Subgroups — C₉×#3126 #11767847

Order 1152.

573 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#540	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#482	C₂	1	2	yes	maximal, complemented
4	C₉×#479	C₂	1	2	yes	maximal, complemented
5	C₉×#460	C₂	1	2	yes	maximal, complemented
6	C₉×#448	C₂	1	2	yes	maximal, complemented
7	C₉×#440	C₂	1	2	yes	maximal, complemented
8	C₄×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
9	C₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3126	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
14	C₉×#127	C₂²	1	1	yes	characteristic
15	C₉×#125	C₂²	1	1	yes	characteristic
16	C₉×#123	—	2	2	no	—
17	C₉×#121	—	2	2	no	—
18	C₉×#121	C₂²	1	2	yes	—
19	C₄×Q₈×C₉	—	2	2	no	—
20	C₄×Q₈×C₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₉	C₂²	1	2	yes	—
22	C₄×D₄×C₉	C₂²	1	2	yes	—
23	C₉×#117	—	2	2	no	—
24	C₉×#117	C₂²	1	1	yes	characteristic
25	C₉×#117	C₂²	1	1	yes	characteristic
26	C₉×#107	C₂²	1	2	yes	—
27	C₉×#106	C₂²	1	2	yes	—
28	C₉×C₄⋊C₈	C₂²	1	2	yes	—
29	C₉×#103	C₂²	1	2	yes	—
30	C₉×#103	C₂²	1	2	yes	—
31	C₉×#102	C₂²	1	2	yes	—
32	C₉×#102	C₂²	1	2	yes	—
33	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₉×#97	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
36	C₃×#540	C₂×C₃	1	1	yes	characteristic
37	C₃×#519	C₂×C₃	1	1	yes	characteristic
38	C₃×#482	C₂×C₃	1	2	yes	—
39	C₃×#479	C₂×C₃	1	2	yes	—
40	C₃×#460	C₂×C₃	1	2	yes	—
41	C₃×#448	C₂×C₃	1	2	yes	—
42	C₃×#440	C₂×C₃	1	2	yes	—
43	C₃×C₄×SD₁₆	C₂×C₃	1	2	yes	—
44	C₃×#433	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	—	2	4	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	C₂³	1	2	yes	—
48	C₂×D₄×C₉	—	2	2	no	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₉×SD₁₆	—	2	4	no	—
55	C₉×SD₁₆	—	2	4	no	—
56	C₂×C₈×C₉	C₂³	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	2	yes	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	D₄	1	2	yes	—
62	C₉×C₄⋊C₄	C₂³	1	2	yes	—
63	C₉×C₄⋊C₄	C₂³	1	2	yes	—
64	C₉×C₂²⋊C₄	—	2	1	no	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	D₄	1	2	yes	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	—	2	1	no	—
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	1	yes	characteristic
73	#3126	C₉	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
76	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
77	C₃×#127	C₂²×C₃	1	1	yes	characteristic
78	C₃×#125	C₂²×C₃	1	1	yes	characteristic
79	C₃×#123	—	2	2	no	—
80	C₃×#121	—	2	2	no	—
81	C₃×#121	C₂²×C₃	1	2	yes	—
82	C₃×C₄×Q₈	—	2	2	no	—
83	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
86	C₃×#117	—	2	2	no	—
87	C₃×#117	C₂²×C₃	1	1	yes	characteristic
88	C₃×#117	C₂²×C₃	1	1	yes	characteristic
89	C₃×#107	C₂²×C₃	1	2	yes	—
90	C₃×#106	C₂²×C₃	1	2	yes	—
91	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
92	C₃×#103	C₂²×C₃	1	2	yes	—
93	C₃×#103	C₂²×C₃	1	2	yes	—
94	C₃×#102	C₂²×C₃	1	2	yes	—
95	C₃×#102	C₂²×C₃	1	2	yes	—
96	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₃×#97	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₂³×C₉	—	2	2	no	—
100	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
101	Q₈×C₉	—	2	4	no	—
102	Q₈×C₉	—	2	2	no	—
103	Q₈×C₉	#41	1	4	yes	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	8	2	no	—
107	D₄×C₉	—	8	2	no	—
108	C₂×C₄×C₉	—	2	4	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	—	2	2	no	—
113	C₂×C₄×C₉	—	8	2	no	—
114	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	—	2	2	no	—
120	#540	C₂×C₉	1	1	yes	characteristic
121	#519	C₂×C₉	1	1	yes	characteristic, complemented
122	#482	C₂×C₉	1	2	yes	complemented
123	#479	C₂×C₉	1	2	yes	complemented
124	#460	C₂×C₉	1	2	yes	complemented
125	#448	C₂×C₉	1	2	yes	complemented
126	#440	C₂×C₉	1	2	yes	complemented
127	C₄×SD₁₆	C₂×C₉	1	2	yes	complemented
128	#433	C₂×C₉	1	1	yes	characteristic, complemented
129	C₂×C₃×Q₈	—	2	4	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	—	2	2	no	—
134	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×SD₁₆	—	2	4	no	—
139	C₃×SD₁₆	—	2	4	no	—
140	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
141	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
146	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
