Subgroups — C₉×#3209 #11767930

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 279 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
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₉×#494	C₂	1	2	yes	maximal, complemented
4	C₉×#487	C₂	1	2	yes	maximal, complemented
5	C₉×#462	C₂	1	4	yes	maximal, complemented
6	C₉×#451	C₂	1	4	yes	maximal
7	C₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₃×#3209	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂×C₉×Q₁₆	C₂²	1	8	yes	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	—	2	1	no	—
13	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₉×#125	—	2	1	no	—
18	C₉×C₂²⋊Q₈	—	2	1	no	—
19	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×Q₈×C₉	—	2	1	no	—
21	C₄×Q₈×C₉	C₂²	1	2	yes	—
22	C₉×#117	—	2	1	no	—
23	C₂×C₉×C₄⋊C₄	—	2	1	no	—
24	C₂×C₉×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×#103	C₂²	1	8	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₃×#533	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₃×#494	C₂×C₃	1	2	yes	—
33	C₃×#487	C₂×C₃	1	2	yes	—
34	C₃×#462	C₂×C₃	1	4	yes	—
35	C₃×#451	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	2	no	—
39	C₂×Q₈×C₉	—	4	4	no	—
40	C₂×Q₈×C₉	C₂³	1	2	yes	—
41	C₂×Q₈×C₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₉	—	2	2	no	—
43	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₉×Q₁₆	—	4	8	no	—
47	C₂×C₈×C₉	C₂³	1	4	yes	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	—	2	2	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	C₂³	1	2	yes	—
58	C₉×C₂²⋊C₄	—	2	2	no	—
59	C₄²×C₉	—	2	2	no	—
60	C₄²×C₉	D₄	1	1	yes	characteristic
61	C₄²×C₉	D₄	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	1	yes	characteristic
63	C₄²×C₉	C₂³	1	1	yes	characteristic
64	#3209	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
66	C₂×C₃×Q₁₆	C₂²×C₃	1	8	yes	—
67	C₃×C₄⋊Q₈	—	2	1	no	—
68	C₃×C₄⋊Q₈	—	2	1	no	—
69	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
70	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
71	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
72	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₃×#125	—	2	1	no	—
74	C₃×C₂²⋊Q₈	—	2	1	no	—
75	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
76	C₃×C₄×Q₈	—	2	1	no	—
77	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
78	C₃×#117	—	2	1	no	—
79	C₂×C₃×C₄⋊C₄	—	2	1	no	—
80	C₂×C₃×C₄⋊C₄	—	2	1	no	—
81	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
83	C₃×#103	C₂²×C₃	1	8	yes	—
84	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
86	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₉	—	2	4	no	—
88	Q₈×C₉	—	2	8	no	—
89	Q₈×C₉	—	4	4	no	—
90	Q₈×C₉	—	4	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	4	4	no	—
97	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
104	C₈×C₉	—	2	4	no	—
105	#533	C₂×C₉	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
107	#494	C₂×C₉	1	2	yes	complemented
108	#487	C₂×C₉	1	2	yes	complemented
109	#462	C₂×C₉	1	4	yes	complemented
110	#451	C₂×C₉	1	4	yes	—
111	#431	C₂×C₉	1	1	yes	characteristic
112	C₂×C₃×Q₈	—	2	1	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	4	4	no	—
115	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
116	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
117	C₂²×C₃×C₄	—	2	2	no	—
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₃×Q₁₆	—	4	8	no	—
122	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₄⋊C₄	—	2	1	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
133	C₃×C₂²⋊C₄	—	2	2	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
136	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
137	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
138	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	4	4	no	—
146	C₄×C₉	—	4	2	no	—
147	C₄×C₉	—	4	2	no	—
148	C₄×C₉	C₂×Q₁₆	1	2	yes	—
149	C₄×C₉	C₂×Q₁₆	1	2	yes	—
150	C₄×C₉	#137	1	2	yes	—
151	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
152	C₂²×Q₈	C₂²×C₉	1	2	yes	—
153	C₂×Q₁₆	C₂²×C₉	1	8	yes	—
154	C₄⋊Q₈	—	2	1	no	—
155	C₄⋊Q₈	—	2	1	no	—
156	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
157	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
158	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
159	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
160	#125	—	2	1	no	—
161	C₂²⋊Q₈	—	2	1	no	—
162	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	C₂²×C₉	1	2	yes	—
165	#117	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₉	1	2	yes	—
170	#103	C₂²×C₉	1	8	yes	—
171	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
172	C₄×C₈	C₂²×C₉	1	2	yes	—
173	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
174	C₃×Q₈	—	2	4	no	—
175	C₃×Q₈	—	2	8	no	—
176	C₃×Q₈	—	4	4	no	—
177	C₃×Q₈	—	4	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	4	4	no	—
184	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
185	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
191	C₃×C₈	—	2	4	no	—
192	C₂×C₉	—	2	2	no	—
193	C₂×C₉	#535	1	1	yes	characteristic
194	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
195	C₂×C₉	C₂×#137	1	1	yes	characteristic
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	C₂³×C₉	1	4	yes	—
200	C₂×Q₈	C₂³×C₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
203	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
205	Q₁₆	—	4	8	no	—
206	C₂×C₈	C₂³×C₉	1	4	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂³×C₉	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	D₄×C₉	1	1	yes	characteristic
220	C₄²	D₄×C₉	1	1	yes	characteristic
221	C₄²	C₂³×C₉	1	1	yes	characteristic
222	C₄²	C₂³×C₉	1	1	yes	characteristic
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
226	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	C₂×C₃×Q₁₆	1	2	yes	—
233	C₃×C₄	C₂×C₃×Q₁₆	1	2	yes	—
234	C₃×C₄	C₃×#137	1	2	yes	—
235	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
236	C₉	#3209	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	8	no	—
240	Q₈	—	4	4	no	—
241	Q₈	—	4	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₉×Q₁₆	1	4	yes	complemented
249	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
254	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	C₃×#535	1	1	yes	characteristic
258	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
259	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	C₂×C₉×Q₁₆	1	2	yes	—
263	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	C₂×C₉×Q₁₆	1	2	yes	—
270	C₄	C₂×C₉×Q₁₆	1	2	yes	—
271	C₄	C₉×#137	1	2	yes	—
272	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
273	C₃	C₃×#3209	1	1	yes	characteristic
274	C₂	—	2	2	no	—
275	C₂	C₉×#535	1	1	yes	characteristic
276	C₂	C₂²×C₉×Q₁₆	1	1	yes	characteristic
277	C₂	C₂×C₉×#137	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	C₂³×C₃	C₂×C₃×D₄
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
3. C₉×#494	C₂	semidirect
4. C₉×#487	C₂	semidirect
5. C₉×#462	C₂	semidirect
64. #3209	C₉	direct (N × H)
97. C₂×C₄×C₉	Q₁₆	semidirect
107. #494	C₂×C₉	semidirect
108. #487	C₂×C₉	semidirect
109. #462	C₂×C₉	semidirect
236. C₉	#3209	direct (N × H)
248. C₂×C₄	C₉×Q₁₆	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×C₄⋊Q₈	2
C₉×#494	2
C₉×#487	2
C₉×#462	2
C₉×#451	2
C₉×#431	2
C₃×#3209	3

Subgroups — C9×#3209 #11767930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3209 #11767930