×C₃	—	8	2	no	—
227	C₂²×C₃	—	8	2	no	—
228	C₂²×C₃	—	8	2	no	—
229	C₂²×C₃	C₃×#120	1	1	yes	characteristic
230	C₂²×C₃	C₃×#121	1	1	yes	characteristic
231	C₂²×C₃	C₃×#124	1	1	yes	characteristic
232	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
233	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
234	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	8	1	no	—
240	C₃×C₄	C₃×#121	1	2	yes	—
241	C₃×C₄	C₃×#123	1	2	yes	—
242	C₉	#2041	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	8	no	—
254	D₄	—	4	8	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	C₉×D₈	1	2	yes	—
268	C₂×C₄	C₉×SD₁₆	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
271	C₂×C₄	C₉×#41	1	2	yes	—
272	C₂×C₄	C₉×#41	1	1	yes	characteristic
273	C₈	—	4	2	no	—
274	C₂×C₃	—	8	4	no	—
275	C₂×C₃	C₃×#394	1	1	yes	characteristic
276	C₂×C₃	C₃×#447	1	1	yes	characteristic
277	C₂×C₃	C₃×#450	1	1	yes	characteristic
278	C₂×C₃	C₃×#459	1	1	yes	characteristic
279	C₂×C₃	C₃×#460	1	1	yes	characteristic
280	C₂×C₃	C₃×#486	1	1	yes	characteristic
281	C₂×C₃	C₃×#490	1	1	yes	characteristic
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	C₉×#120	1	1	yes	characteristic
290	C₂²	C₉×#121	1	1	yes	characteristic
291	C₂²	C₉×#124	1	1	yes	characteristic
292	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
293	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
294	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	1	no	—
300	C₄	C₉×#121	1	2	yes	—
301	C₄	C₉×#123	1	2	yes	—
302	C₃	C₃×#2041	1	1	yes	characteristic
303	C₂	—	8	4	no	—
304	C₂	C₉×#394	1	1	yes	characteristic
305	C₂	C₉×#447	1	1	yes	characteristic
306	C₂	C₉×#450	1	1	yes	characteristic
307	C₂	C₉×#459	1	1	yes	characteristic
308	C₂	C₉×#460	1	1	yes	characteristic
309	C₂	C₉×#486	1	1	yes	characteristic
310	C₂	C₉×#490	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	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#127	C₂×C₃
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
2. C₂×C₉×#102	C₂	semidirect
3. C₂×C₉×#102	C₂	semidirect
4. C₉×#389	C₂	semidirect
5. C₉×#336	C₂	semidirect
48. C₉×C₄⋊C₄	D₄	semidirect
50. #2041	C₉	direct (N × H)
95. C₂×#102	C₂×C₉	semidirect
96. C₂×#102	C₂×C₉	semidirect
97. #389	C₂×C₉	semidirect
98. #336	C₂×C₉	semidirect
220. C₄⋊C₄	D₄×C₉	semidirect
242. C₉	#2041	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×C₉×#127	2
C₂×C₉×#102	2
C₉×#389	2
C₉×#336	2
C₃×#2041	3

Subgroups — C9×#2041 #11766762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2041 #11766762