147	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
148	C₃×C₂²⋊C₄	—	2	1	no	—
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₂²⋊C₄	—	2	2	no	—
151	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
152	C₃×C₄²	—	2	2	no	—
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
156	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	4	2	no	—
160	C₂²×C₉	—	4	2	no	—
161	C₂²×C₉	—	4	2	no	—
162	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
163	C₄×C₉	—	2	4	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	—	4	2	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	C₂×#41	1	2	yes	—
170	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
171	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
172	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
173	#127	C₂²×C₉	1	1	yes	characteristic
174	#125	C₂²×C₉	1	1	yes	characteristic
175	#123	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂²×C₉	1	2	yes	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	C₂²×C₉	1	2	yes	—
180	C₄×Q₈	C₂²×C₉	1	2	yes	—
181	C₄×D₄	C₂²×C₉	1	2	yes	—
182	#117	—	2	2	no	—
183	#117	C₂²×C₉	1	1	yes	characteristic
184	#117	C₂²×C₉	1	1	yes	characteristic
185	#107	C₂²×C₉	1	2	yes	—
186	#106	C₂²×C₉	1	2	yes	—
187	C₄⋊C₈	C₂²×C₉	1	2	yes	—
188	#103	C₂²×C₉	1	2	yes	—
189	#103	C₂²×C₉	1	2	yes	—
190	#102	C₂²×C₉	1	2	yes	—
191	#102	C₂²×C₉	1	2	yes	—
192	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
193	#97	C₂²×C₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
197	C₃×Q₈	—	2	4	no	—
198	C₃×Q₈	—	2	2	no	—
199	C₃×Q₈	C₃×#41	1	4	yes	—
200	C₃×D₄	—	2	2	no	—
201	C₃×D₄	—	2	2	no	—
202	C₃×D₄	—	8	2	no	—
203	C₃×D₄	—	8	2	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	8	2	no	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	2	2	no	—
216	C₂×C₉	—	4	1	no	—
217	C₂×C₉	—	8	2	no	—
218	C₂×C₉	#525	1	1	yes	characteristic
219	C₂×C₉	#577	1	2	yes	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	C₂³×C₉	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂³×C₉	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
229	SD₁₆	—	2	4	no	—
230	SD₁₆	—	2	4	no	—
231	C₂×C₈	C₂³×C₉	1	2	yes	—
232	C₂×C₈	C₂³×C₉	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	D₄×C₉	1	2	yes	—
237	C₄⋊C₄	C₂³×C₉	1	2	yes	—
238	C₄⋊C₄	C₂³×C₉	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	D₄×C₉	1	2	yes	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₂³×C₉	1	1	yes	characteristic
247	C₄²	C₂³×C₉	1	1	yes	characteristic
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	—	4	2	no	—
253	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	C₂×C₃×#41	1	2	yes	—
261	C₉	#3126	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	2	no	—
263	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	2	2	no	—
266	Q₈	C₉×#41	1	4	yes	—
267	D₄	—	2	2	no	—
268	D₄	—	2	2	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
278	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
279	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
280	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₂×C₃	—	4	1	no	—
284	C₂×C₃	—	8	2	no	—
285	C₂×C₃	C₃×#525	1	1	yes	characteristic
286	C₂×C₃	C₃×#577	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	C₂×C₉×#41	1	2	yes	—
300	C₃	C₃×#3126	1	1	yes	characteristic
301	C₂	—	4	1	no	—
302	C₂	—	8	2	no	—
303	C₂	C₉×#525	1	1	yes	characteristic
304	C₂	C₉×#577	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×#540	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#519	C₂	semidirect
3. C₉×#482	C₂	semidirect
4. C₉×#479	C₂	semidirect
5. C₉×#460	C₂	semidirect
6. C₉×#448	C₂	semidirect
7. C₉×#440	C₂	semidirect
8. C₄×C₉×SD₁₆	C₂	semidirect
9. C₉×#433	C₂	semidirect
73. #3126	C₉	direct (N × H)
121. #519	C₂×C₉	semidirect
122. #482	C₂×C₉	semidirect
123. #479	C₂×C₉	semidirect
124. #460	C₂×C₉	semidirect
125. #448	C₂×C₉	semidirect
126. #440	C₂×C₉	semidirect
127. C₄×SD₁₆	C₂×C₉	semidirect
128. #433	C₂×C₉	semidirect
261. C₉	#3126	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×#540	2
C₉×#519	2
C₉×#482	2
C₉×#479	2
C₉×#460	2
C₉×#448	2
C₉×#440	2
C₄×C₉×SD₁₆	2
C₉×#433	2
C₃×#3126	3

Subgroups — C9×#3126 #11767847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3126 #11767847