Subgroups — C₉×#1915 #11766636

Order 1152.

462 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#568	C₂	1	1	yes	maximal, characteristic
2	C₉×#536	C₂	1	1	yes	maximal, characteristic
3	C₉×#413	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×#101	C₂	1	2	yes	maximal, complemented
5	C₉×#344	C₂	1	2	yes	maximal, complemented
6	C₃×#1915	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₉×#131	—	2	4	no	—
10	C₂×C₉×M₁₆	—	2	2	no	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂×C₉×M₁₆	C₂²	1	2	yes	—
13	C₉×C₄⋊Q₈	—	2	1	no	—
14	C₉×#124	—	2	1	no	—
15	C₉×#123	C₄	1	4	yes	—
16	C₉×C₂²⋊Q₈	—	2	2	no	—
17	C₉×#117	C₂²	1	1	yes	characteristic
18	C₉×#101	—	2	1	no	—
19	C₉×#101	—	2	4	no	—
20	C₉×#100	—	2	1	no	—
21	C₃×#568	C₂×C₃	1	1	yes	characteristic
22	C₃×#536	C₂×C₃	1	1	yes	characteristic
23	C₃×#413	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#101	C₂×C₃	1	2	yes	—
25	C₃×#344	C₂×C₃	1	2	yes	—
26	C₉×#41	—	2	2	no	—
27	C₉×#41	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	1	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×Q₈×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
33	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₉	D₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×M₁₆	—	2	4	no	—
39	C₉×M₁₆	—	2	2	no	—
40	C₉×M₁₆	—	2	2	no	—
41	C₉×M₁₆	—	2	2	no	—
42	C₉×M₁₆	D₄	1	4	yes	—
43	C₂×C₈×C₉	—	2	4	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	—	2	2	no	—
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₉×C₂²⋊C₄	—	2	4	no	—
49	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
50	C₄²×C₉	—	2	1	no	—
51	#1915	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₃×#131	—	2	4	no	—
55	C₂×C₃×M₁₆	—	2	2	no	—
56	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
57	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×#124	—	2	1	no	—
60	C₃×#123	C₃×C₄	1	4	yes	—
61	C₃×C₂²⋊Q₈	—	2	2	no	—
62	C₃×#117	C₂²×C₃	1	1	yes	characteristic
63	C₃×#101	—	2	1	no	—
64	C₃×#101	—	2	4	no	—
65	C₃×#100	—	2	1	no	—
66	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
68	Q₈×C₉	—	2	2	no	—
69	Q₈×C₉	—	4	1	no	—
70	Q₈×C₉	—	4	2	no	—
71	Q₈×C₉	—	4	1	no	—
72	D₄×C₉	—	2	4	no	—
73	D₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	#41	1	2	yes	—
84	C₈×C₉	—	2	4	no	—
85	C₈×C₉	—	4	2	no	—
86	#568	C₂×C₉	1	1	yes	characteristic
87	#536	C₂×C₉	1	1	yes	characteristic
88	#413	C₂×C₉	1	1	yes	characteristic
89	C₂×#101	C₂×C₉	1	2	yes	complemented
90	#344	C₂×C₉	1	2	yes	complemented
91	C₃×#41	—	2	2	no	—
92	C₃×#41	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	2	no	—
94	C₂×C₃×Q₈	—	2	1	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×Q₈	—	2	2	no	—
97	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
98	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
101	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₃×M₁₆	—	2	4	no	—
104	C₃×M₁₆	—	2	2	no	—
105	C₃×M₁₆	—	2	2	no	—
106	C₃×M₁₆	—	2	2	no	—
107	C₃×M₁₆	C₃×D₄	1	4	yes	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₂²⋊C₄	—	2	4	no	—
114	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
115	C₃×C₄²	—	2	1	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	4	1	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₉	C₄×D₄	1	2	yes	—
121	C₄×C₉	—	2	2	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	—	4	2	no	—
124	C₄×C₉	#120	1	1	yes	characteristic
125	C₄×C₉	#121	1	2	yes	—
126	C₄×C₉	#123	1	1	yes	characteristic
127	C₂×#41	C₂²×C₉	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
129	#131	—	2	4	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	C₂²×C₉	1	2	yes	—
132	C₂×M₁₆	C₂²×C₉	1	2	yes	—
133	C₄⋊Q₈	—	2	1	no	—
134	#124	—	2	1	no	—
135	#123	C₄×C₉	1	4	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	#117	C₂²×C₉	1	1	yes	characteristic
138	#101	—	2	1	no	—
139	#101	—	2	4	no	—
140	#100	—	2	1	no	—
141	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
142	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₃×Q₈	—	2	2	no	—
144	C₃×Q₈	—	4	1	no	—
145	C₃×Q₈	—	4	2	no	—
146	C₃×Q₈	—	4	1	no	—
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₃×#41	1	2	yes	—
159	C₃×C₈	—	2	4	no	—
160	C₃×C₈	—	4	2	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	4	2	no	—
164	C₂×C₉	#386	1	1	yes	characteristic
165	#41	—	2	2	no	—
166	#41	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
172	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
173	C₂×D₄	D₄×C₉	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
175	C₂²×C₄	D₄×C₉	1	2	yes	—
176	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
177	M₁₆	—	2	4	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	D₄×C₉	1	4	yes	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
189	C₄²	—	2	1	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	C₃×#120	1	1	yes	characteristic
199	C₃×C₄	C₃×#121	1	2	yes	—
200	C₃×C₄	C₃×#123	1	1	yes	characteristic
201	C₉	#1915	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
203	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
204	Q₈	—	2	2	no	—
205	Q₈	—	4	1	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	4	no	—
209	D₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
219	C₂×C₄	C₉×#41	1	2	yes	—
220	C₈	—	2	4	no	—
221	C₈	—	4	2	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	—	4	2	no	—
225	C₂×C₃	C₃×#386	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²	C₄×D₄×C₉	1	2	yes	—
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₉×#120	1	1	yes	characteristic
235	C₄	C₉×#121	1	2	yes	—
236	C₄	C₉×#123	1	1	yes	characteristic
237	C₃	C₃×#1915	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	2	no	—
240	C₂	—	4	2	no	—
241	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₉×#101	C₂	semidirect
5. C₉×#344	C₂	semidirect
51. #1915	C₉	direct (N × H)
89. C₂×#101	C₂×C₉	semidirect
90. #344	C₂×C₉	semidirect
201. C₉	#1915	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×#568	2
C₉×#536	2
C₉×#413	2
C₂×C₉×#101	2
C₉×#344	2
C₃×#1915	3

Subgroups — C9×#1915 #11766636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1915 #11766636