Subgroups — C₂×C₉×#492 #11767864

Order 1152.

984 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
4	C₉×#492	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#492	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂²×D₄×C₉	—	2	2	no	—
9	C₂²×D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₉×SD₁₆	—	2	16	no	—
11	C₂×C₉×SD₁₆	C₂²	1	16	yes	complemented
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₉×C₄⋊Q₈	—	2	2	no	—
14	C₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
15	C₉×#127	—	2	2	no	—
16	C₉×#127	C₂²	1	4	yes	—
17	C₂×C₉×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₉	C₂²	1	4	yes	complemented
20	C₂²×C₃×SD₁₆	C₂×C₃	1	4	yes	—
21	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
23	C₃×#492	C₂×C₃	1	8	yes	—
24	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
25	C₂⁴×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	8	no	—
28	C₂×Q₈×C₉	C₂³	1	4	yes	—
29	C₂×D₄×C₉	—	2	8	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	8	no	—
32	C₂×D₄×C₉	—	4	8	no	—
33	C₂×D₄×C₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	D₄	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₉×SD₁₆	—	2	32	no	—
38	C₂×C₈×C₉	D₄	1	8	yes	complemented
39	C₂×C₈×C₉	C₂³	1	4	yes	—
40	C₉×C₄⋊C₄	—	2	8	no	—
41	C₄²×C₉	D₄	1	2	yes	—
42	C₄²×C₉	C₂³	1	2	yes	—
43	C₂×#492	C₉	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
45	C₂²×C₃×D₄	—	2	2	no	—
46	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×SD₁₆	—	2	16	no	—
48	C₂×C₃×SD₁₆	C₂²×C₃	1	16	yes	—
49	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
50	C₃×C₄⋊Q₈	—	2	2	no	—
51	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
52	C₃×#127	—	2	2	no	—
53	C₃×#127	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₄⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
57	C₂³×C₉	—	2	4	no	—
58	C₂³×C₉	—	4	4	no	—
59	C₂³×C₉	—	4	8	no	—
60	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₉	—	2	8	no	—
62	Q₈×C₉	—	4	4	no	—
63	D₄×C₉	—	2	8	no	—
64	D₄×C₉	—	4	4	no	—
65	D₄×C₉	—	4	16	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	4	4	no	—
68	C₂×C₄×C₉	SD₁₆	1	8	yes	complemented
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
73	C₈×C₉	C₂×D₄	1	8	yes	complemented
74	C₂²×SD₁₆	C₂×C₉	1	4	yes	complemented
75	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
76	C₂×#127	C₂×C₉	1	1	yes	characteristic
77	#492	C₂×C₉	1	8	yes	direct product factor
78	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
79	C₂⁴×C₃	—	2	2	no	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×Q₈	—	2	8	no	—
82	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
83	C₂×C₃×D₄	—	2	8	no	—
84	C₂×C₃×D₄	—	2	2	no	—
85	C₂×C₃×D₄	—	2	8	no	—
86	C₂×C₃×D₄	—	4	8	no	—
87	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₃×SD₁₆	—	2	32	no	—
92	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
93	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
94	C₃×C₄⋊C₄	—	2	8	no	—
95	C₃×C₄²	C₃×D₄	1	2	yes	—
96	C₃×C₄²	C₂³×C₃	1	2	yes	—
97	C₂²×C₉	—	4	4	no	—
98	C₂²×C₉	—	4	8	no	—
99	C₂²×C₉	—	8	8	no	—
100	C₂²×C₉	#127	1	2	yes	—
101	C₂²×C₉	C₂×SD₁₆	1	4	yes	—
102	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₉	—	4	4	no	—
104	C₄×C₉	#127	1	2	yes	—
105	C₄×C₉	C₂×SD₁₆	1	8	yes	complemented
106	C₄×C₉	C₂²×D₄	1	2	yes	—
107	C₂²×Q₈	C₂²×C₉	1	2	yes	—
108	C₂²×D₄	—	2	2	no	—
109	C₂²×D₄	C₂²×C₉	1	2	yes	—
110	C₂×SD₁₆	—	2	16	no	—
111	C₂×SD₁₆	C₂²×C₉	1	16	yes	complemented
112	C₂²×C₈	C₂²×C₉	1	2	yes	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂²×C₉	1	4	yes	complemented
115	#127	—	2	2	no	—
116	#127	C₂²×C₉	1	4	yes	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₉	1	4	yes	complemented
120	C₂³×C₃	—	2	4	no	—
121	C₂³×C₃	—	4	4	no	—
122	C₂³×C₃	—	4	8	no	—
123	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
124	C₃×Q₈	—	2	8	no	—
125	C₃×Q₈	—	4	4	no	—
