Subgroups — C₉×#2972 #11767693

Order 1152.

750 conjugacy classes of subgroups.

The subgroup lattice has 210 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	C₈×D₄×C₉	C₂	1	8	yes	maximal, complemented
3	C₉×#431	C₂	1	2	yes	maximal, complemented
4	C₂×C₉×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₃×#2972	C₃	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	8	yes	complemented
9	C₄×D₄×C₉	C₄	1	4	yes	—
10	C₄×D₄×C₉	C₂²	1	4	yes	—
11	C₂×C₉×C₄⋊C₄	C₄	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	C₂²	1	4	yes	complemented
15	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₉×C₂²⋊C₈	C₂²	1	8	yes	complemented
17	C₄×C₈×C₉	C₂²	1	4	yes	complemented
18	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
19	C₃×C₈×D₄	C₂×C₃	1	8	yes	—
20	C₃×#431	C₂×C₃	1	2	yes	—
21	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	4	yes	—
22	C₂⁴×C₉	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₉	C₈	1	8	yes	complemented
24	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₉	—	4	4	no	—
26	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	4	yes	—
32	C₂×C₈×C₉	—	4	8	no	—
33	C₂×C₈×C₉	C₂³	1	8	yes	—
34	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
36	C₄²×C₉	C₂×C₄	1	2	yes	—
37	C₄²×C₉	C₂³	1	2	yes	—
38	#2972	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃×D₄	C₃×C₄	1	1	yes	characteristic
40	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
41	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
42	C₃×C₄×D₄	C₃×C₄	1	4	yes	—
43	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
45	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	2	yes	—
46	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
47	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
48	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
49	C₃×C₂²⋊C₈	C₂²×C₃	1	8	yes	—
50	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
51	C₂³×C₉	—	4	2	no	—
52	C₂³×C₉	—	4	2	no	—
53	C₂³×C₉	C₂×C₈	1	8	yes	complemented
54	C₂³×C₉	C₂²×C₄	1	4	yes	—
55	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
56	D₄×C₉	C₂×C₈	1	16	yes	complemented
57	C₂×C₄×C₉	—	2	8	no	—
58	C₂×C₄×C₉	—	2	2	no	—
59	C₂×C₄×C₉	—	4	4	no	—
60	C₂×C₄×C₉	—	4	2	no	—
61	C₂×C₄×C₉	—	4	1	no	—
62	C₂×C₄×C₉	C₂×C₈	1	4	yes	complemented
63	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
68	C₈×C₉	—	2	8	no	—
69	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
70	C₈×D₄	C₂×C₉	1	8	yes	complemented
71	#431	C₂×C₉	1	2	yes	complemented
72	C₂×C₂²⋊C₈	C₂×C₉	1	4	yes	complemented
73	C₂⁴×C₃	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₃×D₄	C₃×C₈	1	8	yes	—
75	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
76	C₂²×C₃×C₄	—	4	4	no	—
77	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
79	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
83	C₂×C₃×C₈	—	4	8	no	—
84	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
85	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
86	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
87	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
88	C₃×C₄²	C₂³×C₃	1	2	yes	—
89	C₂²×C₉	—	2	1	no	—
90	C₂²×C₉	—	2	4	no	—
91	C₂²×C₉	—	2	1	no	—
92	C₂²×C₉	—	2	4	no	—
93	C₂²×C₉	—	4	8	no	—
94	C₂²×C₉	C₂²×C₈	1	2	yes	—
95	C₂²×C₉	C₂²×C₈	1	8	yes	—
96	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₉	—	2	1	no	—
98	C₄×C₉	—	2	4	no	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	C₂²×C₈	1	4	yes	—
101	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
102	C₂²×D₄	C₄×C₉	1	1	yes	characteristic
103	C₂³×C₄	C₂²×C₉	1	2	yes	—
104	C₂²×C₈	C₂²×C₉	1	8	yes	complemented
105	C₄×D₄	C₄×C₉	1	4	yes	—
106	C₄×D₄	C₂²×C₉	1	4	yes	—
107	C₂×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	C₄×C₉	1	2	yes	—
109	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
111	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
112	C₂²⋊C₈	C₂²×C₉	1	8	yes	complemented
