Subgroups — C₉×#2014 #11766735

Order 1152.

711 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
4	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2014	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₉	—	2	2	no	—
9	C₂²×D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₉×SD₁₆	—	2	4	no	—
11	C₂×C₉×SD₁₆	—	2	4	no	—
12	C₂×C₉×SD₁₆	—	2	4	no	—
13	C₂²×C₈×C₉	C₂²	1	2	yes	—
14	C₉×#127	—	2	4	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×#102	—	2	4	no	—
18	C₉×#95	—	2	2	no	—
19	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
20	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#102	C₂×C₃	1	2	yes	—
22	C₃×#391	C₂×C₃	1	1	yes	characteristic
23	C₃×#336	C₂×C₃	1	1	yes	characteristic
24	C₂⁴×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	1	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	D₄	1	2	yes	complemented
29	C₂×D₄×C₉	—	2	8	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	2	2	no	—
33	C₂×D₄×C₉	—	2	2	no	—
34	C₂×D₄×C₉	—	2	2	no	—
35	C₂×D₄×C₉	—	4	4	no	—
36	C₂×D₄×C₉	—	4	4	no	—
37	C₂×D₄×C₉	D₄	1	4	yes	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	D₄	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₉×SD₁₆	—	2	16	no	—
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	D₄	1	4	yes	complemented
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₄²×C₉	—	2	2	no	—
48	#2014	C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃×D₄	—	2	2	no	—
51	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
52	C₂×C₃×SD₁₆	—	2	4	no	—
53	C₂×C₃×SD₁₆	—	2	4	no	—
54	C₂×C₃×SD₁₆	—	2	4	no	—
55	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
56	C₃×#127	—	2	4	no	—
57	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₃×#102	—	2	4	no	—
60	C₃×#95	—	2	2	no	—
61	C₂³×C₉	—	2	4	no	—
62	C₂³×C₉	—	4	2	no	—
63	C₂³×C₉	—	4	2	no	—
64	C₂³×C₉	—	4	4	no	—
65	C₂³×C₉	—	4	4	no	—
66	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₉	—	2	4	no	—
68	Q₈×C₉	—	4	2	no	—
69	D₄×C₉	—	2	4	no	—
70	D₄×C₉	—	2	4	no	—
71	D₄×C₉	—	4	16	no	—
72	D₄×C₉	—	4	2	no	—
73	D₄×C₉	—	4	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	—	4	1	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	2	no	—
81	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	#41	1	1	yes	characteristic
86	C₈×C₉	—	2	4	no	—
87	C₂²×SD₁₆	C₂×C₉	1	2	yes	complemented
88	C₂×#127	C₂×C₉	1	1	yes	characteristic
89	C₂×#102	C₂×C₉	1	2	yes	complemented
90	#391	C₂×C₉	1	1	yes	characteristic, complemented
91	#336	C₂×C₉	1	1	yes	characteristic, complemented
92	C₂⁴×C₃	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	1	no	—
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	2	no	—
96	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
97	C₂×C₃×D₄	—	2	8	no	—
98	C₂×C₃×D₄	—	2	2	no	—
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	4	4	no	—
104	C₂×C₃×D₄	—	4	4	no	—
105	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₃×SD₁₆	—	2	16	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄²	—	2	2	no	—
116	C₂²×C₉	—	4	4	no	—
117	C₂²×C₉	—	4	8	no	—
118	C₂²×C₉	—	8	4	no	—
119	C₂²×C₉	—	8	2	no	—
120	C₂²×C₉	—	8	2	no	—
121	C₂²×C₉	#120	1	1	yes	characteristic
122	C₂²×C₉	#121	1	1	yes	characteristic
123	C₂²×C₉	#127	1	1	yes	characteristic
124	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
125	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₉	—	2	4	no	—
127	C₄×C₉	—	4	2	no	—
128	C₄×C₉	—	8	1	no	—
129	C₄×C₉	#120	1	2	yes	—
130	C₄×C₉	#121	1	2	yes	—
131	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
132	C₂²×D₄	—	2	2	no	—
133	C₂²×D₄	C₂²×C₉	1	2	yes	—
134	C₂×SD₁₆	—	2	4	no	—
135	C₂×SD₁₆	—	2	4	no	—
136	C₂×SD₁₆	—	2	4	no	—
