Subgroups — C₂×C₉×#237 #11722611

Order 864.

546 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#67	C₂	1	2	yes	maximal, complemented
3	C₂²×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₉×#237	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#237	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	—	3	1	no	maximal
9	C₂³×C₃×C₉	C₂²	1	2	yes	—
10	C₂²×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₃×D₄×C₉	C₂²	1	4	yes	complemented
12	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₉×#67	C₂²	1	8	yes	complemented
14	C₂×C₉×Dic₃	C₂²	1	2	yes	—
15	C₂×C₉×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₉×Dic₃	C₂²	1	8	yes	complemented
17	C₄×D₃×C₉	C₂²	1	4	yes	complemented
18	C₉×Dic₆	C₂²	1	4	yes	complemented
19	C₂×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#67	C₂×C₃	1	2	yes	—
21	C₂²×C₃×Dic₃	C₂×C₃	1	2	yes	—
22	C₃×#237	C₂×C₃	1	8	yes	—
23	C₂×C₃×C₄×D₃	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
25	C₉×#41	—	3	8	no	—
26	C₂×Q₈×C₉	—	3	1	no	—
27	C₂×D₄×C₉	—	2	1	no	—
28	C₂×D₄×C₉	—	3	2	no	—
29	C₂×D₄×C₉	D₃	1	1	yes	characteristic, complemented
30	C₂²×C₄×C₉	—	3	2	no	—
31	C₂²×C₄×C₉	—	3	1	no	—
32	C₂²×C₃×C₉	—	2	4	no	—
33	C₂²×C₃×C₉	C₂³	1	4	yes	—
34	C₂²×C₃×C₉	C₂³	1	1	yes	characteristic
35	C₂×D₃×C₉	—	2	2	no	—
36	C₂×D₃×C₉	C₂³	1	2	yes	—
37	C₃×C₄×C₉	C₂³	1	2	yes	—
38	C₉×Dic₃	C₂³	1	4	yes	—
39	C₉×Dic₃	C₂³	1	2	yes	—
40	C₂×C₃×#41	—	3	1	no	—
41	C₂×#237	C₉	1	1	yes	characteristic, direct product factor
42	C₂³×C₃²	C₂²×C₃	1	2	yes	—
43	C₂²×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
44	C₃²×D₄	C₂²×C₃	1	4	yes	—
45	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₃×#67	C₂²×C₃	1	8	yes	—
47	C₂×C₃×Dic₃	C₂²×C₃	1	2	yes	—
48	C₂×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×Dic₃	C₂²×C₃	1	8	yes	—
50	C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
51	C₃×Dic₆	C₂²×C₃	1	4	yes	—
52	C₂³×C₉	—	2	2	no	—
53	C₂³×C₉	—	3	1	no	—
54	C₂³×C₉	C₂×D₃	1	2	yes	—
55	Q₈×C₉	—	3	4	no	—
56	D₄×C₉	—	2	4	no	—
57	D₄×C₉	—	3	8	no	—
58	D₄×C₉	C₂×D₃	1	4	yes	complemented
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	3	8	no	—
61	C₂×C₄×C₉	—	3	1	no	—
62	C₂×C₄×C₉	—	3	2	no	—
63	C₂×C₄×C₉	—	3	4	no	—
64	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₉	—	2	4	no	—
66	C₂×C₃×C₉	#41	1	2	yes	complemented
67	C₂×C₃×C₉	C₂⁴	1	1	yes	characteristic
68	D₃×C₉	—	2	2	no	—
69	C₃×#41	—	3	8	no	—
70	C₂×C₃×Q₈	—	3	1	no	—
71	C₂×C₃×D₄	—	2	1	no	—
72	C₂×C₃×D₄	—	3	2	no	—
73	C₂×C₃×D₄	C₂×C₉	1	1	yes	characteristic, complemented
74	C₂×C₃×D₄	C₃×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	3	2	no	—
76	C₂²×C₃×C₄	—	3	1	no	—
77	C₂×#67	C₂×C₉	1	2	yes	complemented
78	C₂²×Dic₃	C₂×C₉	1	2	yes	complemented
