Subgroups — C₉×#3487 #11768208

Order 1152.

2304 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#550	C₂	1	8	yes	maximal, complemented
4	C₉×#548	C₂	1	8	yes	maximal, complemented
5	C₉×#535	C₂	1	6	yes	maximal, complemented
6	C₉×#525	C₂	1	6	yes	maximal, complemented
7	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3487	C₃	1	1	yes	maximal, characteristic
9	C₉×2₋¹⁺⁴	—	2	8	no	—
10	C₂×C₉×#41	—	2	8	no	—
11	C₂×C₉×#41	C₂²	1	6	yes	—
12	C₂×C₉×#41	C₂²	1	8	yes	—
13	C₂×C₉×#41	C₂²	1	4	yes	complemented
14	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₉	C₂²	1	4	yes	complemented
16	C₂²×D₄×C₉	C₂²	1	3	yes	—
17	C₂³×C₄×C₉	C₂²	1	3	yes	—
18	C₉×#127	C₂²	1	12	yes	complemented
19	C₉×#124	C₂²	1	12	yes	complemented
20	C₉×#123	C₂²	1	4	yes	—
21	C₉×C₂²⋊Q₈	C₂²	1	12	yes	complemented
22	C₉×#121	C₂²	1	12	yes	complemented
23	C₉×#121	C₂²	1	24	yes	complemented
24	C₉×#120	C₂²	1	12	yes	—
25	C₄×Q₈×C₉	C₂²	1	8	yes	complemented
26	C₄×D₄×C₉	C₂²	1	24	yes	complemented
27	C₉×#117	C₂²	1	6	yes	—
28	C₂×C₃×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic
29	C₂²×C₃×#41	C₂×C₃	1	1	yes	characteristic
30	C₃×#550	C₂×C₃	1	8	yes	—
31	C₃×#548	C₂×C₃	1	8	yes	—
32	C₃×#535	C₂×C₃	1	6	yes	—
33	C₃×#525	C₂×C₃	1	6	yes	—
34	C₃×#519	C₂×C₃	1	1	yes	characteristic
35	C₂⁴×C₉	C₂³	1	3	yes	—
36	C₉×#41	—	2	32	no	—
37	C₉×#41	—	2	16	no	—
38	C₉×#41	—	2	24	no	—
39	C₂×Q₈×C₉	—	2	16	no	—
40	C₂×Q₈×C₉	D₄	1	8	yes	complemented
41	C₂×Q₈×C₉	C₂³	1	6	yes	—
42	C₂×Q₈×C₉	C₂³	1	4	yes	—
43	C₂×D₄×C₉	—	2	12	no	—
44	C₂×D₄×C₉	C₂³	1	4	yes	—
45	C₂×D₄×C₉	C₂³	1	12	yes	—
46	C₂×D₄×C₉	C₂³	1	6	yes	—
47	C₂×D₄×C₉	C₂³	1	24	yes	—
48	C₂²×C₄×C₉	—	2	12	no	—
49	C₂²×C₄×C₉	C₂³	1	12	yes	—
50	C₂²×C₄×C₉	C₂³	1	12	yes	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₉	C₂³	1	3	yes	—
53	C₉×C₄⋊C₄	C₂³	1	12	yes	—
54	C₉×C₄⋊C₄	C₂³	1	8	yes	—
55	C₉×C₂²⋊C₄	C₂³	1	4	yes	—
56	C₉×C₂²⋊C₄	C₂³	1	8	yes	—
57	C₉×C₂²⋊C₄	C₂³	1	24	yes	—
58	C₄²×C₉	C₂³	1	12	yes	—
59	#3487	C₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×2₋¹⁺⁴	—	2	8	no	—
61	C₂×C₃×#41	—	2	8	no	—
62	C₂×C₃×#41	C₂²×C₃	1	8	yes	—
63	C₂×C₃×#41	C₂²×C₃	1	6	yes	—
64	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
65	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
67	C₂²×C₃×D₄	C₂²×C₃	1	3	yes	—
68	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
69	C₃×#127	C₂²×C₃	1	12	yes	—
70	C₃×#124	C₂²×C₃	1	12	yes	—
71	C₃×#123	C₂²×C₃	1	4	yes	—
72	C₃×C₂²⋊Q₈	C₂²×C₃	1	12	yes	—
73	C₃×#121	C₂²×C₃	1	12	yes	—
74	C₃×#121	C₂²×C₃	1	24	yes	—
75	C₃×#120	C₂²×C₃	1	12	yes	—
76	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	—
77	C₃×C₄×D₄	C₂²×C₃	1	24	yes	—
78	C₃×#117	C₂²×C₃	1	6	yes	—
79	C₂³×C₉	—	2	6	no	—
80	C₂³×C₉	—	4	6	no	—
81	C₂³×C₉	C₂⁴	1	6	yes	—
82	C₂³×C₉	C₂⁴	1	4	yes	—
83	C₂³×C₉	C₂⁴	1	1	yes	characteristic
84	Q₈×C₉	—	2	12	no	—
85	Q₈×C₉	C₂×D₄	1	16	yes	—
86	D₄×C₉	—	2	8	no	—
87	D₄×C₉	—	2	24	no	—
88	D₄×C₉	—	2	12	no	—
89	D₄×C₉	—	4	24	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	24	no	—
92	C₂×C₄×C₉	—	2	24	no	—
93	C₂×C₄×C₉	C₂×D₄	1	12	yes	—
94	C₂×C₄×C₉	C₂⁴	1	8	yes	—
95	C₂×C₄×C₉	C₂⁴	1	4	yes	—
96	C₂×C₄×C₉	C₂⁴	1	2	yes	—
97	C₂×C₄×C₉	C₂⁴	1	6	yes	—
98	C₂×2₋¹⁺⁴	C₂×C₉	1	1	yes	characteristic, complemented
