Subgroups — C₉×#3260 #11767981

Order 1152.

519 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#540	C₂	1	1	yes	maximal, characteristic
3	C₉×#479	C₂	1	2	yes	maximal, complemented
4	C₉×#478	C₂	1	2	yes	maximal, complemented
5	C₉×#468	C₂	1	2	yes	maximal, complemented
6	C₉×#465	C₂	1	2	yes	maximal, complemented
7	C₉×#461	C₂	1	2	yes	maximal, complemented
8	C₉×#460	C₂	1	2	yes	maximal, complemented
9	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3260	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×#127	C₂²	1	1	yes	characteristic
17	C₉×#126	—	2	2	no	—
18	C₉×#125	—	2	2	no	—
19	C₉×#125	C₂²	1	1	yes	characteristic
20	C₉×#125	C₂²	1	1	yes	characteristic
21	C₉×#123	—	2	2	no	—
22	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₉×#121	—	2	2	no	—
24	C₉×#121	C₂²	1	2	yes	—
25	C₄×Q₈×C₉	C₂²	1	2	yes	—
26	C₄×D₄×C₉	C₂²	1	2	yes	—
27	C₉×#117	C₂²	1	1	yes	characteristic
28	C₉×#107	C₂²	1	2	yes	—
29	C₉×#106	C₂²	1	2	yes	—
30	C₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₉×C₄⋊C₈	C₂²	1	2	yes	—
32	C₉×#103	C₂²	1	2	yes	—
33	C₉×#103	C₂²	1	2	yes	—
34	C₉×#102	C₂²	1	2	yes	—
35	C₉×#102	C₂²	1	2	yes	—
36	C₃×#541	C₂×C₃	1	1	yes	characteristic
37	C₃×#540	C₂×C₃	1	1	yes	characteristic
38	C₃×#479	C₂×C₃	1	2	yes	—
39	C₃×#478	C₂×C₃	1	2	yes	—
40	C₃×#468	C₂×C₃	1	2	yes	—
41	C₃×#465	C₂×C₃	1	2	yes	—
42	C₃×#461	C₂×C₃	1	2	yes	—
43	C₃×#460	C₂×C₃	1	2	yes	—
44	C₃×#424	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	C₂³	1	2	yes	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×SD₁₆	—	4	2	no	—
52	C₉×SD₁₆	—	4	2	no	—
53	C₉×M₁₆	—	4	1	no	—
54	C₂×C₈×C₉	—	4	1	no	—
55	C₂×C₈×C₉	C₂³	1	2	yes	—
56	C₂×C₈×C₉	C₂³	1	2	yes	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	D₄	1	2	yes	—
63	C₉×C₄⋊C₄	C₂³	1	2	yes	—
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	D₄	1	2	yes	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	C₂³	1	1	yes	characteristic
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	#3260	C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
74	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
75	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×#127	C₂²×C₃	1	1	yes	characteristic
79	C₃×#126	—	2	2	no	—
80	C₃×#125	—	2	2	no	—
81	C₃×#125	C₂²×C₃	1	1	yes	characteristic
82	C₃×#125	C₂²×C₃	1	1	yes	characteristic
83	C₃×#123	—	2	2	no	—
84	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₃×#121	—	2	2	no	—
86	C₃×#121	C₂²×C₃	1	2	yes	—
87	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
88	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
89	C₃×#117	C₂²×C₃	1	1	yes	characteristic
90	C₃×#107	C₂²×C₃	1	2	yes	—
91	C₃×#106	C₂²×C₃	1	2	yes	—
92	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
93	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	C₃×#103	C₂²×C₃	1	2	yes	—
95	C₃×#103	C₂²×C₃	1	2	yes	—
96	C₃×#102	C₂²×C₃	1	2	yes	—
97	C₃×#102	C₂²×C₃	1	2	yes	—
98	C₂³×C₉	—	2	2	no	—
99	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₉	—	2	2	no	—
101	Q₈×C₉	—	2	2	no	—
102	D₄×C₉	—	2	2	no	—
103	D₄×C₉	—	2	2	no	—
104	D₄×C₉	—	8	2	no	—
105	D₄×C₉	—	8	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	8	2	no	—
112	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
116	C₈×C₉	—	2	2	no	—
117	C₈×C₉	—	2	2	no	—
118	#541	C₂×C₉	1	1	yes	characteristic, complemented
119	#540	C₂×C₉	1	1	yes	characteristic
120	#479	C₂×C₉	1	2	yes	complemented
121	#478	C₂×C₉	1	2	yes	complemented
122	#468	C₂×C₉	1	2	yes	complemented
123	#465	C₂×C₉	1	2	yes	complemented
124	#461	C₂×C₉	1	2	yes	complemented
125	#460	C₂×C₉	1	2	yes	complemented
126	#424	C₂×C₉	1	1	yes	characteristic, complemented
127	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
128	C₂×C₃×D₄	—	2	2	no	—
129	C₂×C₃×D₄	—	2	2	no	—
130	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₃×SD₁₆	—	4	2	no	—
134	C₃×SD₁₆	—	4	2	no	—
135	C₃×M₁₆	—	4	1	no	—
136	C₂×C₃×C₈	—	4	1	no	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
145	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
146	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
