Subgroups — C₉×#3287 #11768008

Order 1152.

756 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	2	yes	maximal, complemented
2	C₉×#550	C₂	1	2	yes	maximal, complemented
3	C₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#466	C₂	1	2	yes	maximal, complemented
5	C₉×#460	C₂	1	2	yes	maximal, complemented
6	C₉×#449	C₂	1	4	yes	maximal, complemented
7	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3287	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	4	yes	—
11	C₉×#135	—	2	8	no	—
12	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₉×D₈	C₂²	1	2	yes	—
15	C₂×C₉×D₈	C₂²	1	2	yes	—
16	C₂²×C₈×C₉	C₂²	1	2	yes	—
17	C₉×#127	—	2	2	no	—
18	C₉×#127	C₂²	1	2	yes	—
19	C₉×#121	—	2	4	no	—
20	C₉×#121	C₂²	1	4	yes	—
21	C₄×Q₈×C₉	C₂²	1	2	yes	—
22	C₄×D₄×C₉	—	2	2	no	—
23	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
24	C₉×#107	C₂²	1	1	yes	characteristic
25	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₉×#103	C₂²	1	2	yes	—
27	C₉×#102	C₂²	1	4	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
30	C₂×C₃×#135	C₂×C₃	1	2	yes	—
31	C₃×#550	C₂×C₃	1	2	yes	—
32	C₃×#493	C₂×C₃	1	1	yes	characteristic
33	C₃×#466	C₂×C₃	1	2	yes	—
34	C₃×#460	C₂×C₃	1	2	yes	—
35	C₃×#449	C₂×C₃	1	4	yes	—
36	C₃×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
37	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic
38	C₉×#41	—	2	8	no	—
39	C₉×#41	—	4	4	no	—
40	C₂×Q₈×C₉	C₂³	1	2	yes	—
41	C₂×D₄×C₉	—	2	4	no	—
42	C₂×D₄×C₉	—	2	4	no	—
43	C₂×D₄×C₉	—	2	2	no	—
44	C₂×D₄×C₉	—	2	2	no	—
45	C₂×D₄×C₉	D₄	1	1	yes	characteristic
46	C₂×D₄×C₉	C₂³	1	4	yes	—
47	C₂²×C₄×C₉	—	2	4	no	—
48	C₂²×C₄×C₉	C₂³	1	2	yes	—
49	C₉×Q₁₆	D₄	1	4	yes	complemented
50	C₉×SD₁₆	—	2	8	no	—
51	C₉×D₈	—	2	4	no	—
52	C₉×D₈	—	4	2	no	—
53	C₂×C₈×C₉	—	2	4	no	—
54	C₂×C₈×C₉	C₂³	1	2	yes	—
55	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₉×C₄⋊C₄	C₂³	1	2	yes	—
60	C₉×C₂²⋊C₄	—	2	4	no	—
61	C₉×C₂²⋊C₄	D₄	1	2	yes	—
62	C₄²×C₉	—	2	2	no	—
63	C₄²×C₉	C₂³	1	1	yes	characteristic
64	#3287	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
66	C₃×#135	—	2	8	no	—
67	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
69	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
70	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
71	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
72	C₃×#127	—	2	2	no	—
73	C₃×#127	C₂²×C₃	1	2	yes	—
74	C₃×#121	—	2	4	no	—
75	C₃×#121	C₂²×C₃	1	4	yes	—
76	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
77	C₃×C₄×D₄	—	2	2	no	—
78	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₃×#107	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₃×#103	C₂²×C₃	1	2	yes	—
82	C₃×#102	C₂²×C₃	1	4	yes	—
83	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂³×C₉	—	2	4	no	—
86	C₂³×C₉	C₂×D₄	1	2	yes	—
87	Q₈×C₉	—	2	2	no	—
88	Q₈×C₉	C₂×D₄	1	4	yes	—
89	D₄×C₉	—	2	2	no	—
90	D₄×C₉	—	2	4	no	—
91	D₄×C₉	—	2	4	no	—
92	D₄×C₉	—	4	8	no	—
93	D₄×C₉	—	4	4	no	—
94	D₄×C₉	—	8	2	no	—
95	D₄×C₉	—	8	4	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	4	8	no	—
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₉	—	2	2	no	—
106	C₈×C₉	—	2	1	no	—
107	C₈×C₉	C₂×D₄	1	2	yes	—
108	C₂×#135	C₂×C₉	1	2	yes	complemented
109	#550	C₂×C₉	1	2	yes	complemented
110	#493	C₂×C₉	1	1	yes	characteristic, complemented
111	#466	C₂×C₉	1	2	yes	complemented
112	#460	C₂×C₉	1	2	yes	complemented
113	#449	C₂×C₉	1	4	yes	complemented
114	C₄×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
115	C₈×D₄	C₂×C₉	1	1	yes	characteristic, complemented
116	C₃×#41	—	2	8	no	—
117	C₃×#41	—	4	4	no	—
118	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
119	C₂×C₃×D₄	—	2	4	no	—
120	C₂×C₃×D₄	—	2	4	no	—
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	—	2	2	no	—
123	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
124	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
127	C₃×Q₁₆	C₃×D₄	1	4	yes	—
128	C₃×SD₁₆	—	2	8	no	—
129	C₃×D₈	—	2	4	no	—
130	C₃×D₈	—	4	2	no	—
131	C₂×C₃×C₈	—	2	4	no	—
132	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
134	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
137	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
138	C₃×C₂²⋊C₄	—	2	4	no	—
139	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
140	C₃×C₄²	—	2	2	no	—
141	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	2	no	—
144	C₂²×C₉	—	2	2	no	—
145	C₂²×C₉	—	4	4	no	—
146	C₂²×C₉	—	4	4	no	—
147	C₂²×C₉	—	4	4	no	—
