Subgroups — C₉×#2285 #11767006

Order 1152.

984 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#117	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#384	C₂	1	2	yes	maximal, complemented
4	C₉×#382	C₂	1	2	yes	maximal, complemented
5	C₉×#380	C₂	1	4	yes	maximal
6	C₄×C₉×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₃×#2285	C₃	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₉	C₂²	1	2	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₉×#117	—	2	8	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄²×C₉	C₄	1	8	yes	complemented
15	C₂×C₄²×C₉	C₂²	1	4	yes	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₉	C₂²	1	2	yes	—
19	C₉×#95	C₂²	1	8	yes	—
20	C₂×C₃×#117	C₂×C₃	1	2	yes	—
21	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
22	C₃×#384	C₂×C₃	1	2	yes	—
23	C₃×#382	C₂×C₃	1	2	yes	—
24	C₃×#380	C₂×C₃	1	4	yes	—
25	C₃×C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
26	C₂⁴×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	—	2	8	no	—
28	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉	D₄	1	4	yes	—
31	C₂²×C₄×C₉	Q₈	1	4	yes	—
32	C₂²×C₄×C₉	C₂³	1	8	yes	—
33	C₂²×C₄×C₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₉×C₄⋊C₄	—	2	8	no	—
38	C₉×C₄⋊C₄	—	2	16	no	—
39	C₉×C₂²⋊C₄	—	2	8	no	—
40	C₄²×C₉	—	2	8	no	—
41	C₄²×C₉	C₂×C₄	1	4	yes	—
42	C₄²×C₉	C₂×C₄	1	8	yes	—
43	C₄²×C₉	C₂×C₄	1	4	yes	—
44	#2285	C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
46	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₃×#117	—	2	8	no	—
48	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
49	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₃×C₄²	C₃×C₄	1	8	yes	—
52	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
53	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₃×#95	C₂²×C₃	1	8	yes	—
57	C₂³×C₉	—	2	4	no	—
58	C₂³×C₉	C₂²×C₄	1	2	yes	—
59	C₂³×C₉	C₂×D₄	1	2	yes	—
60	C₂³×C₉	C₂×Q₈	1	2	yes	—
61	C₂³×C₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₉	—	2	4	no	—
63	C₂×C₄×C₉	—	2	16	no	—
64	C₂×C₄×C₉	—	2	8	no	—
65	C₂×C₄×C₉	—	2	16	no	—
66	C₂×C₄×C₉	C₄⋊C₄	1	16	yes	complemented
67	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₉	#41	1	8	yes	—
76	C₂×#117	C₂×C₉	1	2	yes	—
77	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
78	#384	C₂×C₉	1	2	yes	complemented
79	#382	C₂×C₉	1	2	yes	complemented
80	#380	C₂×C₉	1	4	yes	—
81	C₄×C₄⋊C₄	C₂×C₉	1	4	yes	complemented
82	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	8	no	—
84	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
85	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
86	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
87	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
88	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
89	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
90	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
93	C₃×C₄⋊C₄	—	2	8	no	—
94	C₃×C₄⋊C₄	—	2	16	no	—
95	C₃×C₂²⋊C₄	—	2	8	no	—
96	C₃×C₄²	—	2	8	no	—
97	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
98	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
99	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
100	C₂²×C₉	—	2	4	no	—
101	C₂²×C₉	—	2	4	no	—
102	C₂²×C₉	—	2	4	no	—
103	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₉	C₂×#41	1	4	yes	—
108	C₄×C₉	—	2	4	no	—
109	C₄×C₉	—	4	8	no	—
110	C₄×C₉	C₂×C₄⋊C₄	1	4	yes	—
111	C₄×C₉	C₂×C₄⋊C₄	1	4	yes	—
112	C₄×C₉	#117	1	8	yes	complemented
113	C₂³×C₄	C₂²×C₉	1	2	yes	—
114	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
115	#117	—	2	8	no	—
116	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
117	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
119	C₂×C₄²	C₄×C₉	1	8	yes	complemented
120	C₂×C₄²	C₂²×C₉	1	4	yes	—
121	C₂×C₄²	C₂²×C₉	1	2	yes	—
122	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
124	#95	C₂²×C₉	1	8	yes	—
125	C₂³×C₃	—	2	4	no	—
126	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
127	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
