Subgroups — C₉×#3208 #11767929

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₉×#492	C₂	1	2	yes	maximal, complemented
4	C₉×#486	C₂	1	2	yes	maximal, complemented
5	C₉×#460	C₂	1	4	yes	maximal, complemented
6	C₉×#450	C₂	1	4	yes	maximal, complemented
7	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3208	C₃	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₉	—	2	1	no	—
10	C₂²×D₄×C₉	—	2	1	no	—
11	C₂²×D₄×C₉	C₂²	1	2	yes	—
12	C₂×C₉×SD₁₆	C₂²	1	8	yes	complemented
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₉×#127	—	2	1	no	—
16	C₉×#127	—	2	1	no	—
17	C₉×#127	C₂²	1	2	yes	—
18	C₉×#127	C₂²	1	2	yes	—
19	C₉×#125	—	2	1	no	—
20	C₉×C₂²⋊Q₈	—	2	1	no	—
21	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₉	—	2	1	no	—
23	C₄×Q₈×C₉	C₂²	1	2	yes	complemented
24	C₉×#117	—	2	1	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₉×#102	C₂²	1	8	yes	complemented
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	complemented
30	C₃×#533	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
32	C₃×#492	C₂×C₃	1	2	yes	—
33	C₃×#486	C₂×C₃	1	2	yes	—
34	C₃×#460	C₂×C₃	1	4	yes	—
35	C₃×#450	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₂⁴×C₉	—	2	2	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	C₂³	1	2	yes	—
40	C₂×D₄×C₉	—	2	2	no	—
41	C₂×D₄×C₉	—	2	2	no	—
42	C₂×D₄×C₉	—	2	2	no	—
43	C₂×D₄×C₉	—	2	2	no	—
44	C₂×D₄×C₉	—	2	2	no	—
45	C₂×D₄×C₉	—	4	4	no	—
46	C₂×D₄×C₉	—	4	4	no	—
47	C₂×D₄×C₉	—	4	4	no	—
48	C₂×D₄×C₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₉×SD₁₆	—	4	8	no	—
53	C₂×C₈×C₉	C₂³	1	4	yes	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	—	2	1	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	C₂³	1	2	yes	—
60	C₉×C₂²⋊C₄	—	2	2	no	—
61	C₄²×C₉	—	2	2	no	—
62	C₄²×C₉	D₄	1	1	yes	characteristic
63	C₄²×C₉	D₄	1	1	yes	characteristic
64	C₄²×C₉	C₂³	1	1	yes	characteristic
65	C₄²×C₉	C₂³	1	1	yes	characteristic
66	#3208	C₉	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₃×D₄	—	2	1	no	—
68	C₂²×C₃×D₄	—	2	1	no	—
69	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
70	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
71	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
72	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₃×#127	—	2	1	no	—
74	C₃×#127	—	2	1	no	—
75	C₃×#127	C₂²×C₃	1	2	yes	—
76	C₃×#127	C₂²×C₃	1	2	yes	—
77	C₃×#125	—	2	1	no	—
78	C₃×C₂²⋊Q₈	—	2	1	no	—
79	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
80	C₃×C₄×Q₈	—	2	1	no	—
81	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
82	C₃×#117	—	2	1	no	—
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
85	C₃×#102	C₂²×C₃	1	8	yes	—
86	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
87	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
88	C₂³×C₉	—	2	4	no	—
89	C₂³×C₉	—	4	4	no	—
90	C₂³×C₉	—	4	4	no	—
91	C₂³×C₉	—	4	4	no	—
92	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
93	Q₈×C₉	—	2	4	no	—
94	Q₈×C₉	—	4	2	no	—
95	D₄×C₉	—	2	8	no	—
96	D₄×C₉	—	4	4	no	—
97	D₄×C₉	—	4	4	no	—
98	D₄×C₉	—	4	4	no	—
99	D₄×C₉	—	4	4	no	—
100	D₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
112	C₈×C₉	—	2	4	no	—
113	#533	C₂×C₉	1	1	yes	characteristic, complemented
114	C₂×#127	C₂×C₉	1	1	yes	characteristic
115	#492	C₂×C₉	1	2	yes	complemented
116	#486	C₂×C₉	1	2	yes	complemented
117	#460	C₂×C₉	1	4	yes	complemented
118	#450	C₂×C₉	1	4	yes	complemented
119	#431	C₂×C₉	1	1	yes	characteristic, complemented
120	C₂⁴×C₃	—	2	2	no	—
121	C₂×C₃×Q₈	—	2	1	no	—
122	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
123	C₂×C₃×D₄	—	2	2	no	—
124	C₂×C₃×D₄	—	2	2	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	—	2	2	no	—
128	C₂×C₃×D₄	—	4	4	no	—
129	C₂×C₃×D₄	—	4	4	no	—
130	C₂×C₃×D₄	—	4	4	no	—
131	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
132	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₃×SD₁₆	—	4	8	no	—
136	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
143	C₃×C₂²⋊C₄	—	2	2	no	—
144	C₃×C₄²	—	2	2	no	—
145	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
146	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
147	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
148	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	4	4	no	—
152	C₂²×C₉	—	4	4	no	—
153	C₂²×C₉	—	4	4	no	—
154	C₂²×C₉	—	8	8	no	—
155	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
156	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	—	4	2	no	—
161	C₄×C₉	C₂×SD₁₆	1	2	yes	—
162	C₄×C₉	C₂×SD₁₆	1	2	yes	—
163	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
165	C₂²×D₄	—	2	1	no	—
166	C₂²×D₄	—	2	1	no	—
167	C₂²×D₄	C₂²×C₉	1	2	yes	—
168	C₂×SD₁₆	C₂²×C₉	1	8	yes	complemented
169	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
170	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
171	#127	—	2	1	no	—
172	#127	—	2	1	no	—
173	#127	C₂²×C₉	1	2	yes	—
174	#127	C₂²×C₉	1	2	yes	—
175	#125	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	C₂²×C₉	1	2	yes	complemented
180	#117	—	2	1	no	—
181	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊C₈	C₂²×C₉	1	2	yes	complemented
183	#102	C₂²×C₉	1	8	yes	complemented
184	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
185	C₄×C₈	C₂²×C₉	1	2	yes	complemented
186	C₂³×C₃	—	2	4	no	—
187	C₂³×C₃	—	4	4	no	—
188	C₂³×C₃	—	4	4	no	—
189	C₂³×C₃	—	4	4	no	—
190	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
191	C₃×Q₈	—	2	4	no	—
192	C₃×Q₈	—	4	2	no	—
193	C₃×D₄	—	2	8	no	—
194	C₃×D₄	—	4	4	no	—
195	C₃×D₄	—	4	4	no	—
196	C₃×D₄	—	4	4	no	—
197	C₃×D₄	—	4	4	no	—
198	C₃×D₄	—	4	4	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
210	C₃×C₈	—	2	4	no	—
211	C₂×C₉	—	2	2	no	—
212	C₂×C₉	—	8	4	no	—
213	C₂×C₉	#535	1	1	yes	characteristic
214	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
215	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂⁴	—	2	2	no	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	C₂³×C₉	1	2	yes	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	C₂³×C₉	1	4	yes	—
228	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
229	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
231	SD₁₆	—	4	8	no	—
232	C₂×C₈	C₂³×C₉	1	4	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂³×C₉	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	D₄×C₉	1	1	yes	characteristic
242	C₄²	D₄×C₉	1	1	yes	characteristic
243	C₄²	C₂³×C₉	1	1	yes	characteristic
244	C₄²	C₂³×C₉	1	1	yes	characteristic
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	—	8	8	no	—
251	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
252	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
258	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
259	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
260	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
261	C₉	#3208	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	4	2	no	—
269	D₄	—	2	8	no	—
270	D₄	—	4	4	no	—
271	D₄	—	4	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	C₉×SD₁₆	1	4	yes	complemented
280	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
285	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	8	4	no	—
289	C₂×C₃	C₃×#535	1	1	yes	characteristic
290	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
291	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	8	no	—
298	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
299	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	C₂×C₉×SD₁₆	1	2	yes	—
305	C₄	C₂×C₉×SD₁₆	1	2	yes	—
306	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
307	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
308	C₃	C₃×#3208	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	—	8	4	no	—
311	C₂	C₉×#535	1	1	yes	characteristic
312	C₂	C₂²×C₉×SD₁₆	1	1	yes	characteristic
313	C₂	C₂×C₉×AGL(1,ℤ/8)	1	1	yes	characteristic

