Subgroups — C₉×#2328 #11767049

Order 1152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₉	C₂	1	2	yes	maximal, complemented
3	C₉×#390	C₂	1	2	yes	maximal, complemented
4	C₉×#386	C₂	1	8	yes	maximal, complemented
5	C₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2328	C₃	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₉	C₂²	1	2	yes	—
9	C₂²×D₄×C₉	C₂²	1	4	yes	—
10	C₂³×C₄×C₉	C₂²	1	4	yes	—
11	C₉×#127	C₄	1	8	yes	complemented
12	C₄×D₄×C₉	—	2	8	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	8	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	1	yes	characteristic
19	C₉×#95	C₂²	1	4	yes	—
20	C₉×#95	C₂²	1	4	yes	—
21	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×D₄	C₂×C₃	1	2	yes	—
23	C₃×#390	C₂×C₃	1	2	yes	—
24	C₃×#386	C₂×C₃	1	8	yes	—
25	C₃×#382	C₂×C₃	1	1	yes	characteristic
26	C₃×#376	C₂×C₃	1	1	yes	characteristic
27	C₂⁴×C₉	C₂³	1	4	yes	—
28	C₂×D₄×C₉	—	2	4	no	—
29	C₂×D₄×C₉	—	2	4	no	—
30	C₂×D₄×C₉	—	4	8	no	—
31	C₂×D₄×C₉	C₂×C₄	1	8	yes	complemented
32	C₂×D₄×C₉	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄×C₉	—	2	8	no	—
34	C₂²×C₄×C₉	—	4	8	no	—
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₂²⋊C₄	—	2	8	no	—
42	C₉×C₂²⋊C₄	—	4	8	no	—
43	C₄²×C₉	—	2	2	no	—
44	C₄²×C₉	C₂×C₄	1	4	yes	complemented
45	C₄²×C₉	D₄	1	4	yes	complemented
46	#2328	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
48	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
49	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
50	C₃×#127	C₃×C₄	1	8	yes	—
51	C₃×C₄×D₄	—	2	8	no	—
52	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
55	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₃×#95	C₂²×C₃	1	4	yes	—
59	C₃×#95	C₂²×C₃	1	4	yes	—
60	C₂³×C₉	—	2	4	no	—
61	C₂³×C₉	—	2	4	no	—
62	C₂³×C₉	—	2	8	no	—
63	C₂³×C₉	—	2	8	no	—
64	C₂³×C₉	C₂²×C₄	1	8	yes	—
65	C₂³×C₉	C₂⁴	1	1	yes	characteristic
66	D₄×C₉	—	2	16	no	—
67	D₄×C₉	—	4	16	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	4	no	—
73	C₂×C₄×C₉	—	2	8	no	—
74	C₂×C₄×C₉	—	4	16	no	—
75	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
78	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉	#41	1	2	yes	—
80	C₂×#127	C₂×C₉	1	1	yes	characteristic
81	C₂×C₄×D₄	C₂×C₉	1	2	yes	complemented
82	#390	C₂×C₉	1	2	yes	complemented
83	#386	C₂×C₉	1	8	yes	complemented
84	#382	C₂×C₉	1	1	yes	characteristic, complemented
85	#376	C₂×C₉	1	1	yes	characteristic, complemented
86	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
87	C₂×C₃×D₄	—	2	4	no	—
88	C₂×C₃×D₄	—	2	4	no	—
89	C₂×C₃×D₄	—	4	8	no	—
90	C₂×C₃×D₄	C₂×C₃×C₄	1	8	yes	—
91	C₂×C₃×D₄	C₂×C₃×C₄	1	16	yes	—
92	C₂²×C₃×C₄	—	2	8	no	—
93	C₂²×C₃×C₄	—	4	8	no	—
94	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
95	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
96	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₂²⋊C₄	—	2	8	no	—
101	C₃×C₂²⋊C₄	—	4	8	no	—
102	C₃×C₄²	—	2	2	no	—
103	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
104	C₃×C₄²	C₃×D₄	1	4	yes	—
105	C₂²×C₉	—	2	8	no	—
106	C₂²×C₉	—	2	16	no	—
107	C₂²×C₉	—	4	4	no	—
108	C₂²×C₉	—	4	4	no	—
109	C₂²×C₉	—	4	8	no	—
110	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
111	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₉	C₂×#41	1	1	yes	characteristic
113	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
114	C₄×C₉	—	2	4	no	—
115	C₄×C₉	—	2	4	no	—
116	C₄×C₉	—	4	4	no	—
117	C₄×C₉	—	4	2	no	—
118	C₄×C₉	C₄×D₄	1	4	yes	—
119	C₂²×D₄	C₂²×C₉	1	2	yes	—
120	C₂²×D₄	C₂²×C₉	1	4	yes	—
121	C₂³×C₄	C₂²×C₉	1	4	yes	—
122	#127	C₄×C₉	1	8	yes	complemented
123	C₄×D₄	—	2	8	no	—
124	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
127	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
