Subgroups — C₄×C₉×#67 #11721588

Order 864.

336 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#67	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×#67	C₃	1	1	yes	maximal, characteristic
9	C₄×D₄×C₉	—	3	1	no	maximal
10	C₂³×C₃×C₉	C₂²	1	1	yes	characteristic
11	C₂²×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₉	—	2	2	no	—
13	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₉×#67	C₄	1	4	yes	direct product factor
16	C₂×C₉×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₉×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₉×Dic₃	C₂²	1	1	yes	characteristic
19	C₄×D₃×C₉	—	2	2	no	—
20	C₂²×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#67	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×D₃	C₂×C₃	1	1	yes	characteristic
23	C₃×#217	C₂×C₃	1	1	yes	characteristic
24	C₃×(C₂×D₃)⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₃×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₂×D₄×C₉	—	3	1	no	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	—	3	1	no	—
30	C₂²×C₄×C₉	D₃	1	1	yes	characteristic, complemented
31	C₉×C₄⋊C₄	—	3	1	no	—
32	C₉×C₂²⋊C₄	—	3	1	no	—
33	C₉×C₂²⋊C₄	—	3	1	no	—
34	C₄²×C₉	—	3	1	no	—
35	C₂²×C₃×C₉	—	2	1	no	—
36	C₂²×C₃×C₉	—	2	1	no	—
37	C₂²×C₃×C₉	C₂×C₄	1	2	yes	complemented
38	C₂²×C₃×C₉	C₂³	1	1	yes	characteristic
39	C₂×D₃×C₉	—	2	1	no	—
40	C₂×D₃×C₉	—	2	1	no	—
41	C₂×D₃×C₉	C₂×C₄	1	2	yes	complemented
42	C₃×C₄×C₉	—	2	1	no	—
43	C₃×C₄×C₉	D₄	1	2	yes	complemented
44	C₉×Dic₃	—	2	1	no	—
45	C₉×Dic₃	—	2	1	no	—
46	C₉×Dic₃	C₂×C₄	1	2	yes	complemented
47	C₃×C₄×D₄	—	3	1	no	—
48	C₄×#67	C₉	1	1	yes	characteristic, direct product factor
49	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	2	no	—
52	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₃×#67	C₃×C₄	1	4	yes	—
55	C₂×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄×D₃	—	2	2	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	—	3	1	no	—
61	C₂³×C₉	C₂×D₃	1	1	yes	characteristic
62	D₄×C₉	—	3	4	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	1	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	3	1	no	—
68	C₂×C₄×C₉	—	3	1	no	—
69	C₂×C₄×C₉	—	3	1	no	—
70	C₂×C₄×C₉	—	6	2	no	—
71	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×C₉	—	2	2	no	—
74	C₂×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₉	#41	1	1	yes	characteristic
77	D₃×C₉	—	2	2	no	—
78	C₂×C₃×D₄	—	3	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	3	1	no	—
81	C₂²×C₃×C₄	C₂×C₉	1	1	yes	characteristic, complemented
82	C₂²×C₃×C₄	C₃×D₃	1	1	yes	characteristic
83	C₂×#67	C₂×C₉	1	1	yes	characteristic
84	C₂×C₄×D₃	C₂×C₉	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₄	—	3	1	no	—
86	C₃×C₂²⋊C₄	—	3	1	no	—
87	C₃×C₂²⋊C₄	—	3	1	no	—
88	C₃×C₄²	—	3	1	no	—
89	#217	C₂×C₉	1	1	yes	characteristic, complemented
90	(C₂×D₃)⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