148	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	2	4	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	4	2	no	—
153	C₄×C₉	#135	1	2	yes	complemented
154	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×#41	C₂²×C₉	1	4	yes	—
157	#135	—	2	8	no	—
158	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
159	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
160	C₂×D₈	C₂²×C₉	1	2	yes	—
161	C₂×D₈	C₂²×C₉	1	2	yes	—
162	C₂²×C₈	C₂²×C₉	1	2	yes	—
163	#127	—	2	2	no	—
164	#127	C₂²×C₉	1	2	yes	—
165	#121	—	2	4	no	—
166	#121	C₂²×C₉	1	4	yes	—
167	C₄×Q₈	C₂²×C₉	1	2	yes	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
170	#107	C₂²×C₉	1	1	yes	characteristic
171	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
172	#103	C₂²×C₉	1	2	yes	—
173	#102	C₂²×C₉	1	4	yes	—
174	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
175	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
176	C₂³×C₃	—	2	4	no	—
177	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
178	C₃×Q₈	—	2	2	no	—
179	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
180	C₃×D₄	—	2	2	no	—
181	C₃×D₄	—	2	4	no	—
182	C₃×D₄	—	2	4	no	—
183	C₃×D₄	—	4	8	no	—
184	C₃×D₄	—	4	4	no	—
185	C₃×D₄	—	8	2	no	—
186	C₃×D₄	—	8	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	4	8	no	—
192	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₉	—	4	2	no	—
200	C₂×C₉	—	8	4	no	—
201	C₂×C₉	D₄²	1	1	yes	characteristic
202	C₂×C₉	C₂×#135	1	1	yes	characteristic
203	C₂×C₉	#576	1	1	yes	characteristic
204	#41	—	2	8	no	—
205	#41	—	4	4	no	—
206	C₂×Q₈	C₂³×C₉	1	2	yes	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	D₄×C₉	1	1	yes	characteristic
212	C₂×D₄	C₂³×C₉	1	4	yes	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂³×C₉	1	2	yes	—
215	Q₁₆	D₄×C₉	1	4	yes	complemented
216	SD₁₆	—	2	8	no	—
217	D₈	—	2	4	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₉	1	2	yes	—
222	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₉	1	2	yes	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	D₄×C₉	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	C₂³×C₉	1	1	yes	characteristic
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	4	4	no	—
234	C₂²×C₃	—	4	4	no	—
235	C₂²×C₃	—	4	4	no	—
236	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	C₃×#135	1	2	yes	—
242	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
243	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₉	#3287	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	4	no	—
246	C₂³	C₂×D₄×C₉	1	2	yes	—
247	Q₈	—	2	2	no	—
248	Q₈	C₂×D₄×C₉	1	4	yes	—
249	D₄	—	2	2	no	—
250	D₄	—	2	4	no	—
251	D₄	—	2	4	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	4	no	—
254	D₄	—	8	2	no	—
255	D₄	—	8	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
264	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	C₂×D₄×C₉	1	2	yes	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	—	8	4	no	—
270	C₂×C₃	C₃×D₄²	1	1	yes	characteristic
271	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
272	C₂×C₃	C₃×#576	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	2	no	—
284	C₄	C₉×#135	1	2	yes	complemented
285	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
286	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
287	C₃	C₃×#3287	1	1	yes	characteristic
288	C₂	—	4	2	no	—
289	C₂	—	8	4	no	—
290	C₂	D₄²×C₉	1	1	yes	characteristic
291	C₂	C₂×C₉×#135	1	1	yes	characteristic
292	C₂	C₉×#576	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	C₃×#3287	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×D₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#135	C₂	semidirect
2. C₉×#550	C₂	semidirect
3. C₉×#493	C₂	semidirect
4. C₉×#466	C₂	semidirect
5. C₉×#460	C₂	semidirect
6. C₉×#449	C₂	semidirect
7. C₄×C₉×Q₁₆	C₂	semidirect
8. C₈×D₄×C₉	C₂	semidirect
49. C₉×Q₁₆	D₄	semidirect
64. #3287	C₉	direct (N × H)
108. C₂×#135	C₂×C₉	semidirect
109. #550	C₂×C₉	semidirect
110. #493	C₂×C₉	semidirect
111. #466	C₂×C₉	semidirect
112. #460	C₂×C₉	semidirect
113. #449	C₂×C₉	semidirect
114. C₄×Q₁₆	C₂×C₉	semidirect
115. C₈×D₄	C₂×C₉	semidirect
153. C₄×C₉	#135	semidirect
215. Q₁₆	D₄×C₉	semidirect
244. C₉	#3287	direct (N × H)
284. C₄	C₉×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#135	2
C₉×#550	2
C₉×#493	2
C₉×#466	2
C₉×#460	2
C₉×#449	2
C₄×C₉×Q₁₆	2
C₈×D₄×C₉	2
C₃×#3287	3

Subgroups — C9×#3287 #11768008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3287 #11768008