137	C₂²×C₈	C₂²×C₉	1	2	yes	—
138	#127	—	2	4	no	—
139	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
141	#102	—	2	4	no	—
142	#95	—	2	2	no	—
143	C₂³×C₃	—	2	4	no	—
144	C₂³×C₃	—	4	2	no	—
145	C₂³×C₃	—	4	2	no	—
146	C₂³×C₃	—	4	4	no	—
147	C₂³×C₃	—	4	4	no	—
148	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
149	C₃×Q₈	—	2	4	no	—
150	C₃×Q₈	—	4	2	no	—
151	C₃×D₄	—	2	4	no	—
152	C₃×D₄	—	2	4	no	—
153	C₃×D₄	—	4	16	no	—
154	C₃×D₄	—	4	2	no	—
155	C₃×D₄	—	4	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
164	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
168	C₃×C₈	—	2	4	no	—
169	C₂×C₉	—	8	4	no	—
170	C₂×C₉	#392	1	1	yes	characteristic
171	C₂×C₉	#450	1	2	yes	—
172	C₂×C₉	#460	1	2	yes	—
173	C₂×C₉	#492	1	1	yes	characteristic
174	C₂×C₉	#496	1	1	yes	characteristic
175	C₂⁴	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	D₄×C₉	1	2	yes	complemented
180	C₂×D₄	—	2	8	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	D₄×C₉	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	D₄×C₉	1	2	yes	—
192	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
193	SD₁₆	—	2	16	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	D₄×C₉	1	4	yes	complemented
197	C₄⋊C₄	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₂²×C₃	—	4	4	no	—
200	C₂²×C₃	—	4	8	no	—
201	C₂²×C₃	—	8	4	no	—
202	C₂²×C₃	—	8	2	no	—
203	C₂²×C₃	—	8	2	no	—
204	C₂²×C₃	C₃×#120	1	1	yes	characteristic
205	C₂²×C₃	C₃×#121	1	1	yes	characteristic
206	C₂²×C₃	C₃×#127	1	1	yes	characteristic
207	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
208	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
209	C₃×C₄	—	2	4	no	—
210	C₃×C₄	—	4	2	no	—
211	C₃×C₄	—	8	1	no	—
212	C₃×C₄	C₃×#120	1	2	yes	—
213	C₃×C₄	C₃×#121	1	2	yes	—
214	C₉	#2014	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	4	no	—
225	D₄	—	4	16	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₉×SD₁₆	1	4	yes	complemented
236	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
239	C₂×C₄	C₉×#41	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₂×C₃	—	8	4	no	—
242	C₂×C₃	C₃×#392	1	1	yes	characteristic
243	C₂×C₃	C₃×#450	1	2	yes	—
244	C₂×C₃	C₃×#460	1	2	yes	—
245	C₂×C₃	C₃×#492	1	1	yes	characteristic
246	C₂×C₃	C₃×#496	1	1	yes	characteristic
247	C₂²	—	4	4	no	—
248	C₂²	—	4	8	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₉×#120	1	1	yes	characteristic
253	C₂²	C₉×#121	1	1	yes	characteristic
254	C₂²	C₉×#127	1	1	yes	characteristic
255	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
256	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	4	2	no	—
259	C₄	—	8	1	no	—
260	C₄	C₉×#120	1	2	yes	—
261	C₄	C₉×#121	1	2	yes	—
262	C₃	C₃×#2014	1	1	yes	characteristic
263	C₂	—	8	4	no	—
264	C₂	C₉×#392	1	1	yes	characteristic
265	C₂	C₉×#450	1	2	yes	—
266	C₂	C₉×#460	1	2	yes	—
267	C₂	C₉×#492	1	1	yes	characteristic
268	C₂	C₉×#496	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#127	C₂×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×SD₁₆	C₂	semidirect
3. C₂×C₉×#102	C₂	semidirect
4. C₉×#391	C₂	semidirect
5. C₉×#336	C₂	semidirect
28. C₂×Q₈×C₉	D₄	semidirect
45. C₂×C₈×C₉	D₄	semidirect
48. #2014	C₉	direct (N × H)
81. C₂×C₄×C₉	SD₁₆	semidirect
87. C₂²×SD₁₆	C₂×C₉	semidirect
89. C₂×#102	C₂×C₉	semidirect
90. #391	C₂×C₉	semidirect
91. #336	C₂×C₉	semidirect
179. C₂×Q₈	D₄×C₉	semidirect
196. C₂×C₈	D₄×C₉	semidirect
214. C₉	#2014	direct (N × H)
235. C₂×C₄	C₉×SD₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₉×SD₁₆	2
C₂×C₉×#127	2
C₂×C₉×#102	2
C₉×#391	2
C₉×#336	2
C₃×#2014	3

Subgroups — C9×#2014 #11766735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2014 #11766735