Subgroups — C₉×#1800 #11766521

Order 1152.

504 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×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#423	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#100	C₂	1	2	yes	maximal
4	C₉×#343	C₂	1	2	yes	maximal, complemented
5	C₃×#1800	C₃	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₄	1	2	yes	—
8	C₂×C₉×M₁₆	C₂²	1	4	yes	—
9	C₄×D₄×C₉	—	2	2	no	—
10	C₄×D₄×C₉	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊C₈	—	2	2	no	—
15	C₉×C₄⋊C₈	—	2	2	no	—
16	C₉×#100	—	2	4	no	—
17	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
18	C₃×#423	C₂×C₃	1	2	yes	—
19	C₂×C₃×#100	C₂×C₃	1	2	yes	—
20	C₃×#343	C₂×C₃	1	2	yes	—
21	C₂⁴×C₉	C₂×C₄	1	2	yes	—
22	C₂×D₄×C₉	—	2	2	no	—
23	C₂×D₄×C₉	—	2	2	no	—
24	C₂×D₄×C₉	D₄	1	2	yes	—
25	C₂×D₄×C₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	4	4	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×M₁₆	—	2	4	no	—
34	C₉×M₁₆	—	2	4	no	—
35	C₂×C₈×C₉	—	2	4	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₂²⋊C₄	—	2	4	no	—
38	C₄²×C₉	D₄	1	2	yes	—
39	C₄²×C₉	D₄	1	2	yes	—
40	#1800	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₃×C₄	1	2	yes	—
43	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
44	C₃×C₄×D₄	—	2	2	no	—
45	C₃×C₄×D₄	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	2	yes	—
48	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×C₄⋊C₈	—	2	2	no	—
51	C₃×#100	—	2	4	no	—
52	C₂³×C₉	—	2	2	no	—
53	C₂³×C₉	—	2	2	no	—
54	C₂³×C₉	—	4	2	no	—
55	C₂³×C₉	—	4	2	no	—
56	C₂³×C₉	C₄⋊C₄	1	4	yes	—
57	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
58	D₄×C₉	—	2	4	no	—
59	D₄×C₉	—	2	4	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	4	1	no	—
63	C₂×C₄×C₉	—	4	4	no	—
64	C₂×C₄×C₉	—	4	4	no	—
65	C₂×C₄×C₉	—	4	2	no	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	2	yes	—
73	C₈×C₉	—	4	4	no	—
74	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
75	#423	C₂×C₉	1	2	yes	complemented
76	C₂×#100	C₂×C₉	1	2	yes	—
77	#343	C₂×C₉	1	2	yes	complemented
78	C₂⁴×C₃	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₃×D₄	—	2	2	no	—
80	C₂×C₃×D₄	—	2	2	no	—
81	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
82	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	—	4	4	no	—
86	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₂²⋊C₄	—	2	4	no	—
95	C₃×C₄²	C₃×D₄	1	2	yes	—
96	C₃×C₄²	C₃×D₄	1	2	yes	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	—	2	1	no	—
99	C₂²×C₉	—	2	4	no	—
100	C₂²×C₉	—	4	4	no	—
101	C₂²×C₉	—	4	4	no	—
102	C₂²×C₉	—	4	2	no	—
103	C₂²×C₉	—	4	2	no	—
104	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₉	#117	1	1	yes	characteristic
107	C₄×C₉	—	2	1	no	—
108	C₄×C₉	—	2	2	no	—
109	C₄×C₉	—	4	2	no	—
110	C₄×C₉	#100	1	2	yes	complemented
111	C₄×C₉	#121	1	2	yes	—
112	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
113	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
114	C₂³×C₄	C₄×C₉	1	2	yes	—
115	C₂×M₁₆	C₂²×C₉	1	4	yes	—
116	C₄×D₄	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₄×C₉	1	2	yes	—
120	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	#100	—	2	4	no	—
124	C₂³×C₃	—	2	2	no	—
125	C₂³×C₃	—	2	2	no	—
126	C₂³×C₃	—	4	2	no	—
127	C₂³×C₃	—	4	2	no	—
128	C₂³×C₃	C₃×C₄⋊C₄	1	4	yes	—
129	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₃×D₄	—	2	4	no	—
131	C₃×D₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	4	1	no	—
135	C₂×C₃×C₄	—	4	4	no	—
136	C₂×C₃×C₄	—	4	4	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
139	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
143	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₂×C₃×C₄	C₃×#41	1	2	yes	—
145	C₃×C₈	—	4	4	no	—
146	C₂×C₉	—	2	1	no	—
147	C₂×C₉	—	2	1	no	—
148	C₂×C₉	—	4	4	no	—
149	C₂×C₉	#380	1	1	yes	characteristic
150	C₂×C₉	C₂×#100	1	1	yes	characteristic
151	C₂×C₉	#413	1	1	yes	characteristic
152	C₂⁴	C₂×C₄×C₉	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	D₄×C₉	1	2	yes	—
156	C₂×D₄	Q₈×C₉	1	2	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
162	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	D₄×C₉	1	2	yes	—
170	C₄²	D₄×C₉	1	2	yes	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	4	no	—
174	C₂²×C₃	—	4	4	no	—
175	C₂²×C₃	—	4	4	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₃	C₃×#117	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	C₃×#100	1	2	yes	—
185	C₃×C₄	C₃×#121	1	2	yes	—
186	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
187	C₉	#1800	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₉×C₄⋊C₄	1	4	yes	—
193	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
194	D₄	—	2	4	no	—
195	D₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
203	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
207	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
208	C₂×C₄	C₉×#41	1	2	yes	—
209	C₈	—	4	4	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	4	4	no	—
213	C₂×C₃	C₃×#380	1	1	yes	characteristic
214	C₂×C₃	C₂×C₃×#100	1	1	yes	characteristic
215	C₂×C₃	C₃×#413	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	4	4	no	—
220	C₂²	—	4	4	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
225	C₂²	C₉×#117	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	C₉×#100	1	2	yes	complemented
230	C₄	C₉×#121	1	2	yes	—
231	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
232	C₃	C₃×#1800	1	1	yes	characteristic
233	C₂	—	2	1	no	—
234	C₂	—	2	1	no	—
235	C₂	—	4	4	no	—
236	C₂	C₉×#380	1	1	yes	characteristic
237	C₂	C₂×C₉×#100	1	1	yes	characteristic
238	C₂	C₉×#413	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₉×#423	C₂	semidirect
4. C₉×#343	C₂	semidirect
40. #1800	C₉	direct (N × H)
75. #423	C₂×C₉	semidirect
77. #343	C₂×C₉	semidirect
110. C₄×C₉	#100	semidirect
187. C₉	#1800	direct (N × H)
229. C₄	C₉×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₉	2
C₉×#423	2
C₂×C₉×#100	2
C₉×#343	2
C₃×#1800	3

Subgroups — C9×#1800 #11766521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1800 #11766521