Subgroups — C₉×#1894 #11766615

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₉×#380	C₂	1	1	yes	maximal, characteristic
5	C₉×#340	C₂	1	2	yes	maximal, complemented
6	C₃×#1894	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×M₁₆	—	2	2	no	—
10	C₂×C₉×M₁₆	—	2	4	no	—
11	C₂²×C₈×C₉	C₂²	1	2	yes	—
12	C₉×C₂²⋊Q₈	—	2	2	no	—
13	C₉×C₂²⋊Q₈	C₄	1	4	yes	complemented
14	C₂×C₉×C₄⋊C₄	—	2	1	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
18	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
19	C₉×#103	—	2	4	no	—
20	C₉×#95	—	2	1	no	—
21	C₂²×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#103	C₂×C₃	1	2	yes	—
24	C₃×#380	C₂×C₃	1	1	yes	characteristic
25	C₃×#340	C₂×C₃	1	2	yes	—
26	C₂⁴×C₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	—	2	2	no	—
36	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	D₄	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×M₁₆	—	2	8	no	—
41	C₂×C₈×C₉	—	2	2	no	—
42	C₂×C₈×C₉	—	2	2	no	—
43	C₂×C₈×C₉	D₄	1	4	yes	—
44	C₉×C₄⋊C₄	—	2	4	no	—
45	C₉×C₄⋊C₄	—	2	1	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₉×C₂²⋊C₄	—	2	4	no	—
50	C₉×C₂²⋊C₄	—	4	2	no	—
51	#1894	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×M₁₆	—	2	2	no	—
55	C₂×C₃×M₁₆	—	2	4	no	—
56	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
57	C₃×C₂²⋊Q₈	—	2	2	no	—
58	C₃×C₂²⋊Q₈	C₃×C₄	1	4	yes	—
59	C₂×C₃×C₄⋊C₄	—	2	1	no	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
63	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
64	C₃×#103	—	2	4	no	—
65	C₃×#95	—	2	1	no	—
66	C₂³×C₉	—	2	2	no	—
67	C₂³×C₉	—	2	1	no	—
68	C₂³×C₉	—	2	1	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₉	—	2	4	no	—
73	Q₈×C₉	—	4	1	no	—
74	Q₈×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	4	2	no	—
82	C₂×C₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	—	4	2	no	—
85	C₂×C₄×C₉	—	4	2	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	#41	1	2	yes	—
92	C₈×C₉	—	2	4	no	—
93	C₂²×M₁₆	C₂×C₉	1	1	yes	characteristic
94	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
95	C₂×#103	C₂×C₉	1	2	yes	complemented
96	#380	C₂×C₉	1	1	yes	characteristic
97	#340	C₂×C₉	1	2	yes	complemented
98	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
99	C₂×C₃×Q₈	—	2	1	no	—
100	C₂×C₃×Q₈	—	2	2	no	—
101	C₂×C₃×Q₈	—	2	2	no	—
102	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₃×M₁₆	—	2	8	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
116	C₃×C₄⋊C₄	—	2	4	no	—
117	C₃×C₄⋊C₄	—	2	1	no	—
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
121	C₃×C₂²⋊C₄	—	2	4	no	—
122	C₃×C₂²⋊C₄	—	4	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	4	2	no	—
127	C₂²×C₉	—	4	2	no	—
128	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₉	#121	1	2	yes	—
130	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂²×C₉	#137	1	1	yes	characteristic
132	C₂²×C₉	#137	1	2	yes	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	4	2	no	—
135	C₄×C₉	—	4	2	no	—
136	C₄×C₉	—	8	2	no	—
137	C₄×C₉	C₄×D₄	1	2	yes	—
138	C₄×C₉	#120	1	1	yes	characteristic
139	C₄×C₉	#123	1	1	yes	characteristic
140	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂²×C₈	C₂²×C₉	1	2	yes	—
145	C₂²⋊Q₈	—	2	2	no	—
146	C₂²⋊Q₈	C₄×C₉	1	4	yes	complemented
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₂²⋊C₄	—	2	1	no	—
152	#103	—	2	4	no	—
153	#95	—	2	1	no	—
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	—	2	1	no	—
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	—	2	2	no	—
158	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
159	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	—	4	1	no	—
162	C₃×Q₈	—	4	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	—	4	1	no	—
171	C₂×C₃×C₄	—	4	1	no	—
172	C₂×C₃×C₄	—	4	2	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₃×#41	1	2	yes	—
180	C₃×C₈	—	2	4	no	—
181	C₂×C₉	—	4	2	no	—
182	C₂×C₉	#386	1	1	yes	characteristic
183	C₂×C₉	#417	1	1	yes	characteristic
184	C₂×C₉	#419	1	1	yes	characteristic
185	C₂×C₉	#468	1	2	yes	—
186	C₂×C₉	#470	1	2	yes	—
187	C₂⁴	D₄×C₉	1	1	yes	characteristic
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
198	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
199	C₂²×C₄	D₄×C₉	1	2	yes	—
200	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
201	M₁₆	—	2	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	D₄×C₉	1	4	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	—	4	2	no	—
217	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₃	C₃×#121	1	2	yes	—
219	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂²×C₃	C₃×#137	1	1	yes	characteristic
221	C₂²×C₃	C₃×#137	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	—	4	2	no	—
225	C₃×C₄	—	8	2	no	—
226	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
227	C₃×C₄	C₃×#120	1	1	yes	characteristic
228	C₃×C₄	C₃×#123	1	1	yes	characteristic
229	C₉	#1894	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
235	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	1	no	—
238	Q₈	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
255	C₂×C₄	C₉×#41	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₂×C₃	—	4	2	no	—
258	C₂×C₃	C₃×#386	1	1	yes	characteristic
259	C₂×C₃	C₃×#417	1	1	yes	characteristic
260	C₂×C₃	C₃×#419	1	1	yes	characteristic
261	C₂×C₃	C₃×#468	1	2	yes	—
262	C₂×C₃	C₃×#470	1	2	yes	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²	C₉×#121	1	2	yes	—
270	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₂²	C₉×#137	1	1	yes	characteristic
272	C₂²	C₉×#137	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	2	no	—
277	C₄	C₄×D₄×C₉	1	2	yes	—
278	C₄	C₉×#120	1	1	yes	characteristic
279	C₄	C₉×#123	1	1	yes	characteristic
280	C₃	C₃×#1894	1	1	yes	characteristic
281	C₂	—	4	2	no	—
282	C₂	C₉×#386	1	1	yes	characteristic
283	C₂	C₉×#417	1	1	yes	characteristic
284	C₂	C₉×#419	1	1	yes	characteristic
285	C₂	C₉×#468	1	2	yes	—
286	C₂	C₉×#470	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×#103	C₂	semidirect
5. C₉×#340	C₂	semidirect
13. C₉×C₂²⋊Q₈	C₄	semidirect
51. #1894	C₉	direct (N × H)
95. C₂×#103	C₂×C₉	semidirect
97. #340	C₂×C₉	semidirect
146. C₂²⋊Q₈	C₄×C₉	semidirect
229. C₉	#1894	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂×C₉×C₂²⋊Q₈	2
C₂×C₉×#103	2
C₉×#380	2
C₉×#340	2
C₃×#1894	3

Subgroups — C9×#1894 #11766615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1894 #11766615