Subgroups — C₉×#1912 #11766633

Order 1152.

471 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×#536	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#421	C₂	1	2	yes	maximal
4	C₂×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₃×#1912	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	C₂×C₉×M₁₆	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	C₄	1	2	yes	complemented
11	C₉×#124	C₄	1	2	yes	complemented
12	C₉×#123	—	2	2	no	—
13	C₉×C₂²⋊Q₈	—	2	2	no	—
14	C₄×Q₈×C₉	—	2	4	no	—
15	C₉×#117	—	2	2	no	—
16	C₉×#117	C₂²	1	1	yes	characteristic
17	C₉×#117	C₂²	1	2	yes	—
18	C₉×#104	—	2	4	no	—
19	C₉×#101	—	2	2	no	—
20	C₃×#536	C₂×C₃	1	1	yes	characteristic
21	C₃×#519	C₂×C₃	1	1	yes	characteristic
22	C₃×#421	C₂×C₃	1	2	yes	—
23	C₂×C₃×#101	C₂×C₃	1	1	yes	characteristic
24	C₃×H(3,ℤ/4)	C₂×C₃	1	2	yes	—
25	C₉×#41	—	2	2	no	—
26	C₉×#41	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₉	D₄	1	2	yes	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	D₄	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₉×M₁₆	—	2	2	no	—
38	C₉×M₁₆	—	2	2	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₄⋊C₄	—	2	4	no	—
42	C₉×C₄⋊C₄	—	2	2	no	—
43	C₉×C₄⋊C₄	—	2	1	no	—
44	C₉×C₂²⋊C₄	—	2	2	no	—
45	C₉×C₂²⋊C₄	—	2	1	no	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	C₄²×C₉	—	2	4	no	—
48	C₄²×C₉	C₂×C₄	1	2	yes	—
49	C₄²×C₉	D₄	1	4	yes	—
50	#1912	C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×C₄⋊Q₈	C₃×C₄	1	2	yes	—
55	C₃×#124	C₃×C₄	1	2	yes	—
56	C₃×#123	—	2	2	no	—
57	C₃×C₂²⋊Q₈	—	2	2	no	—
58	C₃×C₄×Q₈	—	2	4	no	—
59	C₃×#117	—	2	2	no	—
60	C₃×#117	C₂²×C₃	1	1	yes	characteristic
61	C₃×#117	C₂²×C₃	1	2	yes	—
62	C₃×#104	—	2	4	no	—
63	C₃×#101	—	2	2	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₉	—	2	4	no	—
67	Q₈×C₉	—	2	4	no	—
68	Q₈×C₉	—	2	2	no	—
69	D₄×C₉	—	2	2	no	—
70	D₄×C₉	—	4	1	no	—
71	D₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	#41	1	2	yes	—
83	C₈×C₉	—	4	2	no	—
84	#536	C₂×C₉	1	1	yes	characteristic
85	#519	C₂×C₉	1	1	yes	characteristic, complemented
86	#421	C₂×C₉	1	2	yes	—
87	C₂×#101	C₂×C₉	1	1	yes	characteristic, complemented
88	H(3,ℤ/4)	C₂×C₉	1	2	yes	complemented
89	C₃×#41	—	2	2	no	—
90	C₃×#41	—	2	2	no	—
91	C₂×C₃×Q₈	—	2	2	no	—
92	C₂×C₃×Q₈	—	2	2	no	—
93	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
96	C₂×C₃×D₄	—	2	1	no	—
97	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₃×M₁₆	—	2	2	no	—
102	C₃×M₁₆	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	1	no	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	—	2	1	no	—
110	C₃×C₂²⋊C₄	—	2	4	no	—
111	C₃×C₄²	—	2	4	no	—
112	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
113	C₃×C₄²	C₃×D₄	1	4	yes	—
114	C₂²×C₉	—	4	1	no	—
115	C₂²×C₉	—	4	2	no	—
116	C₂²×C₉	—	4	1	no	—
117	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₉	#120	1	1	yes	characteristic
119	C₂²×C₉	#123	1	1	yes	characteristic
120	C₄×C₉	—	2	4	no	—
121	C₄×C₉	—	4	4	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	—	4	1	no	—
124	C₄×C₉	C₄×D₄	1	2	yes	—
125	C₄×C₉	#121	1	2	yes	—
126	C₂×#41	C₂²×C₉	1	1	yes	characteristic
127	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
128	C₂×M₁₆	C₂²×C₉	1	2	yes	—
129	C₄⋊Q₈	C₄×C₉	1	2	yes	complemented
130	#124	C₄×C₉	1	2	yes	complemented
131	#123	—	2	2	no	—
132	C₂²⋊Q₈	—	2	2	no	—
133	C₄×Q₈	—	2	4	no	—
134	#117	—	2	2	no	—
135	#117	C₂²×C₉	1	1	yes	characteristic
136	#117	C₂²×C₉	1	2	yes	—
137	#104	—	2	4	no	—
138	#101	—	2	2	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
141	C₃×Q₈	—	2	4	no	—
142	C₃×Q₈	—	2	4	no	—
143	C₃×Q₈	—	2	2	no	—
144	C₃×D₄	—	2	2	no	—
145	C₃×D₄	—	4	1	no	—
146	C₃×D₄	—	4	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
154	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
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	C₃×#41	1	2	yes	—
158	C₃×C₈	—	4	2	no	—
159	C₂×C₉	—	2	1	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	8	1	no	—
163	C₂×C₉	#386	1	1	yes	characteristic
164	#41	—	2	2	no	—
165	#41	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
169	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
170	C₂×Q₈	D₄×C₉	1	2	yes	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	D₄×C₉	1	2	yes	—
175	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	C₂×C₄×C₉	1	2	yes	—
188	C₄²	D₄×C₉	1	4	yes	—
189	C₂²×C₃	—	4	1	no	—
190	C₂²×C₃	—	4	2	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	C₃×#120	1	1	yes	characteristic
194	C₂²×C₃	C₃×#123	1	1	yes	characteristic
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	4	4	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
200	C₃×C₄	C₃×#121	1	2	yes	—
201	C₉	#1912	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	—	2	1	no	—
203	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	1	no	—
209	D₄	—	4	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₉×C₂²⋊C₄	1	4	yes	—
217	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
220	C₂×C₄	C₉×#41	1	2	yes	—
221	C₈	—	4	2	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	8	1	no	—
226	C₂×C₃	C₃×#386	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²	C₉×#120	1	1	yes	characteristic
232	C₂²	C₉×#123	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	4	4	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	C₄×D₄×C₉	1	2	yes	—
238	C₄	C₉×#121	1	2	yes	—
239	C₃	C₃×#1912	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	2	1	no	—
243	C₂	—	8	1	no	—
244	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ 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
2. C₉×#519	C₂	semidirect
4. C₂×C₉×#101	C₂	semidirect
5. C₉×H(3,ℤ/4)	C₂	semidirect
10. C₉×C₄⋊Q₈	C₄	semidirect
11. C₉×#124	C₄	semidirect
50. #1912	C₉	direct (N × H)
85. #519	C₂×C₉	semidirect
87. C₂×#101	C₂×C₉	semidirect
88. H(3,ℤ/4)	C₂×C₉	semidirect
129. C₄⋊Q₈	C₄×C₉	semidirect
130. #124	C₄×C₉	semidirect
201. C₉	#1912	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×#536	2
C₉×#519	2
C₉×#421	2
C₂×C₉×#101	2
C₉×H(3,ℤ/4)	2
C₃×#1912	3

Subgroups — C9×#1912 #11766633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1912 #11766633