Subgroups — C₉×#3256 #11767977

Order 1152.

531 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	C₉×#529	C₂	1	2	yes	maximal, complemented
2	C₉×#479	C₂	1	2	yes	maximal, complemented
3	C₉×#478	C₂	1	2	yes	maximal, complemented
4	C₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#469	C₂	1	1	yes	maximal, characteristic
6	C₉×#468	C₂	1	2	yes	maximal, complemented
7	C₉×#464	C₂	1	2	yes	maximal, complemented
8	C₉×#463	C₂	1	2	yes	maximal, complemented
9	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3256	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₉×#126	—	2	2	no	—
17	C₉×#125	C₂²	1	2	yes	—
18	C₉×#124	C₂²	1	2	yes	—
19	C₉×#123	—	2	2	no	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₉×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₉	C₂²	1	2	yes	—
23	C₄×D₄×C₉	—	2	2	no	—
24	C₄×D₄×C₉	C₂²	1	2	yes	—
25	C₉×#117	—	2	2	no	—
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₉×#107	C₂²	1	2	yes	—
28	C₉×#106	C₂²	1	1	yes	characteristic
29	C₉×#106	C₂²	1	1	yes	characteristic
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×#529	C₂×C₃	1	2	yes	—
37	C₃×#479	C₂×C₃	1	2	yes	—
38	C₃×#478	C₂×C₃	1	2	yes	—
39	C₃×#470	C₂×C₃	1	1	yes	characteristic
40	C₃×#469	C₂×C₃	1	1	yes	characteristic
41	C₃×#468	C₂×C₃	1	2	yes	—
42	C₃×#464	C₂×C₃	1	2	yes	—
43	C₃×#463	C₂×C₃	1	2	yes	—
44	C₃×#423	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	C₂³	1	2	yes	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	D₄	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×Q₁₆	—	4	1	no	—
52	C₉×SD₁₆	—	4	2	no	—
53	C₉×D₈	—	4	1	no	—
54	C₉×M₁₆	—	4	1	no	—
55	C₉×M₁₆	—	4	1	no	—
56	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₉	C₂³	1	2	yes	—
58	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
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	—	2	2	no	—
69	C₉×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₉	—	2	2	no	—
71	C₄²×C₉	D₄	1	1	yes	characteristic
72	C₄²×C₉	D₄	1	1	yes	characteristic
73	C₄²×C₉	C₂³	1	2	yes	—
74	#3256	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
77	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
80	C₃×#126	—	2	2	no	—
81	C₃×#125	C₂²×C₃	1	2	yes	—
82	C₃×#124	C₂²×C₃	1	2	yes	—
83	C₃×#123	—	2	2	no	—
84	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₃×#121	C₂²×C₃	1	2	yes	—
86	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
87	C₃×C₄×D₄	—	2	2	no	—
88	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
89	C₃×#117	—	2	2	no	—
90	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₃×#107	C₂²×C₃	1	2	yes	—
92	C₃×#106	C₂²×C₃	1	1	yes	characteristic
93	C₃×#106	C₂²×C₃	1	1	yes	characteristic
94	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
96	C₃×#103	C₂²×C₃	1	2	yes	—
97	C₃×#103	C₂²×C₃	1	2	yes	—
98	C₃×#102	C₂²×C₃	1	2	yes	—
99	C₃×#102	C₂²×C₃	1	2	yes	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₉	—	2	2	no	—
103	Q₈×C₉	—	2	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	4	4	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	4	2	no	—
113	C₂×C₄×C₉	—	4	4	no	—
114	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₉	—	2	1	no	—
119	C₈×C₉	—	2	2	no	—
120	C₈×C₉	—	2	1	no	—
121	#529	C₂×C₉	1	2	yes	complemented
122	#479	C₂×C₉	1	2	yes	complemented
123	#478	C₂×C₉	1	2	yes	complemented
124	#470	C₂×C₉	1	1	yes	characteristic, complemented
125	#469	C₂×C₉	1	1	yes	characteristic
126	#468	C₂×C₉	1	2	yes	complemented
127	#464	C₂×C₉	1	2	yes	complemented
128	#463	C₂×C₉	1	2	yes	complemented
129	#423	C₂×C₉	1	1	yes	characteristic, complemented
130	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₃×C₄	—	2	2	no	—
134	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₃×Q₁₆	—	4	1	no	—
137	C₃×SD₁₆	—	4	2	no	—
138	C₃×D₈	—	4	1	no	—
139	C₃×M₁₆	—	4	1	no	—
140	C₃×M₁₆	—	4	1	no	—
141	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
143	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
149	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₃×C₂²⋊C₄	—	2	2	no	—
151	C₃×C₂²⋊C₄	—	2	2	no	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₄²	—	2	2	no	—