	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	C₂×C₃×#127	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×#533	C₂	semidirect
3. C₉×#492	C₂	semidirect
4. C₉×#486	C₂	semidirect
5. C₉×#460	C₂	semidirect
6. C₉×#450	C₂	semidirect
7. C₉×#431	C₂	semidirect
12. C₂×C₉×SD₁₆	C₂²	semidirect
13. C₉×C₄⋊Q₈	C₂²	semidirect
14. C₉×C₄⋊Q₈	C₂²	semidirect
23. C₄×Q₈×C₉	C₂²	semidirect
26. C₉×C₄⋊C₈	C₂²	semidirect
27. C₉×#102	C₂²	semidirect
29. C₄×C₈×C₉	C₂²	semidirect
66. #3208	C₉	direct (N × H)
105. C₂×C₄×C₉	SD₁₆	semidirect
113. #533	C₂×C₉	semidirect
115. #492	C₂×C₉	semidirect
116. #486	C₂×C₉	semidirect
117. #460	C₂×C₉	semidirect
118. #450	C₂×C₉	semidirect
119. #431	C₂×C₉	semidirect
168. C₂×SD₁₆	C₂²×C₉	semidirect
169. C₄⋊Q₈	C₂²×C₉	semidirect
170. C₄⋊Q₈	C₂²×C₉	semidirect
179. C₄×Q₈	C₂²×C₉	semidirect
182. C₄⋊C₈	C₂²×C₉	semidirect
183. #102	C₂²×C₉	semidirect
185. C₄×C₈	C₂²×C₉	semidirect
261. C₉	#3208	direct (N × H)
279. C₂×C₄	C₉×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₂×C₉×#127	2
C₉×#492	2
C₉×#486	2
C₉×#460	2
C₉×#450	2
C₉×#431	2
C₃×#3208	3

Subgroups — C9×#3208 #11767929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3208 #11767929