79	#237	C₂×C₉	1	8	yes	direct product factor
80	C₂×C₄×D₃	C₂×C₉	1	1	yes	characteristic, complemented
81	C₂×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
82	C₂²×C₃²	—	2	4	no	—
83	C₂²×C₃²	C₂³×C₃	1	4	yes	—
84	C₂²×C₃²	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₃×D₃	—	2	2	no	—
86	C₂×C₃×D₃	C₂³×C₃	1	2	yes	—
87	C₃²×C₄	C₂³×C₃	1	2	yes	—
88	C₃×Dic₃	C₂³×C₃	1	4	yes	—
89	C₃×Dic₃	C₂³×C₃	1	2	yes	—
90	C₂²×C₉	—	2	4	no	—
91	C₂²×C₉	—	2	1	no	—
92	C₂²×C₉	—	2	4	no	—
93	C₂²×C₉	—	3	2	no	—
94	C₂²×C₉	—	4	4	no	—
95	C₂²×C₉	—	6	2	no	—
96	C₂²×C₉	C₂²×D₃	1	4	yes	—
97	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
98	C₄×C₉	—	2	2	no	—
99	C₄×C₉	—	3	2	no	—
100	C₄×C₉	—	3	4	no	—
101	C₄×C₉	C₂²×D₃	1	2	yes	—
102	C₂×#41	—	3	1	no	Sylow
103	C₃×C₉	C₂×#41	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	3	1	no	—
106	C₂³×C₃	C₂²×C₉	1	2	yes	—
107	C₂³×C₃	C₂×C₃×D₃	1	2	yes	—
108	C₂²×D₃	C₂²×C₉	1	1	yes	characteristic
109	C₃×Q₈	—	3	4	no	—
110	C₃×D₄	—	2	4	no	—
111	C₃×D₄	—	3	8	no	—
112	C₃×D₄	C₂²×C₉	1	4	yes	complemented
113	C₃×D₄	C₂×C₃×D₃	1	4	yes	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	3	1	no	—
116	C₂×C₃×C₄	—	3	2	no	—
117	C₂×C₃×C₄	—	3	8	no	—
118	C₂×C₃×C₄	—	3	4	no	—
119	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
121	#67	C₂²×C₉	1	8	yes	complemented
122	C₂×Dic₃	C₂²×C₉	1	2	yes	—
123	C₂×Dic₃	C₂²×C₉	1	1	yes	characteristic
124	C₂×Dic₃	C₂²×C₉	1	8	yes	complemented
125	C₄×D₃	C₂²×C₉	1	4	yes	complemented
126	Dic₆	C₂²×C₉	1	4	yes	complemented
127	C₂×C₃²	—	2	4	no	—
128	C₂×C₃²	C₃×#41	1	2	yes	—
129	C₂×C₃²	C₂⁴×C₃	1	1	yes	characteristic
130	C₃×D₃	—	2	2	no	—
131	C₂×C₉	—	2	4	no	—
132	C₂×C₉	—	2	1	no	—
133	C₂×C₉	—	2	2	no	—
134	C₂×C₉	—	4	4	no	—
135	C₂×C₉	—	6	2	no	—
136	C₂×C₉	#237	1	2	yes	direct product factor
137	C₂×C₉	C₂³×D₃	1	1	yes	characteristic
138	#41	—	3	8	no	—
139	C₂×Q₈	—	3	1	no	—
140	C₂×D₄	—	3	2	no	—
141	C₂×D₄	D₃×C₉	1	1	yes	characteristic, complemented
142	C₂²×C₄	—	3	2	no	—
143	C₂²×C₄	—	3	1	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	—	2	1	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	3	2	no	—
149	C₂²×C₃	—	4	4	no	—
150	C₂²×C₃	—	6	2	no	—
151	C₂²×C₃	C₂³×C₉	1	4	yes	—
152	C₂²×C₃	C₂³×C₉	1	1	yes	characteristic
153	C₂²×C₃	C₂²×C₃×D₃	1	4	yes	—
154	C₂²×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
155	C₂×D₃	—	2	2	no	—
156	C₂×D₃	C₂³×C₉	1	2	yes	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	3	2	no	—
159	C₃×C₄	—	3	4	no	—
160	C₃×C₄	C₂³×C₉	1	2	yes	—
161	C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
162	Dic₃	C₂³×C₉	1	4	yes	—
163	Dic₃	C₂³×C₉	1	2	yes	—
164	C₃²	C₂×C₃×#41	1	1	yes	characteristic
165	C₉	—	2	1	no	—