99	C₂²×#41	C₂×C₉	1	1	yes	characteristic, complemented
100	#550	C₂×C₉	1	8	yes	complemented
101	#548	C₂×C₉	1	8	yes	complemented
102	#535	C₂×C₉	1	6	yes	complemented
103	#525	C₂×C₉	1	6	yes	complemented
104	#519	C₂×C₉	1	1	yes	characteristic, complemented
105	C₂⁴×C₃	C₂³×C₃	1	3	yes	—
106	C₃×#41	—	2	32	no	—
107	C₃×#41	—	2	16	no	—
108	C₃×#41	—	2	24	no	—
109	C₂×C₃×Q₈	—	2	16	no	—
110	C₂×C₃×Q₈	C₃×D₄	1	8	yes	—
111	C₂×C₃×Q₈	C₂³×C₃	1	6	yes	—
112	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
113	C₂×C₃×D₄	—	2	12	no	—
114	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
115	C₂×C₃×D₄	C₂³×C₃	1	12	yes	—
116	C₂×C₃×D₄	C₂³×C₃	1	6	yes	—
117	C₂×C₃×D₄	C₂³×C₃	1	24	yes	—
118	C₂²×C₃×C₄	—	2	12	no	—
119	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	12	yes	—
124	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
125	C₃×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
126	C₃×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
127	C₃×C₂²⋊C₄	C₂³×C₃	1	24	yes	—
128	C₃×C₄²	C₂³×C₃	1	12	yes	—
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	2	4	no	—
131	C₂²×C₉	—	2	6	no	—
132	C₂²×C₉	—	2	8	no	—
133	C₂²×C₉	—	2	12	no	—
134	C₂²×C₉	—	4	12	no	—
135	C₂²×C₉	C₂²×D₄	1	2	yes	—
136	C₂²×C₉	C₂⁵	1	1	yes	characteristic
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	—	2	8	no	—
140	C₄×C₉	C₂²×D₄	1	12	yes	—
141	2₋¹⁺⁴	—	2	8	no	—
142	C₂×#41	—	2	8	no	—
143	C₂×#41	C₂²×C₉	1	6	yes	—
144	C₂×#41	C₂²×C₉	1	8	yes	—
145	C₂×#41	C₂²×C₉	1	4	yes	complemented
146	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
147	C₂²×Q₈	C₂²×C₉	1	4	yes	complemented
148	C₂²×D₄	C₂²×C₉	1	3	yes	—
149	C₂³×C₄	C₂²×C₉	1	3	yes	—
150	#127	C₂²×C₉	1	12	yes	complemented
151	#124	C₂²×C₉	1	12	yes	complemented
152	#123	C₂²×C₉	1	4	yes	—
153	C₂²⋊Q₈	C₂²×C₉	1	12	yes	complemented
154	#121	C₂²×C₉	1	12	yes	complemented
155	#121	C₂²×C₉	1	24	yes	complemented
156	#120	C₂²×C₉	1	12	yes	—
157	C₄×Q₈	C₂²×C₉	1	8	yes	complemented
158	C₄×D₄	C₂²×C₉	1	24	yes	complemented
159	#117	C₂²×C₉	1	6	yes	—
160	C₂³×C₃	—	2	6	no	—
161	C₂³×C₃	—	4	6	no	—
162	C₂³×C₃	C₂⁴×C₃	1	6	yes	—
163	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
164	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
165	C₃×Q₈	—	2	12	no	—
166	C₃×Q₈	C₂×C₃×D₄	1	16	yes	—
167	C₃×D₄	—	2	8	no	—
168	C₃×D₄	—	2	24	no	—
169	C₃×D₄	—	2	12	no	—
170	C₃×D₄	—	4	24	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	24	no	—
173	C₂×C₃×C₄	—	2	24	no	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
175	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
177	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
178	C₂×C₃×C₄	C₂⁴×C₃	1	6	yes	—
179	C₂×C₉	—	2	2	no	—
180	C₂×C₉	—	4	4	no	—
181	C₂×C₉	—	4	6	no	—
182	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
183	C₂×C₉	#585	1	2	yes	—
184	C₂⁴	C₂³×C₉	1	3	yes	—
185	#41	—	2	32	no	—
186	#41	—	2	16	no	—
187	#41	—	2	24	no	—
188	C₂×Q₈	—	2	16	no	—
189	C₂×Q₈	D₄×C₉	1	8	yes	complemented
190	C₂×Q₈	C₂³×C₉	1	6	yes	—
191	C₂×Q₈	C₂³×C₉	1	4	yes	—
192	C₂×D₄	—	2	12	no	—
193	C₂×D₄	C₂³×C₉	1	4	yes	—
194	C₂×D₄	C₂³×C₉	1	12	yes	—
195	C₂×D₄	C₂³×C₉	1	6	yes	—
196	C₂×D₄	C₂³×C₉	1	24	yes	—
197	C₂²×C₄	—	2	12	no	—
