Subgroups — C₄×C₉×#208 #13142373

Order 1728.

534 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	2	yes	maximal
3	C₉×#803	C₂	1	2	yes	maximal, complemented
4	C₄×C₉×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#208	C₃	1	1	yes	maximal, characteristic
6	C₄×C₉×M₁₆	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×#208	C₄	1	8	yes	direct product factor
13	C₂×C₉×C₃⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃×#208	C₂×C₃	1	2	yes	—
16	C₃×#803	C₂×C₃	1	2	yes	—
17	C₃×C₄×C₃⋊C₈	C₂×C₃	1	2	yes	—
18	C₂×C₉×M₁₆	—	3	2	no	—
19	C₂×C₄²×C₉	—	2	1	no	—
20	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
21	C₉×#97	—	3	2	no	—
22	C₄×C₈×C₉	—	3	2	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₉×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
31	C₃×C₄×M₁₆	—	3	1	no	—
32	C₄×#208	C₉	1	1	yes	characteristic, direct product factor
33	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
34	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₃²×C₄²	C₃×C₄	1	2	yes	—
36	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
37	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
38	C₃×#208	C₃×C₄	1	8	yes	—
39	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	Dic₃	1	2	yes	—
43	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
44	C₉×M₁₆	—	3	8	no	—
45	C₂×C₈×C₉	—	3	4	no	—
46	C₄²×C₉	—	2	2	no	—
47	C₄²×C₉	—	2	1	no	—
48	C₄²×C₉	—	2	1	no	—
49	C₄²×C₉	Dic₃	1	2	yes	—
50	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
51	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃×C₉	—	2	2	no	—
53	C₂²×C₃×C₉	C₄²	1	2	yes	—
54	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₄×C₉	—	2	2	no	—
56	C₃×C₄×C₉	C₄²	1	2	yes	—
57	C₃×C₄×C₉	M₁₆	1	4	yes	complemented
58	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
59	C₂×C₃×M₁₆	—	3	2	no	—
60	C₂×C₃×C₄²	—	2	1	no	—
61	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
63	C₂×#208	C₂×C₉	1	2	yes	—
64	C₃×#97	—	3	2	no	—
65	C₃×C₄×C₈	—	3	2	no	—
66	#803	C₂×C₉	1	2	yes	complemented
67	C₄×C₃⋊C₈	C₂×C₉	1	2	yes	complemented
68	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	—	2	4	no	—
70	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
77	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
78	C₂×C₄×C₉	—	2	8	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
87	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
88	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₈×C₉	—	3	8	no	—
91	C₄×M₁₆	—	3	1	no	Sylow
92	C₂×C₃×C₉	—	2	2	no	—
93	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
98	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
100	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
101	C₃×M₁₆	—	3	8	no	—
102	C₂×C₃×C₈	—	3	4	no	—
103	C₃×C₄²	—	2	2	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	C₄×C₉	1	2	yes	—
107	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
108	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
109	C₃×C₄²	C₃×Dic₃	1	2	yes	—
110	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
112	#208	C₄×C₉	1	8	yes	direct product factor
113	C₂×C₃⋊C₈	C₂²×C₉	1	4	yes	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	C₃×C₄²	1	2	yes	—
116	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	C₃×C₄²	1	2	yes	—
119	C₃²×C₄	C₃×M₁₆	1	4	yes	—
120	C₃²×C₄	C₂²×C₃×C₄	1	2	yes	—
121	C₂²×C₉	—	2	4	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	C₄×Dic₃	1	2	yes	—
126	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
127	C₄×C₉	—	2	4	no	—
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	—	2	4	no	—
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	#208	1	4	yes	direct product factor
133	C₄×C₉	C₄×Dic₃	1	2	yes	—
134	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
135	C₂×M₁₆	—	3	2	no	—
136	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
137	#97	—	3	2	no	—
138	C₄×C₈	—	3	2	no	—
139	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
142	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	8	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
152	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
153	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
155	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
157	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
158	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
159	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
161	C₃×C₈	—	3	8	no	—
162	C₃⋊C₈	C₂×C₄×C₉	1	8	yes	complemented
163	C₂×C₃²	—	2	2	no	—
164	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
165	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
166	C₂×C₉	—	2	4	no	—
167	C₂×C₉	—	2	2	no	—
168	C₂×C₉	—	2	2	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	C₂×#208	1	2	yes	—
171	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
172	C₂²×C₄	C₉×Dic₃	1	2	yes	—
173	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
174	M₁₆	—	3	8	no	—
175	C₂×C₈	—	3	4	no	—
176	C₄²	C₉×Dic₃	1	2	yes	—
177	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
178	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	C₄²×C₉	1	2	yes	—
185	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
186	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
187	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	C₄²×C₉	1	2	yes	—
195	C₃×C₄	C₉×M₁₆	1	4	yes	complemented
196	C₃×C₄	C₂²×C₄×C₉	1	2	yes	—
197	C₃×C₄	C₃×#208	1	4	yes	—
198	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
199	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
200	C₃²	C₃×C₄×M₁₆	1	1	yes	characteristic
201	C₉	—	2	1	no	—
202	C₉	C₄×#208	1	1	yes	characteristic, direct product factor
203	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	C₄×D₃×C₉	1	4	yes	—
206	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
207	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
208	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
210	C₈	—	3	8	no	—
211	C₂×C₃	—	2	4	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
217	C₂×C₃	C₂×C₉×M₁₆	1	2	yes	—
218	C₂×C₃	C₂×C₃×#208	1	2	yes	—
219	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	C₄×C₉×Dic₃	1	2	yes	—
222	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	C₉×#208	1	4	yes	direct product factor
225	C₄	C₄×C₉×Dic₃	1	2	yes	—
226	C₄	C₂×C₄×D₃×C₉	1	2	yes	—
227	C₃	—	2	1	no	—
228	C₃	C₄×C₉×M₁₆	1	1	yes	characteristic, complemented
229	C₃	C₃×C₄×#208	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×C₉×#208	1	2	yes	—
232	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₉	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₄×C₉×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×#803	C₂	semidirect
4. C₄×C₉×C₃⋊C₈	C₂	semidirect
12. C₉×#208	C₄	direct (N × H)
30. C₉×C₃⋊C₈	C₂×C₄	semidirect
32. C₄×#208	C₉	direct (N × H)
57. C₃×C₄×C₉	M₁₆	semidirect
66. #803	C₂×C₉	semidirect
67. C₄×C₃⋊C₈	C₂×C₉	semidirect
112. #208	C₄×C₉	direct (N × H)
132. C₄×C₉	#208	direct (N × H)
139. C₃×C₉	C₄×M₁₆	semidirect
162. C₃⋊C₈	C₂×C₄×C₉	semidirect
195. C₃×C₄	C₉×M₁₆	semidirect
202. C₉	C₄×#208	direct (N × H)
224. C₄	C₉×#208	direct (N × H)
228. C₃	C₄×C₉×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₉×#208	2
C₉×#803	2
C₄×C₉×C₃⋊C₈	2
C₃×C₄×#208	3
C₄×C₉×M₁₆	3

Subgroups — C4×C9×#208 #13142373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×#208 #13142373