130	#95	C₂²×C₉	1	4	yes	—
131	#95	C₂²×C₉	1	4	yes	—
132	C₂³×C₃	—	2	4	no	—
133	C₂³×C₃	—	2	4	no	—
134	C₂³×C₃	—	2	8	no	—
135	C₂³×C₃	—	2	8	no	—
136	C₂³×C₃	C₂²×C₃×C₄	1	8	yes	—
137	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
138	C₃×D₄	—	2	16	no	—
139	C₃×D₄	—	4	16	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	8	no	—
146	C₂×C₃×C₄	—	4	16	no	—
147	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
148	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₃×C₄	C₃×#41	1	2	yes	—
152	C₂×C₉	—	4	8	no	—
153	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×C₉	#518	1	2	yes	—
155	C₂×C₉	#535	1	2	yes	—
156	C₂×C₉	#540	1	2	yes	—
157	C₂⁴	C₂³×C₉	1	4	yes	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	4	8	no	—
161	C₂×D₄	C₂×C₄×C₉	1	8	yes	complemented
162	C₂×D₄	C₂×C₄×C₉	1	16	yes	complemented
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	C₂³×C₉	1	2	yes	—
166	C₂²×C₄	C₂³×C₉	1	4	yes	—
167	C₂²×C₄	C₂³×C₉	1	2	yes	—
168	C₂²×C₄	C₂³×C₉	1	2	yes	—
169	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₂²⋊C₄	—	4	8	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×C₄×C₉	1	4	yes	complemented
175	C₄²	D₄×C₉	1	4	yes	complemented
176	C₂²×C₃	—	2	8	no	—
177	C₂²×C₃	—	2	16	no	—
178	C₂²×C₃	—	4	4	no	—
179	C₂²×C₃	—	4	4	no	—
180	C₂²×C₃	—	4	8	no	—
181	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
182	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
183	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
184	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
190	C₉	#2328	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	8	no	—
195	C₂³	C₂²×C₄×C₉	1	8	yes	—
196	C₂³	C₂⁴×C₉	1	1	yes	characteristic
197	D₄	—	2	16	no	—
198	D₄	—	4	16	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	16	no	—
206	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
207	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
209	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
210	C₂×C₄	C₉×#41	1	2	yes	—
211	C₂×C₃	—	4	8	no	—
212	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
213	C₂×C₃	C₃×#518	1	2	yes	—
214	C₂×C₃	C₃×#535	1	2	yes	—
215	C₂×C₃	C₃×#540	1	2	yes	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	16	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	4	no	—
220	C₂²	—	4	8	no	—
221	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
222	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
223	C₂²	C₂×C₉×#41	1	1	yes	characteristic
224	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	4	no	—
228	C₄	—	4	2	no	—
229	C₄	C₄×D₄×C₉	1	4	yes	—
230	C₃	C₃×#2328	1	1	yes	characteristic
231	C₂	—	4	8	no	—
232	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
233	C₂	C₉×#518	1	2	yes	—
234	C₂	C₉×#535	1	2	yes	—
235	C₂	C₉×#540	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#127	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. C₂×C₄×D₄×C₉	C₂	semidirect
3. C₉×#390	C₂	semidirect
4. C₉×#386	C₂	semidirect
5. C₉×#382	C₂	semidirect
6. C₉×#376	C₂	semidirect
11. C₉×#127	C₄	semidirect
31. C₂×D₄×C₉	C₂×C₄	semidirect
32. C₂×D₄×C₉	C₂×C₄	semidirect
44. C₄²×C₉	C₂×C₄	semidirect
45. C₄²×C₉	D₄	semidirect
46. #2328	C₉	direct (N × H)
81. C₂×C₄×D₄	C₂×C₉	semidirect
82. #390	C₂×C₉	semidirect
83. #386	C₂×C₉	semidirect
84. #382	C₂×C₉	semidirect
85. #376	C₂×C₉	semidirect
122. #127	C₄×C₉	semidirect
161. C₂×D₄	C₂×C₄×C₉	semidirect
162. C₂×D₄	C₂×C₄×C₉	semidirect
174. C₄²	C₂×C₄×C₉	semidirect
175. C₄²	D₄×C₉	semidirect
190. C₉	#2328	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×#127	2
C₂×C₄×D₄×C₉	2
C₉×#390	2
C₉×#386	2
C₉×#382	2
C₉×#376	2
C₃×#2328	3

Subgroups — C9×#2328 #11767049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2328 #11767049