Subgroups — C₉×#2072 #11766793

Order 1152.

357 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#103	C₂	1	2	yes	maximal
2	C₉×#391	C₂	1	1	yes	maximal, characteristic
3	C₉×#384	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	2	yes	maximal
5	C₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₃×#2072	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×C₉×C₄⋊C₄	—	2	1	no	—
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#103	—	2	4	no	—
15	C₉×#95	—	2	1	no	—
16	C₉×#95	—	2	1	no	—
17	C₂×C₃×#103	C₂×C₃	1	2	yes	—
18	C₃×#391	C₂×C₃	1	1	yes	characteristic
19	C₃×#384	C₂×C₃	1	1	yes	characteristic
20	C₃×#340	C₂×C₃	1	2	yes	—
21	C₃×#336	C₂×C₃	1	1	yes	characteristic
22	C₂×Q₈×C₉	—	2	1	no	—
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	—	2	2	no	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	4	2	no	—
39	C₉×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₉	—	2	2	no	—
41	#2072	C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
44	C₂×C₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×#103	—	2	4	no	—
50	C₃×#95	—	2	1	no	—
51	C₃×#95	—	2	1	no	—
52	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
53	Q₈×C₉	—	2	4	no	—
54	Q₈×C₉	—	4	2	no	—
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	4	1	no	—
61	C₂×C₄×C₉	—	4	1	no	—
62	C₂×C₄×C₉	—	4	2	no	—
63	C₂×C₄×C₉	—	4	1	no	—
64	C₂×C₄×C₉	—	4	2	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	—	4	1	no	—
67	C₂×C₄×C₉	—	4	1	no	—
68	C₂×C₄×C₉	SD₁₆	1	2	yes	—
69	C₂×C₄×C₉	Q₁₆	1	2	yes	—
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	#41	1	2	yes	—
72	C₂×C₄×C₉	#41	1	2	yes	—
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₈×C₉	—	4	2	no	—
75	C₂×#103	C₂×C₉	1	2	yes	—
76	#391	C₂×C₉	1	1	yes	characteristic
77	#384	C₂×C₉	1	1	yes	characteristic
78	#340	C₂×C₉	1	2	yes	—
79	#336	C₂×C₉	1	1	yes	characteristic
80	C₂×C₃×Q₈	—	2	1	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	—	2	2	no	—
83	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	4	2	no	—
97	C₃×C₄⋊C₄	—	4	2	no	—
98	C₃×C₄²	—	2	2	no	—
99	C₂²×C₉	#121	1	1	yes	characteristic
100	C₂²×C₉	#123	1	1	yes	characteristic
101	C₂²×C₉	#124	1	1	yes	characteristic
102	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×C₉	#137	1	1	yes	characteristic
106	C₄×C₉	—	2	2	no	—
107	C₄×C₉	—	2	2	no	—
108	C₄×C₉	—	4	2	no	—
109	C₄×C₉	—	8	2	no	—
110	C₄×C₉	—	8	1	no	—
111	C₄×C₉	#123	1	2	yes	—
112	C₄×C₉	#126	1	2	yes	—
113	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₉	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
119	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
120	#103	—	2	4	no	—
121	#95	—	2	1	no	—
122	#95	—	2	1	no	—
123	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	4	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	4	1	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	—	4	1	no	—
139	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
140	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₃×#41	1	2	yes	—
143	C₂×C₃×C₄	C₃×#41	1	2	yes	—
144	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
145	C₃×C₈	—	4	2	no	—
146	C₂×C₉	#397	1	1	yes	characteristic
147	C₂×C₉	#460	1	1	yes	characteristic
148	C₂×C₉	#462	1	1	yes	characteristic
149	C₂×C₉	#483	1	1	yes	characteristic
150	C₂×C₉	#484	1	1	yes	characteristic
151	C₂×C₉	#487	1	1	yes	characteristic
152	C₂×C₉	#489	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	D₄×C₉	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
161	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄²	—	2	2	no	—
172	C₂²×C₃	C₃×#121	1	1	yes	characteristic
173	C₂²×C₃	C₃×#123	1	1	yes	characteristic
174	C₂²×C₃	C₃×#124	1	1	yes	characteristic
175	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
176	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
177	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂²×C₃	C₃×#137	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	8	2	no	—
183	C₃×C₄	—	8	1	no	—
184	C₃×C₄	C₃×#123	1	2	yes	—
185	C₃×C₄	C₃×#126	1	2	yes	—
186	C₉	#2072	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	C₉×SD₁₆	1	2	yes	—
204	C₂×C₄	C₉×Q₁₆	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
206	C₂×C₄	C₉×#41	1	2	yes	—
207	C₂×C₄	C₉×#41	1	2	yes	—
208	C₂×C₄	C₉×#41	1	1	yes	characteristic
209	C₈	—	4	2	no	—
210	C₂×C₃	C₃×#397	1	1	yes	characteristic
211	C₂×C₃	C₃×#460	1	1	yes	characteristic
212	C₂×C₃	C₃×#462	1	1	yes	characteristic
213	C₂×C₃	C₃×#483	1	1	yes	characteristic
214	C₂×C₃	C₃×#484	1	1	yes	characteristic
215	C₂×C₃	C₃×#487	1	1	yes	characteristic
216	C₂×C₃	C₃×#489	1	1	yes	characteristic
217	C₂²	C₉×#121	1	1	yes	characteristic
218	C₂²	C₉×#123	1	1	yes	characteristic
219	C₂²	C₉×#124	1	1	yes	characteristic
220	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
221	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
222	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²	C₉×#137	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	8	2	no	—
228	C₄	—	8	1	no	—
229	C₄	C₉×#123	1	2	yes	—
230	C₄	C₉×#126	1	2	yes	—
231	C₃	C₃×#2072	1	1	yes	characteristic
232	C₂	C₉×#397	1	1	yes	characteristic
233	C₂	C₉×#460	1	1	yes	characteristic
234	C₂	C₉×#462	1	1	yes	characteristic
235	C₂	C₉×#483	1	1	yes	characteristic
236	C₂	C₉×#484	1	1	yes	characteristic
237	C₂	C₉×#487	1	1	yes	characteristic
238	C₂	C₉×#489	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, prime radical	C₂³×C₃	C₂×C₃×D₄
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
41. #2072	C₉	direct (N × H)
186. C₉	#2072	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#103	2
C₉×#391	2
C₉×#384	2
C₉×#340	2
C₉×#336	2
C₃×#2072	3

Subgroups — C9×#2072 #11766793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2072 #11766793