156	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
157	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₃	1	2	yes	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	4	2	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	—	4	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
173	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
174	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
175	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
176	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
177	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
178	#126	—	2	2	no	—
179	#125	C₂²×C₉	1	2	yes	—
180	#124	C₂²×C₉	1	2	yes	—
181	#123	—	2	2	no	—
182	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
183	#121	C₂²×C₉	1	2	yes	—
184	C₄×Q₈	C₂²×C₉	1	2	yes	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	C₂²×C₉	1	2	yes	—
187	#117	—	2	2	no	—
188	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
189	#107	C₂²×C₉	1	2	yes	—
190	#106	C₂²×C₉	1	1	yes	characteristic
191	#106	C₂²×C₉	1	1	yes	characteristic
192	C₄⋊C₈	C₂²×C₉	1	2	yes	—
193	C₄⋊C₈	C₂²×C₉	1	2	yes	—
194	#103	C₂²×C₉	1	2	yes	—
195	#103	C₂²×C₉	1	2	yes	—
196	#102	C₂²×C₉	1	2	yes	—
197	#102	C₂²×C₉	1	2	yes	—
198	C₂³×C₃	—	2	2	no	—
199	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
200	C₃×Q₈	—	2	2	no	—
201	C₃×Q₈	—	2	2	no	—
202	C₃×D₄	—	2	2	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	4	4	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	—	4	4	no	—
212	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	2	1	no	—
219	C₂×C₉	—	4	1	no	—
220	C₂×C₉	—	8	2	no	—
221	C₂×C₉	#537	1	1	yes	characteristic
222	C₂×C₉	#575	1	2	yes	—
223	C₂×Q₈	C₂³×C₉	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂³×C₉	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	D₄×C₉	1	2	yes	—
228	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
229	Q₁₆	—	4	1	no	—
230	SD₁₆	—	4	2	no	—
231	D₈	—	4	1	no	—
232	M₁₆	—	4	1	no	—
233	M₁₆	—	4	1	no	—
234	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
235	C₂×C₈	C₂³×C₉	1	2	yes	—
236	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	C₂³×C₉	1	2	yes	—
242	C₄⋊C₄	C₂³×C₉	1	2	yes	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	D₄×C₉	1	1	yes	characteristic
250	C₄²	D₄×C₉	1	1	yes	characteristic
251	C₄²	C₂³×C₉	1	2	yes	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	—	4	2	no	—
256	C₂²×C₃	—	4	2	no	—
257	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	4	2	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
266	C₉	#3256	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	2	2	no	—
268	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	D₄	—	2	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	4	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
284	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	2	no	—
287	C₈	—	2	1	no	—
288	C₂×C₃	—	4	1	no	—
289	C₂×C₃	—	8	2	no	—
290	C₂×C₃	C₃×#537	1	1	yes	characteristic
291	C₂×C₃	C₃×#575	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
305	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
306	C₃	C₃×#3256	1	1	yes	characteristic
307	C₂	—	4	1	no	—
308	C₂	—	8	2	no	—
309	C₂	C₉×#537	1	1	yes	characteristic
310	C₂	C₉×#575	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×#469	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#529	C₂	semidirect
2. C₉×#479	C₂	semidirect
3. C₉×#478	C₂	semidirect
4. C₉×#470	C₂	semidirect
6. C₉×#468	C₂	semidirect
7. C₉×#464	C₂	semidirect
8. C₉×#463	C₂	semidirect
9. C₉×#423	C₂	semidirect
74. #3256	C₉	direct (N × H)
121. #529	C₂×C₉	semidirect
122. #479	C₂×C₉	semidirect
123. #478	C₂×C₉	semidirect
124. #470	C₂×C₉	semidirect
126. #468	C₂×C₉	semidirect
127. #464	C₂×C₉	semidirect
128. #463	C₂×C₉	semidirect
129. #423	C₂×C₉	semidirect
266. C₉	#3256	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×#529	2
C₉×#479	2
C₉×#478	2
C₉×#470	2
C₉×#469	2
C₉×#468	2
C₉×#464	2
C₉×#463	2
C₉×#423	2
C₃×#3256	3

Subgroups — C9×#3256 #11767977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3256 #11767977