166	C₉	C₂×#237	1	1	yes	characteristic, direct product factor
167	C₂³	—	3	1	no	—
168	C₂³	C₂×D₃×C₉	1	2	yes	—
169	Q₈	—	3	4	no	—
170	D₄	—	3	8	no	—
171	D₄	C₂×D₃×C₉	1	4	yes	complemented
172	C₂×C₄	—	3	1	no	—
173	C₂×C₄	—	3	2	no	—
174	C₂×C₄	—	3	8	no	—
175	C₂×C₄	—	3	4	no	—
176	C₂×C₄	C₂×D₃×C₉	1	1	yes	characteristic
177	C₂×C₃	—	2	4	no	—
178	C₂×C₃	—	2	4	no	—
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	—	4	4	no	—
182	C₂×C₃	—	6	2	no	—
183	C₂×C₃	C₉×#41	1	2	yes	complemented
184	C₂×C₃	C₂⁴×C₉	1	1	yes	characteristic
185	C₂×C₃	C₃×#237	1	2	yes	—
186	C₂×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
187	D₃	—	2	2	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	3	2	no	—
190	C₂²	—	6	2	no	—
191	C₂²	C₂²×D₃×C₉	1	4	yes	—
192	C₂²	C₂²×D₃×C₉	1	1	yes	characteristic
193	C₄	—	3	2	no	—
194	C₄	—	3	4	no	—
195	C₄	C₂²×D₃×C₉	1	2	yes	—
196	C₃	—	2	1	no	—
197	C₃	C₂×C₉×#41	1	1	yes	characteristic, complemented
198	C₃	C₂×C₃×#237	1	1	yes	characteristic
199	C₂	—	2	4	no	—
200	C₂	—	6	2	no	—
201	C₂	C₉×#237	1	2	yes	direct product factor
202	C₂	C₂³×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₃
commutator	C₂×C₃	C₂⁴×C₉
Frattini	C₂×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×D₄×C₉	C₂
socle	C₂²×C₃²	C₂³×C₃
prime radical	C₂×C₃×#237	C₃
cosocle	C₂×C₃²	C₂⁴×C₃
hypercenter	C₂×D₄×C₉	D₃
hypocenter	C₃	C₂×C₉×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×D₄×C₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄×C₉	C₂	semidirect
2. C₂×C₉×#67	C₂	semidirect
3. C₂²×C₉×Dic₃	C₂	semidirect
4. C₉×#237	C₂	direct (N × H)
5. C₂×C₄×D₃×C₉	C₂	semidirect
6. C₂×C₉×Dic₆	C₂	semidirect
11. C₃×D₄×C₉	C₂²	semidirect
13. C₉×#67	C₂²	semidirect
16. C₂×C₉×Dic₃	C₂²	semidirect
17. C₄×D₃×C₉	C₂²	semidirect
18. C₉×Dic₆	C₂²	semidirect
29. C₂×D₄×C₉	D₃	semidirect
41. C₂×#237	C₉	direct (N × H)
58. D₄×C₉	C₂×D₃	semidirect
66. C₂×C₃×C₉	#41	semidirect
73. C₂×C₃×D₄	C₂×C₉	semidirect
77. C₂×#67	C₂×C₉	semidirect
78. C₂²×Dic₃	C₂×C₉	semidirect
79. #237	C₂×C₉	direct (N × H)
80. C₂×C₄×D₃	C₂×C₉	semidirect
81. C₂×Dic₆	C₂×C₉	semidirect
103. C₃×C₉	C₂×#41	semidirect
112. C₃×D₄	C₂²×C₉	semidirect
121. #67	C₂²×C₉	semidirect
124. C₂×Dic₃	C₂²×C₉	semidirect
125. C₄×D₃	C₂²×C₉	semidirect
126. Dic₆	C₂²×C₉	semidirect
136. C₂×C₉	#237	direct (N × H)
141. C₂×D₄	D₃×C₉	semidirect
166. C₉	C₂×#237	direct (N × H)
171. D₄	C₂×D₃×C₉	semidirect
183. C₂×C₃	C₉×#41	semidirect
197. C₃	C₂×C₉×#41	semidirect
201. C₂	C₉×#237	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×C₃×D₄×C₉	2
C₂×C₉×#67	2
C₂²×C₉×Dic₃	2
C₉×#237	2
C₂×C₄×D₃×C₉	2
C₂×C₉×Dic₆	2
C₂×C₃×#237	3
C₂×C₉×#41	3

Subgroups — C2×C9×#237 #11722611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#237 #11722611