128	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
129	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	16	no	—
132	C₂×C₃×C₄	—	2	8	no	—
133	C₂×C₃×C₄	—	2	16	no	—
134	C₂×C₃×C₄	C₃×C₄⋊C₄	1	16	yes	—
135	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
136	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
137	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
138	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
143	C₂×C₃×C₄	C₃×#41	1	8	yes	—
144	C₂×C₉	—	2	4	no	—
145	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₉	C₂×#117	1	2	yes	—
147	C₂×C₉	#532	1	2	yes	—
148	C₂×C₉	#533	1	2	yes	—
149	C₂⁴	C₂³×C₉	1	1	yes	characteristic
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	C₂×C₄×C₉	1	8	yes	—
152	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
153	C₂²×C₄	D₄×C₉	1	4	yes	—
154	C₂²×C₄	Q₈×C₉	1	4	yes	—
155	C₂²×C₄	C₂³×C₉	1	8	yes	—
156	C₂²×C₄	C₂³×C₉	1	2	yes	—
157	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₉	1	2	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	16	no	—
162	C₂²⋊C₄	—	2	8	no	—
163	C₄²	—	2	8	no	—
164	C₄²	C₂×C₄×C₉	1	4	yes	—
165	C₄²	C₂×C₄×C₉	1	8	yes	—
166	C₄²	C₂×C₄×C₉	1	4	yes	—
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	—	2	4	no	—
170	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
171	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
172	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
173	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
174	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	4	8	no	—
177	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
178	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
179	C₃×C₄	C₃×#117	1	8	yes	—
180	C₉	#2285	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×C₄×C₉	1	2	yes	—
183	C₂³	C₂×D₄×C₉	1	2	yes	—
184	C₂³	C₂×Q₈×C₉	1	2	yes	—
185	C₂³	C₂⁴×C₉	1	1	yes	characteristic
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	C₉×C₄⋊C₄	1	16	yes	complemented
191	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
192	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
193	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
194	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
195	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
196	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
197	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
198	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
199	C₂×C₄	C₉×#41	1	8	yes	—
200	C₂×C₃	—	2	4	no	—
201	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃	C₂×C₃×#117	1	2	yes	—
203	C₂×C₃	C₃×#532	1	2	yes	—
204	C₂×C₃	C₃×#533	1	2	yes	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	C₂×C₉×C₄⋊C₄	1	4	yes	—
209	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
210	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
211	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
212	C₂²	C₂×C₉×#41	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	4	8	no	—
215	C₄	C₂×C₉×C₄⋊C₄	1	4	yes	—
216	C₄	C₂×C₉×C₄⋊C₄	1	4	yes	—
217	C₄	C₉×#117	1	8	yes	complemented
218	C₃	C₃×#2285	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	C₂²×C₉×C₄⋊C₄	1	1	yes	characteristic
221	C₂	C₂×C₉×#117	1	2	yes	—
222	C₂	C₉×#532	1	2	yes	—
223	C₂	C₉×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₉×#384	C₂	semidirect
4. C₉×#382	C₂	semidirect
6. C₄×C₉×C₄⋊C₄	C₂	semidirect
14. C₂×C₄²×C₉	C₄	semidirect
44. #2285	C₉	direct (N × H)
66. C₂×C₄×C₉	C₄⋊C₄	semidirect
78. #384	C₂×C₉	semidirect
79. #382	C₂×C₉	semidirect
81. C₄×C₄⋊C₄	C₂×C₉	semidirect
112. C₄×C₉	#117	semidirect
119. C₂×C₄²	C₄×C₉	semidirect
180. C₉	#2285	direct (N × H)
190. C₂×C₄	C₉×C₄⋊C₄	semidirect
217. C₄	C₉×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#117	2
C₂²×C₄²×C₉	2
C₉×#384	2
C₉×#382	2
C₉×#380	2
C₄×C₉×C₄⋊C₄	2
C₃×#2285	3

Subgroups — C9×#2285 #11767006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2285 #11767006