Subgroups — C₉×#3227 #11767948

Order 1152.

606 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#498	C₂	1	2	yes	maximal, complemented
4	C₉×#486	C₂	1	2	yes	maximal, complemented
5	C₉×#481	C₂	1	4	yes	maximal, complemented
6	C₉×#476	C₂	1	4	yes	maximal, complemented
7	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3227	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	—	2	1	no	—
10	C₂²×Q₈×C₉	—	2	1	no	—
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×#127	C₂²	1	1	yes	characteristic
17	C₉×#124	—	2	1	no	—
18	C₉×#121	—	2	1	no	—
19	C₉×#121	C₂²	1	2	yes	—
20	C₄×D₄×C₉	—	2	1	no	—
21	C₄×D₄×C₉	C₂²	1	2	yes	—
22	C₂×C₉×C₄⋊C₄	—	2	1	no	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×#106	C₂²	1	8	yes	—
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×#102	C₂²	1	8	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₃×#532	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₃×#498	C₂×C₃	1	2	yes	—
33	C₃×#486	C₂×C₃	1	2	yes	—
34	C₃×#481	C₂×C₃	1	4	yes	—
35	C₃×#476	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₉×#41	—	4	4	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	—	2	2	no	—
40	C₂×Q₈×C₉	—	2	2	no	—
41	C₂×Q₈×C₉	—	4	4	no	—
42	C₂×D₄×C₉	—	2	2	no	—
43	C₂×D₄×C₉	—	2	1	no	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	—	2	2	no	—
48	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₉	C₂³	1	4	yes	—
52	C₉×C₄⋊C₄	—	2	2	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	C₂³	1	4	yes	—
57	C₉×C₄⋊C₄	C₂³	1	2	yes	—
58	C₉×C₂²⋊C₄	—	2	2	no	—
59	C₉×C₂²⋊C₄	—	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	#3227	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	—	2	1	no	—
66	C₂²×C₃×Q₈	—	2	1	no	—
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×#127	C₂²×C₃	1	1	yes	characteristic
73	C₃×#124	—	2	1	no	—
74	C₃×#121	—	2	1	no	—
75	C₃×#121	C₂²×C₃	1	2	yes	—
76	C₃×C₄×D₄	—	2	1	no	—
77	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	—	2	1	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
81	C₃×#106	C₂²×C₃	1	8	yes	—
82	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
83	C₃×#102	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	—	2	2	no	—
87	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₉	—	4	4	no	—
89	Q₈×C₉	—	4	2	no	—
90	Q₈×C₉	—	4	4	no	—
91	D₄×C₉	—	2	4	no	—
92	D₄×C₉	—	4	4	no	—
93	D₄×C₉	—	4	2	no	—
94	D₄×C₉	—	4	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	4	4	no	—
100	C₂×C₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	SD₁₆	1	4	yes	—
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
108	C₈×C₉	—	2	4	no	—
109	#532	C₂×C₉	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
111	#498	C₂×C₉	1	2	yes	complemented
112	#486	C₂×C₉	1	2	yes	complemented
113	#481	C₂×C₉	1	4	yes	complemented
114	#476	C₂×C₉	1	4	yes	complemented
115	#431	C₂×C₉	1	1	yes	characteristic, complemented
116	C₃×#41	—	4	4	no	—
117	C₂×C₃×Q₈	—	2	1	no	—
118	C₂×C₃×Q₈	—	2	2	no	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	4	4	no	—
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
125	C₂²×C₃×C₄	—	2	2	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
136	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
137	C₃×C₂²⋊C₄	—	2	2	no	—
138	C₃×C₂²⋊C₄	—	2	2	no	—
139	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
140	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
141	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
142	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	4	2	no	—
146	C₂²×C₉	—	4	2	no	—
147	C₂²×C₉	—	4	2	no	—
148	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
149	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	4	2	no	—
153	C₄×C₉	—	4	4	no	—
154	C₄×C₉	C₂×SD₁₆	1	2	yes	—
155	C₄×C₉	C₂×SD₁₆	1	2	yes	—
156	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
157	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
158	C₂×#41	—	2	1	no	—
159	C₂²×Q₈	—	2	1	no	—
160	C₄⋊Q₈	—	2	1	no	—
161	C₄⋊Q₈	—	2	1	no	—
162	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
163	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
164	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
165	#127	C₂²×C₉	1	1	yes	characteristic
166	#124	—	2	1	no	—
167	#121	—	2	1	no	—
168	#121	C₂²×C₉	1	2	yes	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂²×C₉	1	2	yes	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
173	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
174	#106	C₂²×C₉	1	8	yes	—
175	C₄⋊C₈	C₂²×C₉	1	2	yes	—
176	#102	C₂²×C₉	1	8	yes	—
177	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
178	C₄×C₈	C₂²×C₉	1	2	yes	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
181	C₃×Q₈	—	4	4	no	—
182	C₃×Q₈	—	4	2	no	—
183	C₃×Q₈	—	4	4	no	—
184	C₃×D₄	—	2	4	no	—
185	C₃×D₄	—	4	4	no	—
186	C₃×D₄	—	4	2	no	—
187	C₃×D₄	—	4	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	—	2	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	—	4	4	no	—
194	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
201	C₃×C₈	—	2	4	no	—
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	8	2	no	—
204	C₂×C₉	#536	1	1	yes	characteristic
205	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
206	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₂³×C₉	1	2	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
219	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₉	1	4	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	C₂³×C₉	1	4	yes	—
227	C₄⋊C₄	C₂³×C₉	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	D₄×C₉	1	1	yes	characteristic
231	C₄²	D₄×C₉	1	1	yes	characteristic
232	C₄²	C₂³×C₉	1	1	yes	characteristic
233	C₄²	C₂³×C₉	1	1	yes	characteristic
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	4	2	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
240	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	4	4	no	—
245	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
246	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
247	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
248	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
249	C₉	#3227	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₉×SD₁₆	1	4	yes	—
266	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
271	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	8	2	no	—
275	C₂×C₃	C₃×#536	1	1	yes	characteristic
276	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
277	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
284	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	C₂×C₉×SD₁₆	1	2	yes	—
290	C₄	C₂×C₉×SD₁₆	1	2	yes	—
291	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
292	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
293	C₃	C₃×#3227	1	1	yes	characteristic
294	C₂	—	2	2	no	—
295	C₂	—	8	2	no	—
296	C₂	C₉×#536	1	1	yes	characteristic
297	C₂	C₂²×C₉×SD₁₆	1	1	yes	characteristic
298	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	C₃×#532	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
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#498	C₂	semidirect
4. C₉×#486	C₂	semidirect
5. C₉×#481	C₂	semidirect
6. C₉×#476	C₂	semidirect
7. C₉×#431	C₂	semidirect
64. #3227	C₉	direct (N × H)
110. C₂×C₄⋊Q₈	C₂×C₉	semidirect
111. #498	C₂×C₉	semidirect
112. #486	C₂×C₉	semidirect
113. #481	C₂×C₉	semidirect
114. #476	C₂×C₉	semidirect
115. #431	C₂×C₉	semidirect
249. C₉	#3227	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×#532	2
C₂×C₉×C₄⋊Q₈	2
C₉×#498	2
C₉×#486	2
C₉×#481	2
C₉×#476	2
C₉×#431	2
C₃×#3227	3

Subgroups — C9×#3227 #11767948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3227 #11767948