Subgroups — C₉×#2586 #11767307

Order 1152.

618 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#398	C₂	1	1	yes	maximal, characteristic
2	C₉×#395	C₂	1	2	yes	maximal
3	C₉×#395	C₂	1	1	yes	maximal, characteristic
4	C₉×#391	C₂	1	2	yes	maximal
5	C₉×#391	C₂	1	1	yes	maximal, characteristic
6	C₉×#389	C₂	1	2	yes	maximal
7	C₉×#387	C₂	1	2	yes	maximal
8	C₉×#387	C₂	1	2	yes	maximal
9	C₄×C₉×C₄⋊C₄	C₂	1	2	yes	maximal
10	C₃×#2586	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₉×#95	C₂²	1	2	yes	—
24	C₉×#95	C₂²	1	2	yes	—
25	C₉×#95	C₂²	1	2	yes	—
26	C₉×#95	C₂²	1	1	yes	characteristic
27	C₉×#95	C₂²	1	1	yes	characteristic
28	C₉×#95	C₂²	1	2	yes	—
29	C₉×#95	C₂²	1	2	yes	—
30	C₉×#95	C₂²	1	2	yes	—
31	C₉×#95	C₂²	1	2	yes	—
32	C₃×#398	C₂×C₃	1	1	yes	characteristic
33	C₃×#395	C₂×C₃	1	1	yes	characteristic
34	C₃×#395	C₂×C₃	1	2	yes	—
35	C₃×#391	C₂×C₃	1	2	yes	—
36	C₃×#391	C₂×C₃	1	1	yes	characteristic
37	C₃×#389	C₂×C₃	1	2	yes	—
38	C₃×#387	C₂×C₃	1	2	yes	—
39	C₃×#387	C₂×C₃	1	2	yes	—
40	C₃×C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
41	C₂×Q₈×C₉	—	2	4	no	—
42	C₂×Q₈×C₉	—	2	2	no	—
43	C₂×Q₈×C₉	—	4	1	no	—
44	C₂×Q₈×C₉	—	4	1	no	—
45	C₂×Q₈×C₉	—	4	1	no	—
46	C₂²×C₄×C₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₉	C₂³	1	2	yes	—
55	C₉×C₄⋊C₄	—	2	4	no	—
56	C₉×C₄⋊C₄	—	2	4	no	—
57	C₉×C₄⋊C₄	—	4	2	no	—
58	C₉×C₄⋊C₄	—	4	2	no	—
59	C₉×C₄⋊C₄	—	4	1	no	—
60	C₉×C₄⋊C₄	—	4	1	no	—
61	C₉×C₄⋊C₄	Q₈	1	8	yes	complemented
62	C₄²×C₉	—	2	4	no	—
63	C₄²×C₉	—	4	1	no	—
64	C₄²×C₉	—	4	2	no	—
65	#2586	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
76	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
78	C₃×#95	C₂²×C₃	1	2	yes	—
79	C₃×#95	C₂²×C₃	1	2	yes	—
80	C₃×#95	C₂²×C₃	1	1	yes	characteristic
81	C₃×#95	C₂²×C₃	1	2	yes	—
82	C₃×#95	C₂²×C₃	1	1	yes	characteristic
83	C₃×#95	C₂²×C₃	1	2	yes	—
84	C₃×#95	C₂²×C₃	1	2	yes	—
85	C₃×#95	C₂²×C₃	1	2	yes	—
86	C₃×#95	C₂²×C₃	1	2	yes	—
87	C₂³×C₉	C₂⁴	1	1	yes	characteristic
88	Q₈×C₉	—	4	4	no	—
89	Q₈×C₉	—	4	4	no	—
90	C₂×C₄×C₉	—	2	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	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	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	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
111	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
112	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
113	C₂×C₄×C₉	#41	1	4	yes	—
114	C₂×C₄×C₉	#41	1	2	yes	—
115	#398	C₂×C₉	1	1	yes	characteristic
116	#395	C₂×C₉	1	1	yes	characteristic
117	#395	C₂×C₉	1	2	yes	—
118	#391	C₂×C₉	1	2	yes	—
119	#391	C₂×C₉	1	1	yes	characteristic
120	#389	C₂×C₉	1	2	yes	—
121	#387	C₂×C₉	1	2	yes	—
122	#387	C₂×C₉	1	2	yes	—
123	C₄×C₄⋊C₄	C₂×C₉	1	2	yes	—
124	C₂×C₃×Q₈	—	2	4	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×Q₈	—	4	1	no	—
127	C₂×C₃×Q₈	—	4	1	no	—
128	C₂×C₃×Q₈	—	4	1	no	—
129	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	2	4	no	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	4	2	no	—
141	C₃×C₄⋊C₄	—	4	2	no	—
142	C₃×C₄⋊C₄	—	4	1	no	—
143	C₃×C₄⋊C₄	—	4	1	no	—
144	C₃×C₄⋊C₄	C₃×Q₈	1	8	yes	—
145	C₃×C₄²	—	2	4	no	—
146	C₃×C₄²	—	4	1	no	—
147	C₃×C₄²	—	4	2	no	—
148	C₂²×C₉	C₂²×Q₈	1	2	yes	—
149	C₂²×C₉	C₂×#41	1	1	yes	characteristic
150	C₂²×C₉	C₂×#41	1	2	yes	—