147	C₃×C₂²⋊C₄	—	2	2	no	—
148	C₃×C₂²⋊C₄	—	2	2	no	—
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
151	C₃×C₄²	—	2	2	no	—
152	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	—	4	2	no	—
157	C₂²×C₉	—	4	2	no	—
158	C₂²×C₉	—	4	2	no	—
159	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	4	2	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
169	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
170	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
172	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
173	#127	C₂²×C₉	1	1	yes	characteristic
174	#126	—	2	2	no	—
175	#125	—	2	2	no	—
176	#125	C₂²×C₉	1	1	yes	characteristic
177	#125	C₂²×C₉	1	1	yes	characteristic
178	#123	—	2	2	no	—
179	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
180	#121	—	2	2	no	—
181	#121	C₂²×C₉	1	2	yes	—
182	C₄×Q₈	C₂²×C₉	1	2	yes	—
183	C₄×D₄	C₂²×C₉	1	2	yes	—
184	#117	C₂²×C₉	1	1	yes	characteristic
185	#107	C₂²×C₉	1	2	yes	—
186	#106	C₂²×C₉	1	2	yes	—
187	C₄⋊C₈	C₂²×C₉	1	2	yes	—
188	C₄⋊C₈	C₂²×C₉	1	2	yes	—
189	#103	C₂²×C₉	1	2	yes	—
190	#103	C₂²×C₉	1	2	yes	—
191	#102	C₂²×C₉	1	2	yes	—
192	#102	C₂²×C₉	1	2	yes	—
193	C₂³×C₃	—	2	2	no	—
194	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
195	C₃×Q₈	—	2	2	no	—
196	C₃×Q₈	—	2	2	no	—
197	C₃×D₄	—	2	2	no	—
198	C₃×D₄	—	2	2	no	—
199	C₃×D₄	—	8	2	no	—
200	C₃×D₄	—	8	2	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	8	2	no	—
207	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	—	2	2	no	—
213	C₂×C₉	—	4	1	no	—
214	C₂×C₉	—	8	2	no	—
215	C₂×C₉	#537	1	1	yes	characteristic
216	C₂×C₉	#577	1	2	yes	—
217	C₂×Q₈	C₂³×C₉	1	2	yes	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	C₂³×C₉	1	2	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
223	SD₁₆	—	4	2	no	—
224	SD₁₆	—	4	2	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₂³×C₉	1	2	yes	—
228	C₂×C₈	C₂³×C₉	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	D₄×C₉	1	2	yes	—
235	C₄⋊C₄	C₂³×C₉	1	2	yes	—
236	C₄⋊C₄	C₂³×C₉	1	2	yes	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	D₄×C₉	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	C₂³×C₉	1	1	yes	characteristic
243	C₄²	C₂³×C₉	1	1	yes	characteristic
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	2	no	—
249	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	4	2	no	—
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
257	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
258	C₉	#3260	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
261	Q₈	—	2	2	no	—
262	Q₈	—	2	2	no	—
263	D₄	—	2	2	no	—
264	D₄	—	2	2	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
274	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
275	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
276	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₂×C₃	—	4	1	no	—
280	C₂×C₃	—	8	2	no	—
281	C₂×C₃	C₃×#537	1	1	yes	characteristic
282	C₂×C₃	C₃×#577	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
296	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
297	C₃	C₃×#3260	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	—	8	2	no	—
300	C₂	C₉×#537	1	1	yes	characteristic
301	C₂	C₉×#577	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#540	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#541	C₂	semidirect
3. C₉×#479	C₂	semidirect
4. C₉×#478	C₂	semidirect
5. C₉×#468	C₂	semidirect
6. C₉×#465	C₂	semidirect
7. C₉×#461	C₂	semidirect
8. C₉×#460	C₂	semidirect
9. C₉×#424	C₂	semidirect
72. #3260	C₉	direct (N × H)
118. #541	C₂×C₉	semidirect
120. #479	C₂×C₉	semidirect
121. #478	C₂×C₉	semidirect
122. #468	C₂×C₉	semidirect
123. #465	C₂×C₉	semidirect
124. #461	C₂×C₉	semidirect
125. #460	C₂×C₉	semidirect
126. #424	C₂×C₉	semidirect
258. C₉	#3260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#541	2
C₉×#540	2
C₉×#479	2
C₉×#478	2
C₉×#468	2
C₉×#465	2
C₉×#461	2
C₉×#460	2
C₉×#424	2
C₃×#3260	3

Subgroups — C9×#3260 #11767981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3260 #11767981