91	Dic₃⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
92	C₄×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
93	C₂²×C₃²	—	2	1	no	—
94	C₂²×C₃²	—	2	1	no	—
95	C₂²×C₃²	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₃²	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃×D₃	—	2	1	no	—
98	C₂×C₃×D₃	—	2	1	no	—
99	C₂×C₃×D₃	C₂×C₃×C₄	1	2	yes	—
100	C₃²×C₄	—	2	1	no	—
101	C₃²×C₄	C₃×D₄	1	2	yes	—
102	C₃×Dic₃	—	2	1	no	—
103	C₃×Dic₃	—	2	1	no	—
104	C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
105	C₂²×C₉	—	2	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	1	no	—
111	C₂²×C₉	—	3	2	no	—
112	C₂²×C₉	—	6	1	no	—
113	C₂²×C₉	—	6	1	no	—
114	C₂²×C₉	C₄×D₃	1	2	yes	complemented
115	C₂²×C₉	C₂²×D₃	1	1	yes	characteristic
116	C₄×C₉	—	2	2	no	—
117	C₄×C₉	—	2	1	no	—
118	C₄×C₉	—	2	2	no	—
119	C₄×C₉	—	3	2	no	—
120	C₄×C₉	—	6	1	no	—
121	C₄×C₉	—	6	1	no	—
122	C₄×C₉	#67	1	2	yes	direct product factor
123	C₄×D₄	—	3	1	no	Sylow
124	C₃×C₉	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	3	1	no	—
127	C₂³×C₃	C₂²×C₉	1	1	yes	characteristic
128	C₂³×C₃	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂²×D₃	C₂²×C₉	1	1	yes	characteristic
130	C₃×D₄	—	3	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	3	1	no	—
137	C₂×C₃×C₄	—	3	1	no	—
138	C₂×C₃×C₄	—	3	1	no	—
139	C₂×C₃×C₄	—	6	2	no	—
140	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
144	#67	C₄×C₉	1	4	yes	direct product factor
145	C₂×Dic₃	C₂²×C₉	1	1	yes	characteristic
146	C₂×Dic₃	C₂²×C₉	1	1	yes	characteristic
147	C₂×Dic₃	C₂²×C₉	1	1	yes	characteristic
148	C₄×D₃	—	2	2	no	—
149	C₂×C₃²	—	2	2	no	—
150	C₂×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₂×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₃²	C₃×#41	1	1	yes	characteristic
153	C₃×D₃	—	2	2	no	—
154	C₂×C₉	—	2	4	no	—
155	C₂×C₉	—	2	2	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	—	2	1	no	—
158	C₂×C₉	—	2	1	no	—
159	C₂×C₉	—	6	2	no	—
160	C₂×C₉	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₉	#235	1	1	yes	characteristic
162	C₂×C₉	C₂×#67	1	1	yes	characteristic
163	C₂×D₄	—	3	1	no	—
164	C₂²×C₄	—	3	1	no	—
165	C₂²×C₄	D₃×C₉	1	1	yes	characteristic, complemented
166	C₄⋊C₄	—	3	1	no	—
167	C₂²⋊C₄	—	3	1	no	—
168	C₂²⋊C₄	—	3	1	no	—
169	C₄²	—	3	1	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	3	2	no	—
179	C₂²×C₃	—	6	1	no	—
180	C₂²×C₃	—	6	1	no	—
181	C₂²×C₃	C₂×C₄×C₉	1	2	yes	complemented
182	C₂²×C₃	C₂³×C₉	1	1	yes	characteristic
183	C₂²×C₃	C₃×C₄×D₃	1	2	yes	—
184	C₂²×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₂×D₃	—	2	1	no	—
186	C₂×D₃	—	2	1	no	—
187	C₂×D₃	C₂×C₄×C₉	1	2	yes	complemented
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	3	2	no	—
193	C₃×C₄	—	6	1	no	—
194	C₃×C₄	—	6	1	no	—
195	C₃×C₄	D₄×C₉	1	2	yes	complemented
196	C₃×C₄	C₃×#67	1	2	yes	—
197	Dic₃	—	2	1	no	—
198	Dic₃	—	2	1	no	—
199	Dic₃	C₂×C₄×C₉	1	2	yes	complemented
