Subgroups — C₃×C₉×#408 #13144982

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₃²×#408	C₃	1	1	yes	maximal, characteristic
6	C₉×#408	C₃	1	3	yes	maximal, direct product factor
7	C₃×C₉×#41	—	2	2	no	—
8	C₃×C₉×#41	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
10	C₂×C₃×D₄×C₉	C₄	1	2	yes	—
11	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×M₁₆	—	2	2	no	—
15	C₂×C₃×C₈×C₉	—	2	2	no	—
16	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
17	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
18	C₂×C₃²×#41	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×#41	C₂×C₃	1	3	yes	—
22	C₃²×C₂²⋊C₈	C₂×C₃	1	4	yes	—
23	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	complemented
24	C₂²×C₈×C₉	C₂×C₃	1	3	yes	complemented
25	C₉×C₂²⋊C₈	C₂×C₃	1	12	yes	complemented
26	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
27	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×D₄×C₉	—	2	4	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
36	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃×#408	C₉	1	3	yes	direct product factor
41	C₃×#408	C₃²	1	1	yes	characteristic
42	C₃²×#41	—	2	2	no	—
43	C₃²×#41	—	2	2	no	—
44	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
45	C₂×C₃²×D₄	C₃×C₄	1	2	yes	—
46	C₂×C₃²×D₄	C₃×C₄	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₃²×M₁₆	—	2	2	no	—
50	C₂×C₃²×C₈	—	2	2	no	—
51	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
53	C₉×#41	—	2	6	no	—
54	C₉×#41	—	2	6	no	—
55	C₂×Q₈×C₉	C₃×C₄	1	3	yes	—
56	C₂×D₄×C₉	C₃×C₄	1	6	yes	—
57	C₂×D₄×C₉	C₃×C₄	1	3	yes	—
58	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
59	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
60	C₉×M₁₆	—	2	6	no	—
61	C₂×C₈×C₉	—	2	6	no	—
62	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
63	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	—	2	2	no	—
66	C₂²×C₃×C₉	—	2	4	no	—
67	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	—	2	2	no	—
70	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
71	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×#41	C₂×C₉	1	3	yes	—
74	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic
75	C₂×C₃×M₁₆	C₂×C₉	1	3	yes	complemented
76	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
77	C₂²×C₃×C₈	C₂×C₉	1	3	yes	complemented
78	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
79	C₃×C₂²⋊C₈	C₂×C₉	1	12	yes	complemented
80	C₃×C₂²⋊C₈	C₂×C₃²	1	4	yes	—
81	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
82	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
83	C₃²×Q₈	—	2	2	no	—
84	C₃²×D₄	—	2	4	no	—
85	C₃²×D₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
91	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
92	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
94	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
95	C₃²×C₈	—	2	2	no	—
96	C₃²×C₈	—	2	2	no	—
97	Q₈×C₉	—	2	6	no	—
98	D₄×C₉	—	2	12	no	—
99	D₄×C₉	—	2	6	no	—
100	C₂×C₄×C₉	—	2	6	no	—
101	C₂×C₄×C₉	—	2	6	no	—
102	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
103	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
104	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
105	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
106	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
107	C₈×C₉	—	2	6	no	—
108	C₈×C₉	—	2	6	no	—
109	#408	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₃×C₉	—	2	2	no	—
111	C₂×C₃×C₉	—	4	2	no	—
112	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₉	#131	1	2	yes	—
114	C₃×#41	—	2	6	no	—
115	C₃×#41	—	2	6	no	—
116	C₃×#41	—	2	2	no	—
117	C₃×#41	—	2	2	no	—
118	C₂×C₃×Q₈	C₄×C₉	1	3	yes	—
119	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
120	C₂×C₃×D₄	C₄×C₉	1	6	yes	—
121	C₂×C₃×D₄	C₄×C₉	1	3	yes	—
122	C₂×C₃×D₄	C₃²×C₄	1	2	yes	—
123	C₂×C₃×D₄	C₃²×C₄	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
125	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
126	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
128	C₃×M₁₆	—	2	6	no	—
129	C₃×M₁₆	—	2	2	no	—
130	C₂×C₃×C₈	—	2	6	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
133	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
134	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
135	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	—	2	4	no	—
139	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₃²×C₄	—	2	2	no	—
142	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
143	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
144	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂²×C₉	—	2	6	no	—
146	C₂²×C₉	—	2	6	no	—
147	C₂²×C₉	—	2	12	no	—
