Subgroups — C₉×#2985 #11767706

Order 1152.

483 conjugacy classes of subgroups.

The subgroup lattice has 246 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	C₉×#446	C₂	1	2	yes	maximal, complemented
3	C₉×#446	C₂	1	4	yes	maximal, complemented
4	C₈×Q₈×C₉	C₂	1	2	yes	maximal, complemented
5	C₉×#433	C₂	1	4	yes	maximal
6	C₉×#432	C₂	1	1	yes	maximal, characteristic
7	C₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₃×#2985	C₃	1	1	yes	maximal, characteristic
9	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₉×#125	C₄	1	2	yes	—
12	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₄×Q₈×C₉	C₂²	1	2	yes	—
15	C₄×Q₈×C₉	C₂²	1	2	yes	—
16	C₉×#117	C₂²	1	2	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₉×C₄⋊C₈	C₂²	1	2	yes	—
20	C₉×C₄⋊C₈	C₂²	1	4	yes	—
21	C₉×C₄⋊C₈	C₂²	1	4	yes	—
22	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₉×#97	C₂²	1	2	yes	—
25	C₉×#97	C₂²	1	4	yes	—
26	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₉	C₂²	1	4	yes	—
29	C₃×#533	C₂×C₃	1	1	yes	characteristic
30	C₃×#446	C₂×C₃	1	2	yes	—
31	C₃×#446	C₂×C₃	1	4	yes	—
32	C₃×C₈×Q₈	C₂×C₃	1	2	yes	—
33	C₃×#433	C₂×C₃	1	4	yes	—
34	C₃×#432	C₂×C₃	1	1	yes	characteristic
35	C₃×#431	C₂×C₃	1	1	yes	characteristic
36	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₉	C₂³	1	2	yes	—
42	C₂×C₈×C₉	C₂³	1	2	yes	—
43	C₂×C₈×C₉	C₂³	1	4	yes	—
44	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
51	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₉	C₂³	1	1	yes	characteristic
54	C₄²×C₉	C₂³	1	1	yes	characteristic
55	C₄²×C₉	C₂³	1	4	yes	—
56	#2985	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
59	C₃×#125	C₃×C₄	1	2	yes	—
60	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
61	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
62	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
63	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
64	C₃×#117	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
67	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
68	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
69	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
70	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
71	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
72	C₃×#97	C₂²×C₃	1	2	yes	—
73	C₃×#97	C₂²×C₃	1	4	yes	—
74	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
77	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
78	Q₈×C₉	—	2	4	no	—
79	Q₈×C₉	—	4	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
91	C₈×C₉	—	2	2	no	—
92	C₈×C₉	—	2	2	no	—
93	C₈×C₉	—	2	4	no	—
94	#533	C₂×C₉	1	1	yes	characteristic
95	#446	C₂×C₉	1	2	yes	complemented
96	#446	C₂×C₉	1	4	yes	complemented
97	C₈×Q₈	C₂×C₉	1	2	yes	complemented
98	#433	C₂×C₉	1	4	yes	—
99	#432	C₂×C₉	1	1	yes	characteristic
100	#431	C₂×C₉	1	1	yes	characteristic
101	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
103	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
107	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
108	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
109	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
110	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
111	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
112	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
113	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
114	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
115	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
116	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
117	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
119	C₃×C₄²	C₂³×C₃	1	4	yes	—
120	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
121	C₂²×C₉	—	2	1	no	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	—	2	1	no	—
124	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₉	—	2	1	no	—
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	4	no	—
128	C₄×C₉	—	2	1	no	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	#131	1	2	yes	—
132	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
133	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
134	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
135	#125	C₄×C₉	1	2	yes	—
136	C₂²⋊Q₈	C₄×C₉	1	2	yes	—
137	C₂²⋊Q₈	C₄×C₉	1	2	yes	—
138	C₄×Q₈	C₂²×C₉	1	2	yes	—
139	C₄×Q₈	C₂²×C₉	1	2	yes	—
140	#117	C₂²×C₉	1	2	yes	—
141	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₉	1	2	yes	—
143	C₄⋊C₈	C₂²×C₉	1	2	yes	—
144	C₄⋊C₈	C₂²×C₉	1	4	yes	—
145	C₄⋊C₈	C₂²×C₉	1	4	yes	—
146	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
147	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
148	#97	C₂²×C₉	1	2	yes	—
149	#97	C₂²×C₉	1	4	yes	—
150	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₉	1	4	yes	—
153	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₃×Q₈	—	2	4	no	—
155	C₃×Q₈	—	4	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	4	1	no	—
159	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
164	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
165	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
167	C₃×C₈	—	2	2	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	2	4	no	—
170	C₂×C₉	—	4	1	no	—
171	C₂×C₉	#519	1	1	yes	characteristic
172	C₂×C₉	C₂×#131	1	1	yes	characteristic
173	C₂×C₉	#568	1	1	yes	characteristic
174	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
175	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
176	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₉	1	2	yes	—
180	C₂×C₈	C₂³×C₉	1	2	yes	—
181	C₂×C₈	C₂³×C₉	1	4	yes	—
182	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
183	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
184	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
185	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
186	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
187	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
189	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
190	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
191	C₄²	C₂³×C₉	1	1	yes	characteristic
192	C₄²	C₂³×C₉	1	4	yes	—
193	C₄²	C₂³×C₉	1	1	yes	characteristic
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	C₃×#131	1	2	yes	—
205	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
206	C₉	#2985	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
218	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
219	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	4	no	—
224	C₂×C₃	—	4	1	no	—
225	C₂×C₃	C₃×#519	1	1	yes	characteristic
226	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
227	C₂×C₃	C₃×#568	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	2	1	no	—
238	C₄	C₉×#131	1	2	yes	—
239	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
240	C₃	C₃×#2985	1	1	yes	characteristic
241	C₂	—	4	1	no	—
242	C₂	C₉×#519	1	1	yes	characteristic
243	C₂	C₂×C₉×#131	1	1	yes	characteristic
244	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
3. C₉×#446	C₂	semidirect
4. C₈×Q₈×C₉	C₂	semidirect
56. #2985	C₉	direct (N × H)
95. #446	C₂×C₉	semidirect
96. #446	C₂×C₉	semidirect
97. C₈×Q₈	C₂×C₉	semidirect
206. C₉	#2985	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×#533	2
C₉×#446	2
C₈×Q₈×C₉	2
C₉×#433	2
C₉×#432	2
C₉×#431	2
C₃×#2985	3

Subgroups — C9×#2985 #11767706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2985 #11767706