200	C₃²	C₃×C₄×D₄	1	1	yes	characteristic
201	C₉	—	2	1	no	—
202	C₉	C₄×#67	1	1	yes	characteristic, direct product factor
203	C₂³	—	3	1	no	—
204	C₂³	C₂×D₃×C₉	1	1	yes	characteristic
205	D₄	—	3	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	3	1	no	—
208	C₂×C₄	—	3	1	no	—
209	C₂×C₄	—	3	1	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	C₂×D₃×C₉	1	1	yes	characteristic
212	C₂×C₄	C₂×D₃×C₉	1	1	yes	characteristic
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	6	2	no	—
220	C₂×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
221	C₂×C₃	C₂×D₄×C₉	1	1	yes	characteristic
222	C₂×C₃	C₉×#41	1	1	yes	characteristic
223	C₂×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
224	C₂×C₃	C₃×#235	1	1	yes	characteristic
225	C₂×C₃	C₂×C₃×#67	1	1	yes	characteristic
226	D₃	—	2	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	3	2	no	—
230	C₂²	—	6	1	no	—
231	C₂²	—	6	1	no	—
232	C₂²	C₄×D₃×C₉	1	2	yes	complemented
233	C₂²	C₂²×D₃×C₉	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	3	2	no	—
236	C₄	—	6	1	no	—
237	C₄	—	6	1	no	—
238	C₄	C₉×#67	1	2	yes	direct product factor
239	C₃	—	2	1	no	—
240	C₃	C₄×D₄×C₉	1	1	yes	characteristic, complemented
241	C₃	C₃×C₄×#67	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	6	2	no	—
244	C₂	C₂×C₄×D₃×C₉	1	1	yes	characteristic
245	C₂	C₉×#235	1	1	yes	characteristic
246	C₂	C₂×C₉×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₉
Frattini	C₂²×C₃	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₄×C₉	C₂
socle, cosocle	C₂²×C₃²	C₂³×C₃
prime radical	C₂×C₃×#67	C₂×C₃
hypercenter	C₂²×C₄×C₉	D₃
hypocenter	C₃	C₄×D₄×C₉
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄×C₉	C₂	semidirect
3. C₂×C₄×D₃×C₉	C₂	semidirect
4. C₉×#217	C₂	semidirect
5. C₉×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₉×Dic₃⋊C₄	C₂	semidirect
7. C₄×C₉×Dic₃	C₂	semidirect
15. C₉×#67	C₄	direct (N × H)
30. C₂²×C₄×C₉	D₃	semidirect
37. C₂²×C₃×C₉	C₂×C₄	semidirect
41. C₂×D₃×C₉	C₂×C₄	semidirect
43. C₃×C₄×C₉	D₄	semidirect
46. C₉×Dic₃	C₂×C₄	semidirect
48. C₄×#67	C₉	direct (N × H)
81. C₂²×C₃×C₄	C₂×C₉	semidirect
84. C₂×C₄×D₃	C₂×C₉	semidirect
89. #217	C₂×C₉	semidirect
90. (C₂×D₃)⋊C₄	C₂×C₉	semidirect
91. Dic₃⋊C₄	C₂×C₉	semidirect
92. C₄×Dic₃	C₂×C₉	semidirect
114. C₂²×C₉	C₄×D₃	semidirect
122. C₄×C₉	#67	direct (N × H)
124. C₃×C₉	C₄×D₄	semidirect
144. #67	C₄×C₉	direct (N × H)
165. C₂²×C₄	D₃×C₉	semidirect
181. C₂²×C₃	C₂×C₄×C₉	semidirect
187. C₂×D₃	C₂×C₄×C₉	semidirect
195. C₃×C₄	D₄×C₉	semidirect
199. Dic₃	C₂×C₄×C₉	semidirect
202. C₉	C₄×#67	direct (N × H)
232. C₂²	C₄×D₃×C₉	semidirect
238. C₄	C₉×#67	direct (N × H)
240. C₃	C₄×D₄×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₉	2
C₂×C₉×#67	2
C₂×C₄×D₃×C₉	2
C₉×#217	2
C₉×(C₂×D₃)⋊C₄	2
C₉×Dic₃⋊C₄	2
C₄×C₉×Dic₃	2
C₃×C₄×#67	3
C₄×D₄×C₉	3

Subgroups — C4×C9×#67 #11721588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×#67 #11721588