Subgroups — C₉×#2308 #11767029

Order 1152.

1740 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₉×#386	C₂	1	8	yes	maximal, complemented
4	C₉×#381	C₂	1	2	yes	maximal, complemented
5	C₉×#379	C₂	1	2	yes	maximal, complemented
6	C₃×#2308	C₃	1	1	yes	maximal, characteristic
7	C₂⁵×C₉	C₂²	1	1	yes	characteristic
8	C₂⁵×C₉	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₉	—	2	8	no	—
10	C₂²×D₄×C₉	C₄	1	8	yes	complemented
11	C₂²×D₄×C₉	C₂²	1	4	yes	—
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₉	C₂²	1	4	yes	complemented
14	C₂×C₉×C₂²⋊C₄	—	2	8	no	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₉×#95	C₂²	1	8	yes	complemented
19	C₂³×C₃×D₄	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
21	C₃×#386	C₂×C₃	1	8	yes	—
22	C₃×#381	C₂×C₃	1	2	yes	—
23	C₃×#379	C₂×C₃	1	2	yes	—
24	C₂⁴×C₉	—	2	4	no	—
25	C₂⁴×C₉	—	4	4	no	—
26	C₂⁴×C₉	—	4	4	no	—
27	C₂⁴×C₉	C₂×C₄	1	4	yes	—
28	C₂⁴×C₉	C₂×C₄	1	4	yes	complemented
29	C₂⁴×C₉	D₄	1	8	yes	—
30	C₂⁴×C₉	C₂³	1	2	yes	—
31	C₂⁴×C₉	C₂³	1	1	yes	characteristic
32	C₂⁴×C₉	C₂³	1	2	yes	—
33	C₂×D₄×C₉	—	2	32	no	—
34	C₂×D₄×C₉	—	2	8	no	—
35	C₂×D₄×C₉	—	2	8	no	—
36	C₂×D₄×C₉	C₂×C₄	1	16	yes	complemented
37	C₂²×C₄×C₉	—	2	4	no	—
38	C₂²×C₄×C₉	—	2	8	no	—
39	C₂²×C₄×C₉	—	4	8	no	—
40	C₂²×C₄×C₉	C₂×C₄	1	4	yes	complemented
41	C₂²×C₄×C₉	C₂³	1	8	yes	—
42	C₂²×C₄×C₉	C₂³	1	2	yes	—
43	C₉×C₂²⋊C₄	—	2	8	no	—
44	C₉×C₂²⋊C₄	—	2	8	no	—
45	C₉×C₂²⋊C₄	—	4	8	no	—
46	#2308	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
48	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃×D₄	—	2	8	no	—
50	C₂²×C₃×D₄	C₃×C₄	1	8	yes	—
51	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
52	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
55	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
58	C₃×#95	C₂²×C₃	1	8	yes	—
59	C₂³×C₉	—	2	8	no	—
60	C₂³×C₉	—	2	4	no	—
61	C₂³×C₉	—	2	8	no	—
62	C₂³×C₉	—	2	4	no	—
63	C₂³×C₉	—	2	8	no	—
64	C₂³×C₉	—	4	8	no	—
65	C₂³×C₉	—	4	8	no	—
66	C₂³×C₉	—	4	16	no	—
67	C₂³×C₉	—	4	16	no	—
68	C₂³×C₉	C₂²⋊C₄	1	16	yes	complemented
69	C₂³×C₉	C₂²×C₄	1	4	yes	—
70	C₂³×C₉	C₂²×C₄	1	2	yes	—
71	C₂³×C₉	C₂²×C₄	1	4	yes	—
72	C₂³×C₉	C₂×D₄	1	4	yes	—
73	C₂³×C₉	C₂×D₄	1	4	yes	—
74	C₂³×C₉	C₂×D₄	1	4	yes	—
75	C₂³×C₉	C₂⁴	1	1	yes	characteristic
76	D₄×C₉	—	2	32	no	—
77	C₂×C₄×C₉	—	2	8	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	8	no	—
80	C₂×C₄×C₉	—	2	16	no	—
81	C₂×C₄×C₉	—	4	16	no	—
82	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
83	C₂³×D₄	C₂×C₉	1	1	yes	characteristic
84	C₂²×C₂²⋊C₄	C₂×C₉	1	2	yes	complemented
85	#386	C₂×C₉	1	8	yes	complemented
86	#381	C₂×C₉	1	2	yes	complemented
87	#379	C₂×C₉	1	2	yes	complemented
88	C₂⁴×C₃	—	2	4	no	—
89	C₂⁴×C₃	—	4	4	no	—
90	C₂⁴×C₃	—	4	4	no	—
91	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
92	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
93	C₂⁴×C₃	C₃×D₄	1	8	yes	—
94	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
95	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
96	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
