Subgroups — C₉×#2056 #11766777

Order 1152.

432 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#103	C₂	1	2	yes	maximal
4	C₉×#391	C₂	1	2	yes	maximal
5	C₉×#340	C₂	1	1	yes	maximal, characteristic
6	C₃×#2056	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂×C₉×Q₁₆	—	2	4	no	—
9	C₂×C₉×Q₁₆	—	2	2	no	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊C₈	—	2	2	no	—
15	C₉×#103	—	2	4	no	—
16	C₉×#95	—	2	2	no	—
17	C₉×#95	—	2	2	no	—
18	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#103	C₂×C₃	1	2	yes	—
21	C₃×#391	C₂×C₃	1	2	yes	—
22	C₃×#340	C₂×C₃	1	1	yes	characteristic
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	D₄	1	4	yes	—
29	C₂²×C₄×C₉	—	2	2	no	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×Q₁₆	—	2	8	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	2	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	D₄	1	2	yes	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	—	2	2	no	—
41	C₄²×C₉	—	2	1	no	—
42	C₄²×C₉	—	2	1	no	—
43	#2056	C₉	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
45	C₂×C₃×Q₁₆	—	2	4	no	—
46	C₂×C₃×Q₁₆	—	2	2	no	—
47	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₃×C₄⋊C₈	—	2	2	no	—
52	C₃×#103	—	2	4	no	—
53	C₃×#95	—	2	2	no	—
54	C₃×#95	—	2	2	no	—
55	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₉	—	2	4	no	—
57	Q₈×C₉	—	2	4	no	—
58	Q₈×C₉	—	4	2	no	—
59	Q₈×C₉	—	4	2	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	—	4	2	no	—
70	C₂×C₄×C₉	Q₁₆	1	2	yes	complemented
71	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₂×C₄×C₉	#41	1	2	yes	—
75	C₈×C₉	—	2	2	no	—
76	C₈×C₉	—	4	1	no	—
77	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic
78	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
79	C₂×#103	C₂×C₉	1	2	yes	—
80	#391	C₂×C₉	1	2	yes	—
81	#340	C₂×C₉	1	1	yes	characteristic
82	C₂×C₃×Q₈	—	2	2	no	—
83	C₂×C₃×Q₈	—	2	2	no	—
84	C₂×C₃×Q₈	—	2	2	no	—
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₂×C₃×Q₈	—	2	2	no	—
87	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
88	C₂²×C₃×C₄	—	2	2	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×Q₁₆	—	2	8	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄⋊C₄	—	2	2	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₂²×C₉	#121	1	2	yes	—
103	C₂²×C₉	#121	1	1	yes	characteristic
104	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
105	C₂²×C₉	#135	1	1	yes	characteristic
106	C₂²×C₉	#137	1	2	yes	—
107	C₄×C₉	—	2	2	no	—
108	C₄×C₉	—	4	4	no	—
109	C₄×C₉	—	4	1	no	—
110	C₄×C₉	—	8	2	no	—
111	C₄×C₉	#120	1	1	yes	characteristic
112	C₄×C₉	#123	1	2	yes	—
113	C₄×C₉	#124	1	1	yes	characteristic
114	C₂²×Q₈	C₂²×C₉	1	2	yes	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
120	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
121	C₄⋊C₈	—	2	2	no	—
122	#103	—	2	4	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
126	C₃×Q₈	—	2	4	no	—
127	C₃×Q₈	—	2	4	no	—
128	C₃×Q₈	—	4	2	no	—
129	C₃×Q₈	—	4	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
144	C₂×C₃×C₄	C₃×#41	1	2	yes	—
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	—	4	1	no	—
147	C₂×C₉	#394	1	1	yes	characteristic
148	C₂×C₉	#462	1	2	yes	—
149	C₂×C₉	#464	1	2	yes	—
150	C₂×C₉	#467	1	1	yes	characteristic
151	C₂×C₉	#470	1	1	yes	characteristic
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	D₄×C₉	1	4	yes	—
158	C₂²×C₄	—	2	2	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	Q₁₆	—	2	8	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	D₄×C₉	1	2	yes	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₂²×C₃	C₃×#121	1	2	yes	—
173	C₂²×C₃	C₃×#121	1	1	yes	characteristic
174	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
175	C₂²×C₃	C₃×#135	1	1	yes	characteristic
176	C₂²×C₃	C₃×#137	1	2	yes	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	4	4	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	8	2	no	—
181	C₃×C₄	C₃×#120	1	1	yes	characteristic
182	C₃×C₄	C₃×#123	1	2	yes	—
183	C₃×C₄	C₃×#124	1	1	yes	characteristic
184	C₉	#2056	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	2	4	no	—
188	Q₈	—	4	2	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	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₉×Q₁₆	1	2	yes	complemented
201	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
202	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
203	C₂×C₄	C₉×#41	1	1	yes	characteristic
204	C₂×C₄	C₉×#41	1	2	yes	—
205	C₈	—	2	2	no	—
206	C₈	—	4	1	no	—
207	C₂×C₃	C₃×#394	1	1	yes	characteristic
208	C₂×C₃	C₃×#462	1	2	yes	—
209	C₂×C₃	C₃×#464	1	2	yes	—
210	C₂×C₃	C₃×#467	1	1	yes	characteristic
211	C₂×C₃	C₃×#470	1	1	yes	characteristic
212	C₂²	C₉×#121	1	2	yes	—
213	C₂²	C₉×#121	1	1	yes	characteristic
214	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
215	C₂²	C₉×#135	1	1	yes	characteristic
216	C₂²	C₉×#137	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	4	1	no	—
220	C₄	—	8	2	no	—
221	C₄	C₉×#120	1	1	yes	characteristic
222	C₄	C₉×#123	1	2	yes	—
223	C₄	C₉×#124	1	1	yes	characteristic
224	C₃	C₃×#2056	1	1	yes	characteristic
225	C₂	C₉×#394	1	1	yes	characteristic
226	C₂	C₉×#462	1	2	yes	—
227	C₂	C₉×#464	1	2	yes	—
228	C₂	C₉×#467	1	1	yes	characteristic
229	C₂	C₉×#470	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
43. #2056	C₉	direct (N × H)
70. C₂×C₄×C₉	Q₁₆	semidirect
184. C₉	#2056	direct (N × H)
200. C₂×C₄	C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#103	2
C₉×#391	2
C₉×#340	2
C₃×#2056	3

Subgroups — C9×#2056 #11766777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2056 #11766777