126	C₃×D₄	—	2	8	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	4	16	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	4	4	no	—
131	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	—
132	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
134	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
135	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
136	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
137	C₂×C₉	—	8	4	no	—
138	C₂×C₉	#492	1	4	yes	direct product factor
139	C₂×C₉	C₂×#127	1	1	yes	characteristic
140	C₂×C₉	C₂²×SD₁₆	1	2	yes	—
141	C₂⁴	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	8	no	—
144	C₂×Q₈	C₂³×C₉	1	4	yes	—
145	C₂×D₄	—	2	8	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	4	8	no	—
149	C₂×D₄	C₂³×C₉	1	4	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	D₄×C₉	1	2	yes	—
152	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
153	SD₁₆	—	2	32	no	—
154	C₂×C₈	D₄×C₉	1	8	yes	complemented
155	C₂×C₈	C₂³×C₉	1	4	yes	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄²	D₄×C₉	1	2	yes	—
158	C₄²	C₂³×C₉	1	2	yes	—
159	C₂²×C₃	—	4	8	no	—
160	C₂²×C₃	—	4	4	no	—
161	C₂²×C₃	—	8	8	no	—
162	C₂²×C₃	C₃×#127	1	2	yes	—
163	C₂²×C₃	C₂×C₃×SD₁₆	1	4	yes	—
164	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
165	C₃×C₄	—	4	4	no	—
166	C₃×C₄	C₃×#127	1	2	yes	—
167	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	—
168	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
169	C₉	C₂×#492	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³	—	2	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	4	8	no	—
173	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	4	4	no	—
176	D₄	—	2	8	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	16	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₉×SD₁₆	1	8	yes	complemented
182	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
184	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
185	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
186	C₈	C₂×D₄×C₉	1	8	yes	complemented
187	C₂×C₃	—	8	4	no	—
188	C₂×C₃	C₃×#492	1	4	yes	—
189	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
190	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
191	C₂²	—	4	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	—	8	8	no	—
194	C₂²	C₉×#127	1	2	yes	—
195	C₂²	C₂×C₉×SD₁₆	1	4	yes	—
196	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
197	C₄	—	4	4	no	—
198	C₄	C₉×#127	1	2	yes	—
199	C₄	C₂×C₉×SD₁₆	1	8	yes	complemented
200	C₄	C₂²×D₄×C₉	1	2	yes	—
201	C₃	C₂×C₃×#492	1	1	yes	characteristic
202	C₂	—	8	4	no	—
203	C₂	C₉×#492	1	4	yes	direct product factor
204	C₂	C₂×C₉×#127	1	1	yes	characteristic
205	C₂	C₂²×C₉×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#127	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×SD₁₆	C₂	semidirect
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₉×#492	C₂	direct (N × H)
5. C₂×C₄×C₈×C₉	C₂	semidirect
11. C₂×C₉×SD₁₆	C₂²	semidirect
14. C₉×C₄⋊Q₈	C₂²	semidirect
19. C₄×C₈×C₉	C₂²	semidirect
38. C₂×C₈×C₉	D₄	semidirect
43. C₂×#492	C₉	direct (N × H)
68. C₂×C₄×C₉	SD₁₆	semidirect
73. C₈×C₉	C₂×D₄	semidirect
74. C₂²×SD₁₆	C₂×C₉	semidirect
75. C₂×C₄⋊Q₈	C₂×C₉	semidirect
77. #492	C₂×C₉	direct (N × H)
78. C₂×C₄×C₈	C₂×C₉	semidirect
105. C₄×C₉	C₂×SD₁₆	semidirect
111. C₂×SD₁₆	C₂²×C₉	semidirect
114. C₄⋊Q₈	C₂²×C₉	semidirect
119. C₄×C₈	C₂²×C₉	semidirect
138. C₂×C₉	#492	direct (N × H)
154. C₂×C₈	D₄×C₉	semidirect
169. C₉	C₂×#492	direct (N × H)
181. C₂×C₄	C₉×SD₁₆	semidirect
186. C₈	C₂×D₄×C₉	semidirect
199. C₄	C₂×C₉×SD₁₆	semidirect
203. C₂	C₉×#492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×SD₁₆	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#127	2
C₉×#492	2
C₂×C₄×C₈×C₉	2
C₂×C₃×#492	3

Subgroups — C2×C9×#492 #11767864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#492 #11767864