97	C₂×C₃×D₄	—	2	32	no	—
98	C₂×C₃×D₄	—	2	8	no	—
99	C₂×C₃×D₄	—	2	8	no	—
100	C₂×C₃×D₄	C₂×C₃×C₄	1	16	yes	—
101	C₂²×C₃×C₄	—	2	4	no	—
102	C₂²×C₃×C₄	—	2	8	no	—
103	C₂²×C₃×C₄	—	4	8	no	—
104	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
105	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
107	C₃×C₂²⋊C₄	—	2	8	no	—
108	C₃×C₂²⋊C₄	—	2	8	no	—
109	C₃×C₂²⋊C₄	—	4	8	no	—
110	C₂²×C₉	—	2	4	no	—
111	C₂²×C₉	—	2	8	no	—
112	C₂²×C₉	—	2	8	no	—
113	C₂²×C₉	—	2	8	no	—
114	C₂²×C₉	—	2	8	no	—
115	C₂²×C₉	—	2	4	no	—
116	C₂²×C₉	—	2	8	no	—
117	C₂²×C₉	—	4	16	no	—
118	C₂²×C₉	—	4	16	no	—
119	C₂²×C₉	—	4	8	no	—
120	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂²×C₉	C₂×C₂²⋊C₄	1	8	yes	—
122	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₉	C₂²×D₄	1	2	yes	—
124	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
125	C₄×C₉	—	2	4	no	—
126	C₄×C₉	—	4	8	no	—
127	C₂⁵	C₂²×C₉	1	1	yes	characteristic
128	C₂⁵	C₂²×C₉	1	1	yes	characteristic
129	C₂²×D₄	—	2	8	no	—
130	C₂²×D₄	C₄×C₉	1	8	yes	complemented
131	C₂²×D₄	C₂²×C₉	1	4	yes	—
132	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₉	1	4	yes	complemented
134	C₂×C₂²⋊C₄	—	2	8	no	—
135	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	complemented
137	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	complemented
138	#95	C₂²×C₉	1	8	yes	complemented
139	C₂³×C₃	—	2	8	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	8	no	—
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	—	2	8	no	—
144	C₂³×C₃	—	4	8	no	—
145	C₂³×C₃	—	4	8	no	—
146	C₂³×C₃	—	4	16	no	—
147	C₂³×C₃	—	4	16	no	—
148	C₂³×C₃	C₃×C₂²⋊C₄	1	16	yes	—
149	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
150	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
151	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
152	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
153	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
154	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
155	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
156	C₃×D₄	—	2	32	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	16	no	—
161	C₂×C₃×C₄	—	4	16	no	—
162	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
163	C₂×C₉	—	2	4	no	—
164	C₂×C₉	—	2	8	no	—
165	C₂×C₉	—	4	4	no	—
166	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₉	#518	1	2	yes	—
168	C₂×C₉	#534	1	4	yes	—
169	C₂⁴	—	2	4	no	—
170	C₂⁴	—	4	4	no	—
171	C₂⁴	—	4	4	no	—
172	C₂⁴	C₂×C₄×C₉	1	4	yes	—
173	C₂⁴	C₂×C₄×C₉	1	4	yes	complemented
174	C₂⁴	D₄×C₉	1	8	yes	—
175	C₂⁴	C₂³×C₉	1	2	yes	—
176	C₂⁴	C₂³×C₉	1	1	yes	characteristic
177	C₂⁴	C₂³×C₉	1	2	yes	—
178	C₂×D₄	—	2	32	no	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	8	no	—
181	C₂×D₄	C₂×C₄×C₉	1	16	yes	complemented
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	C₂×C₄×C₉	1	4	yes	complemented
186	C₂²×C₄	C₂³×C₉	1	8	yes	—
187	C₂²×C₄	C₂³×C₉	1	2	yes	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	—	4	8	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	8	no	—
193	C₂²×C₃	—	2	8	no	—
