Subgroups — C₉×#1986 #11766707

Order 1152.

390 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	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₉×#339	C₂	1	4	yes	maximal
5	C₃×#1986	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×M₁₆	C₂²	1	2	yes	—
7	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	C₉×#125	C₄	1	2	yes	complemented
10	C₉×C₂²⋊Q₈	—	2	1	no	—
11	C₉×C₂²⋊Q₈	—	2	1	no	—
12	C₄×Q₈×C₉	—	2	1	no	—
13	C₄×Q₈×C₉	—	2	1	no	—
14	C₉×#117	C₂²	1	2	yes	—
15	C₂×C₉×C₄⋊C₄	—	2	1	no	—
16	C₂×C₄²×C₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₉×#97	—	2	1	no	—
19	C₄×C₈×C₉	—	2	1	no	—
20	C₉×#95	—	2	1	no	—
21	C₃×#533	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
24	C₃×#339	C₂×C₃	1	4	yes	—
25	C₂×Q₈×C₉	—	2	1	no	—
26	C₂×Q₈×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×M₁₆	—	2	2	no	—
32	C₉×M₁₆	Q₈	1	4	yes	complemented
33	C₂×C₈×C₉	—	2	2	no	—
34	C₉×C₄⋊C₄	—	2	4	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₉×C₂²⋊C₄	—	2	2	no	—
42	C₄²×C₉	—	2	2	no	—
43	C₄²×C₉	—	2	4	no	—
44	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₉	D₄	1	1	yes	characteristic
47	C₄²×C₉	D₄	1	1	yes	characteristic
48	#1986	C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
50	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
51	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
52	C₃×#125	C₃×C₄	1	2	yes	—
53	C₃×C₂²⋊Q₈	—	2	1	no	—
54	C₃×C₂²⋊Q₈	—	2	1	no	—
55	C₃×C₄×Q₈	—	2	1	no	—
56	C₃×C₄×Q₈	—	2	1	no	—
57	C₃×#117	C₂²×C₃	1	2	yes	—
58	C₂×C₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
60	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₃×#97	—	2	1	no	—
62	C₃×C₄×C₈	—	2	1	no	—
63	C₃×#95	—	2	1	no	—
64	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₉	—	4	2	no	—
66	Q₈×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	4	2	no	—
73	C₂×C₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₉	#41	1	2	yes	—
82	C₈×C₉	—	2	4	no	—
83	#533	C₂×C₉	1	1	yes	characteristic
84	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic
85	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
86	#339	C₂×C₉	1	4	yes	—
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂×C₃×Q₈	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	C₃×Q₈	1	4	yes	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄⋊C₄	—	2	1	no	—
100	C₃×C₄⋊C₄	—	2	1	no	—
101	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
102	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
103	C₃×C₂²⋊C₄	—	2	2	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	4	no	—
106	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
107	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
108	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
109	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₉	—	2	1	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₉	#124	1	1	yes	characteristic
114	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
115	C₄×C₉	—	2	1	no	—
116	C₄×C₉	—	2	1	no	—
117	C₄×C₉	—	2	1	no	—
118	C₄×C₉	—	4	4	no	—
119	C₄×C₉	—	4	2	no	—
120	C₄×C₉	—	4	2	no	—
121	C₄×C₉	#104	1	2	yes	—
122	C₄×C₉	C₄×Q₈	1	2	yes	—
123	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
124	C₂×M₁₆	C₂²×C₉	1	2	yes	—
125	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic, complemented
126	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic, complemented
127	#125	C₄×C₉	1	2	yes	complemented
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	—	2	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	#117	C₂²×C₉	1	2	yes	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄²	C₂²×C₉	1	2	yes	—
135	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
136	#97	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	#95	—	2	1	no	—
139	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
140	C₃×Q₈	—	4	2	no	—
141	C₃×Q₈	—	4	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
156	C₂×C₃×C₄	C₃×#41	1	2	yes	—
157	C₃×C₈	—	2	4	no	—
158	C₂×C₉	—	2	1	no	—
159	C₂×C₉	—	2	1	no	—
160	C₂×C₉	#391	1	1	yes	characteristic
161	C₂×C₉	C₂×#104	1	1	yes	characteristic
162	C₂×C₉	#421	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
167	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
169	M₁₆	—	2	2	no	—
170	M₁₆	Q₈×C₉	1	4	yes	complemented
171	C₂×C₈	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
178	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	4	no	—
182	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
183	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
184	C₄²	D₄×C₉	1	1	yes	characteristic
185	C₄²	D₄×C₉	1	1	yes	characteristic
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₃×#124	1	1	yes	characteristic
190	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	4	4	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	C₃×#104	1	2	yes	—
198	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
199	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
200	C₉	#1986	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
217	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
218	C₂×C₄	C₉×#41	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	C₃×#391	1	1	yes	characteristic
223	C₂×C₃	C₂×C₃×#104	1	1	yes	characteristic
224	C₂×C₃	C₃×#421	1	1	yes	characteristic
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²	C₉×#124	1	1	yes	characteristic
229	C₂²	C₉×C₄⋊Q₈	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₉×#104	1	2	yes	—
237	C₄	C₄×Q₈×C₉	1	2	yes	—
238	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
239	C₃	C₃×#1986	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	C₉×#391	1	1	yes	characteristic
243	C₂	C₂×C₉×#104	1	1	yes	characteristic
244	C₂	C₉×#421	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₉×C₄⋊Q₈	C₄	semidirect
8. C₉×C₄⋊Q₈	C₄	semidirect
9. C₉×#125	C₄	semidirect
32. C₉×M₁₆	Q₈	semidirect
48. #1986	C₉	direct (N × H)
125. C₄⋊Q₈	C₄×C₉	semidirect
126. C₄⋊Q₈	C₄×C₉	semidirect
127. #125	C₄×C₉	semidirect
170. M₁₆	Q₈×C₉	semidirect
200. C₉	#1986	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
C₄×C₉×M₁₆	2
C₄×C₉×C₄⋊C₄	2
C₉×#339	2
C₃×#1986	3

Subgroups — C9×#1986 #11766707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1986 #11766707