151	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₄×C₉	—	2	4	no	—
154	C₄×C₉	—	2	2	no	—
155	C₄×C₉	—	2	4	no	—
156	C₄×C₉	—	2	4	no	—
157	C₄×C₉	—	2	4	no	—
158	C₄×C₉	—	4	2	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	—	4	1	no	—
161	C₄×C₉	—	4	1	no	—
162	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
163	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
171	C₂×C₄²	C₂²×C₉	1	2	yes	—
172	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₉	1	2	yes	—
174	#95	C₂²×C₉	1	2	yes	—
175	#95	C₂²×C₉	1	2	yes	—
176	#95	C₂²×C₉	1	1	yes	characteristic
177	#95	C₂²×C₉	1	2	yes	—
178	#95	C₂²×C₉	1	1	yes	characteristic
179	#95	C₂²×C₉	1	2	yes	—
180	#95	C₂²×C₉	1	2	yes	—
181	#95	C₂²×C₉	1	2	yes	—
182	#95	C₂²×C₉	1	2	yes	—
183	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
184	C₃×Q₈	—	4	4	no	—
185	C₃×Q₈	—	4	4	no	—
186	C₂×C₃×C₄	—	2	2	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	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	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	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
208	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
209	C₂×C₃×C₄	C₃×#41	1	4	yes	—
210	C₂×C₃×C₄	C₃×#41	1	2	yes	—
211	C₂×C₉	#533	1	2	yes	—
212	C₂×C₉	#538	1	1	yes	characteristic
213	C₂×C₉	#551	1	1	yes	characteristic
214	C₂×C₉	#554	1	2	yes	—
215	C₂×C₉	Q₈²	1	1	yes	characteristic
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₉	1	2	yes	—
223	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₉	1	2	yes	—
225	C₂²×C₄	C₂³×C₉	1	2	yes	—
226	C₂²×C₄	C₂³×C₉	1	2	yes	—
227	C₂²×C₄	C₂³×C₉	1	2	yes	—
228	C₂²×C₄	C₂³×C₉	1	2	yes	—
229	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	Q₈×C₉	1	8	yes	complemented
237	C₄²	—	2	4	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
241	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
242	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
243	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	—	4	1	no	—
254	C₉	#2586	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	C₂⁴×C₉	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	4	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
279	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
280	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
281	C₂×C₄	C₉×#41	1	4	yes	—
282	C₂×C₄	C₉×#41	1	2	yes	—
283	C₂×C₃	C₃×#533	1	2	yes	—
284	C₂×C₃	C₃×#538	1	1	yes	characteristic
285	C₂×C₃	C₃×#551	1	1	yes	characteristic
286	C₂×C₃	C₃×#554	1	2	yes	—
287	C₂×C₃	C₃×Q₈²	1	1	yes	characteristic
288	C₂²	C₂²×Q₈×C₉	1	2	yes	—
289	C₂²	C₂×C₉×#41	1	1	yes	characteristic
290	C₂²	C₂×C₉×#41	1	2	yes	—
291	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₃	C₃×#2586	1	1	yes	characteristic
303	C₂	C₉×#533	1	2	yes	—
304	C₂	C₉×#538	1	1	yes	characteristic
305	C₂	C₉×#551	1	1	yes	characteristic
306	C₂	C₉×#554	1	2	yes	—
307	C₂	Q₈²×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
61. C₉×C₄⋊C₄	Q₈	semidirect
65. #2586	C₉	direct (N × H)
236. C₄⋊C₄	Q₈×C₉	semidirect
254. C₉	#2586	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×#398	2
C₉×#395	2
C₉×#391	2
C₉×#389	2
C₉×#387	2
C₄×C₉×C₄⋊C₄	2
C₃×#2586	3

Subgroups — C9×#2586 #11767307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2586 #11767307