194	C₂²×C₃	—	2	8	no	—
195	C₂²×C₃	—	2	8	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	8	no	—
198	C₂²×C₃	—	4	16	no	—
199	C₂²×C₃	—	4	16	no	—
200	C₂²×C₃	—	4	8	no	—
201	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	4	yes	—
202	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	8	yes	—
203	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
204	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
205	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	4	8	no	—
208	C₉	#2308	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³	—	2	8	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	8	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	4	8	no	—
215	C₂³	—	4	8	no	—
216	C₂³	—	4	16	no	—
217	C₂³	—	4	16	no	—
218	C₂³	C₉×C₂²⋊C₄	1	16	yes	complemented
219	C₂³	C₂²×C₄×C₉	1	4	yes	—
220	C₂³	C₂²×C₄×C₉	1	2	yes	—
221	C₂³	C₂²×C₄×C₉	1	4	yes	—
222	C₂³	C₂×D₄×C₉	1	4	yes	—
223	C₂³	C₂×D₄×C₉	1	4	yes	—
224	C₂³	C₂×D₄×C₉	1	4	yes	—
225	C₂³	C₂⁴×C₉	1	1	yes	characteristic
226	D₄	—	2	32	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	4	16	no	—
232	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	2	8	no	—
235	C₂×C₃	—	4	4	no	—
236	C₂×C₃	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₃	C₃×#518	1	2	yes	—
238	C₂×C₃	C₃×#534	1	4	yes	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	16	no	—
247	C₂²	—	4	16	no	—
248	C₂²	—	4	8	no	—
249	C₂²	C₂×C₉×C₂²⋊C₄	1	4	yes	—
250	C₂²	C₂×C₉×C₂²⋊C₄	1	8	yes	—
251	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
252	C₂²	C₂²×D₄×C₉	1	2	yes	—
253	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	4	8	no	—
256	C₃	C₃×#2308	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	2	8	no	—
259	C₂	—	4	4	no	—
260	C₂	C₂²×C₉×C₂²⋊C₄	1	1	yes	characteristic
261	C₂	C₉×#518	1	2	yes	—
262	C₂	C₉×#534	1	4	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂³×C₃×D₄	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×C₂²⋊C₄	C₂	semidirect
3. C₉×#386	C₂	semidirect
4. C₉×#381	C₂	semidirect
5. C₉×#379	C₂	semidirect
10. C₂²×D₄×C₉	C₄	semidirect
13. C₂³×C₄×C₉	C₂²	semidirect
16. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
18. C₉×#95	C₂²	semidirect
28. C₂⁴×C₉	C₂×C₄	semidirect
36. C₂×D₄×C₉	C₂×C₄	semidirect
40. C₂²×C₄×C₉	C₂×C₄	semidirect
46. #2308	C₉	direct (N × H)
68. C₂³×C₉	C₂²⋊C₄	semidirect
84. C₂²×C₂²⋊C₄	C₂×C₉	semidirect
85. #386	C₂×C₉	semidirect
86. #381	C₂×C₉	semidirect
87. #379	C₂×C₉	semidirect
130. C₂²×D₄	C₄×C₉	semidirect
133. C₂³×C₄	C₂²×C₉	semidirect
136. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
137. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
138. #95	C₂²×C₉	semidirect
173. C₂⁴	C₂×C₄×C₉	semidirect
181. C₂×D₄	C₂×C₄×C₉	semidirect
185. C₂²×C₄	C₂×C₄×C₉	semidirect
208. C₉	#2308	direct (N × H)
218. C₂³	C₉×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₉	2
C₂²×C₉×C₂²⋊C₄	2
C₉×#386	2
C₉×#381	2
C₉×#379	2
C₃×#2308	3

Subgroups — C9×#2308 #11767029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2308 #11767029