113	C₄×C₈	C₂²×C₉	1	4	yes	complemented
114	C₂³×C₃	—	4	2	no	—
115	C₂³×C₃	—	4	2	no	—
116	C₂³×C₃	C₂×C₃×C₈	1	8	yes	—
117	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
118	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₃×D₄	C₂×C₃×C₈	1	16	yes	—
120	C₂×C₃×C₄	—	2	8	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	4	4	no	—
123	C₂×C₃×C₄	—	4	2	no	—
124	C₂×C₃×C₄	—	4	1	no	—
125	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
126	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
128	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
129	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
130	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
131	C₃×C₈	—	2	8	no	—
132	C₂×C₉	—	2	2	no	—
133	C₂×C₉	—	2	8	no	—
134	C₂×C₉	#518	1	1	yes	characteristic
135	C₂×C₉	C₂³×C₈	1	1	yes	characteristic
136	C₂×C₉	#568	1	1	yes	characteristic
137	C₂⁴	C₂×C₄×C₉	1	2	yes	—
138	C₂×D₄	C₈×C₉	1	8	yes	complemented
139	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
143	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
144	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₉	1	4	yes	—
147	C₂×C₈	—	4	8	no	—
148	C₂×C₈	C₂³×C₉	1	8	yes	—
149	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
150	C₂²⋊C₄	C₂×C₄×C₉	1	8	yes	—
151	C₄²	C₂×C₄×C₉	1	2	yes	—
152	C₄²	C₂³×C₉	1	2	yes	—
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	—	4	8	no	—
158	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
159	C₂²×C₃	C₂²×C₃×C₈	1	8	yes	—
160	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
165	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
166	C₉	#2972	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³	—	4	2	no	—
168	C₂³	—	4	2	no	—
169	C₂³	C₂×C₈×C₉	1	8	yes	complemented
170	C₂³	C₂²×C₄×C₉	1	4	yes	—
171	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
172	D₄	C₂×C₈×C₉	1	16	yes	complemented
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₈×C₉	1	4	yes	complemented
179	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
181	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
182	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
183	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
184	C₈	—	2	8	no	—
185	C₂×C₃	—	2	2	no	—
186	C₂×C₃	—	2	8	no	—
187	C₂×C₃	C₃×#518	1	1	yes	characteristic
188	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
189	C₂×C₃	C₃×#568	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	8	no	—
195	C₂²	C₂²×C₈×C₉	1	2	yes	—
196	C₂²	C₂²×C₈×C₉	1	8	yes	—
197	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	4	no	—
201	C₄	C₂²×C₈×C₉	1	4	yes	—
202	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
203	C₃	C₃×#2972	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	—	2	8	no	—
206	C₂	C₉×#518	1	1	yes	characteristic
207	C₂	C₂³×C₈×C₉	1	1	yes	characteristic
208	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂	C₂³×C₈×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₂²×C₃×D₄	C₃×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×D₄×C₉	C₂	semidirect
3. C₉×#431	C₂	semidirect
4. C₂×C₉×C₂²⋊C₈	C₂	semidirect
8. C₂²×C₈×C₉	C₂²	semidirect
14. C₉×C₄⋊C₈	C₂²	semidirect
16. C₉×C₂²⋊C₈	C₂²	semidirect
17. C₄×C₈×C₉	C₂²	semidirect
23. C₂×D₄×C₉	C₈	semidirect
38. #2972	C₉	direct (N × H)
53. C₂³×C₉	C₂×C₈	semidirect
56. D₄×C₉	C₂×C₈	semidirect
62. C₂×C₄×C₉	C₂×C₈	semidirect
70. C₈×D₄	C₂×C₉	semidirect
71. #431	C₂×C₉	semidirect
72. C₂×C₂²⋊C₈	C₂×C₉	semidirect
104. C₂²×C₈	C₂²×C₉	semidirect
110. C₄⋊C₈	C₂²×C₉	semidirect
112. C₂²⋊C₈	C₂²×C₉	semidirect
113. C₄×C₈	C₂²×C₉	semidirect
138. C₂×D₄	C₈×C₉	semidirect
166. C₉	#2972	direct (N × H)
169. C₂³	C₂×C₈×C₉	semidirect
172. D₄	C₂×C₈×C₉	semidirect
178. C₂×C₄	C₂×C₈×C₉	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
C₈×D₄×C₉	2
C₉×#431	2
C₂×C₉×C₂²⋊C₈	2
C₃×#2972	3

Subgroups — C9×#2972 #11767693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2972 #11767693