198	C₂²×C₄	C₂³×C₉	1	12	yes	—
199	C₂²×C₄	C₂³×C₉	1	12	yes	—
200	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₉	1	3	yes	—
202	C₄⋊C₄	C₂³×C₉	1	12	yes	—
203	C₄⋊C₄	C₂³×C₉	1	8	yes	—
204	C₂²⋊C₄	C₂³×C₉	1	4	yes	—
205	C₂²⋊C₄	C₂³×C₉	1	8	yes	—
206	C₂²⋊C₄	C₂³×C₉	1	24	yes	—
207	C₄²	C₂³×C₉	1	12	yes	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	6	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	—	2	12	no	—
213	C₂²×C₃	—	4	12	no	—
214	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
215	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	C₂²×C₃×D₄	1	12	yes	—
220	C₉	#3487	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	—	2	6	no	—
222	C₂³	—	4	6	no	—
223	C₂³	C₂⁴×C₉	1	6	yes	—
224	C₂³	C₂⁴×C₉	1	4	yes	—
225	C₂³	C₂⁴×C₉	1	1	yes	characteristic
226	Q₈	—	2	12	no	—
227	Q₈	C₂×D₄×C₉	1	16	yes	—
228	D₄	—	2	8	no	—
229	D₄	—	2	24	no	—
230	D₄	—	2	12	no	—
231	D₄	—	4	24	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	24	no	—
235	C₂×C₄	C₂×D₄×C₉	1	12	yes	—
236	C₂×C₄	C₂⁴×C₉	1	8	yes	—
237	C₂×C₄	C₂⁴×C₉	1	4	yes	—
238	C₂×C₄	C₂⁴×C₉	1	2	yes	—
239	C₂×C₄	C₂⁴×C₉	1	6	yes	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	4	6	no	—
242	C₂×C₃	—	4	4	no	—
243	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
244	C₂×C₃	C₃×#585	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	6	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	2	12	no	—
250	C₂²	—	4	12	no	—
251	C₂²	C₂²×D₄×C₉	1	2	yes	—
252	C₂²	C₂⁵×C₉	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	8	no	—
256	C₄	C₂²×D₄×C₉	1	12	yes	—
257	C₃	C₃×#3487	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	6	no	—
260	C₂	—	4	4	no	—
261	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
262	C₂	C₉×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂⁵
commutator	C₂²	C₂⁵×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁵×C₃
prime radical	C₃×#3487	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×2₋¹⁺⁴	C₂	semidirect
2. C₂²×C₉×#41	C₂	semidirect
3. C₉×#550	C₂	semidirect
4. C₉×#548	C₂	semidirect
5. C₉×#535	C₂	semidirect
6. C₉×#525	C₂	semidirect
7. C₉×#519	C₂	semidirect
13. C₂×C₉×#41	C₂²	semidirect
15. C₂²×Q₈×C₉	C₂²	semidirect
18. C₉×#127	C₂²	semidirect
19. C₉×#124	C₂²	semidirect
21. C₉×C₂²⋊Q₈	C₂²	semidirect
22. C₉×#121	C₂²	semidirect
23. C₉×#121	C₂²	semidirect
25. C₄×Q₈×C₉	C₂²	semidirect
26. C₄×D₄×C₉	C₂²	semidirect
40. C₂×Q₈×C₉	D₄	semidirect
59. #3487	C₉	direct (N × H)
98. C₂×2₋¹⁺⁴	C₂×C₉	semidirect
99. C₂²×#41	C₂×C₉	semidirect
100. #550	C₂×C₉	semidirect
101. #548	C₂×C₉	semidirect
102. #535	C₂×C₉	semidirect
103. #525	C₂×C₉	semidirect
104. #519	C₂×C₉	semidirect
145. C₂×#41	C₂²×C₉	semidirect
147. C₂²×Q₈	C₂²×C₉	semidirect
150. #127	C₂²×C₉	semidirect
151. #124	C₂²×C₉	semidirect
153. C₂²⋊Q₈	C₂²×C₉	semidirect
154. #121	C₂²×C₉	semidirect
155. #121	C₂²×C₉	semidirect
157. C₄×Q₈	C₂²×C₉	semidirect
158. C₄×D₄	C₂²×C₉	semidirect
189. C₂×Q₈	D₄×C₉	semidirect
220. C₉	#3487	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×C₉×2₋¹⁺⁴	2
C₂²×C₉×#41	2
C₉×#550	2
C₉×#548	2
C₉×#535	2
C₉×#525	2
C₉×#519	2
C₃×#3487	3

Subgroups — C9×#3487 #11768208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3487 #11768208