148	C₂²×C₉	C₃×C₂²⋊C₄	1	6	yes	—
149	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
150	C₄×C₉	—	2	6	no	—
151	C₄×C₉	C₃×C₂²⋊C₄	1	6	yes	—
152	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
153	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
154	C₂×#41	C₂×C₃×C₉	1	1	yes	characteristic
155	C₂×M₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
156	C₂²×C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
157	C₂²⋊C₈	C₂×C₃×C₉	1	4	yes	complemented
158	C₃×C₉	#408	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
160	C₂³×C₃	C₂×C₄×C₉	1	3	yes	—
161	C₂³×C₃	C₂×C₃²×C₄	1	2	yes	—
162	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
163	C₃×Q₈	—	2	6	no	—
164	C₃×Q₈	—	2	2	no	—
165	C₃×D₄	—	2	12	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	6	no	—
168	C₃×D₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	6	no	—
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
174	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
175	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
176	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
177	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
178	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
182	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
183	C₃×C₈	—	2	6	no	—
184	C₃×C₈	—	2	6	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₂×C₃²	—	2	2	no	—
188	C₂×C₃²	—	4	2	no	—
189	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₃²	C₃×#131	1	2	yes	—
191	C₂×C₉	—	2	6	no	—
192	C₂×C₉	—	4	6	no	—
193	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
194	C₂×C₉	C₃×#131	1	6	yes	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₃×C₄×C₉	1	1	yes	characteristic
198	C₂×D₄	C₃×C₄×C₉	1	2	yes	—
199	C₂×D₄	C₃×C₄×C₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₃×C₉	1	2	yes	—
201	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
205	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
206	C₂²×C₃	—	2	6	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	6	no	—
209	C₂²×C₃	—	2	12	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	C₉×C₂²⋊C₄	1	6	yes	—
213	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
214	C₂²×C₃	C₃²×C₂²⋊C₄	1	2	yes	—
215	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
216	C₃×C₄	—	2	6	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	C₉×C₂²⋊C₄	1	6	yes	—
219	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
220	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
221	C₃×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
222	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
223	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
224	C₃²	C₃×#408	1	1	yes	characteristic
225	C₉	C₃×#408	1	3	yes	direct product factor
226	C₂³	C₂×C₃×C₄×C₉	1	2	yes	—
227	C₂³	C₂×C₃×C₄×C₉	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
234	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
235	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
236	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
237	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₂×C₃	—	2	6	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	4	6	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
245	C₂×C₃	C₉×#131	1	6	yes	—
246	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₃	C₃²×#131	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	C₃×C₉×C₂²⋊C₄	1	2	yes	—
252	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	C₃×C₉×C₂²⋊C₄	1	2	yes	—
255	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
256	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
257	C₃	C₉×#408	1	3	yes	direct product factor
258	C₃	C₃²×#408	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	—	4	2	no	—
261	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
262	C₂	C₃×C₉×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂×C₃²×#41	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×M₁₆	C₂	semidirect
3. C₂²×C₃×C₈×C₉	C₂	semidirect
4. C₃×C₉×C₂²⋊C₈	C₂	semidirect
6. C₉×#408	C₃	direct (N × H)
23. C₂×C₉×M₁₆	C₂×C₃	semidirect
24. C₂²×C₈×C₉	C₂×C₃	semidirect
25. C₉×C₂²⋊C₈	C₂×C₃	semidirect
40. C₃×#408	C₉	direct (N × H)
75. C₂×C₃×M₁₆	C₂×C₉	semidirect
77. C₂²×C₃×C₈	C₂×C₉	semidirect
79. C₃×C₂²⋊C₈	C₂×C₉	semidirect
109. #408	C₃×C₉	direct (N × H)
155. C₂×M₁₆	C₂×C₃×C₉	semidirect
156. C₂²×C₈	C₂×C₃×C₉	semidirect
157. C₂²⋊C₈	C₂×C₃×C₉	semidirect
158. C₃×C₉	#408	direct (N × H)
225. C₉	C₃×#408	direct (N × H)
257. C₃	C₉×#408	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×C₉×#41	2
C₂×C₃×C₉×M₁₆	2
C₂²×C₃×C₈×C₉	2
C₃×C₉×C₂²⋊C₈	2
C₃²×#408	3
C₉×#408	3

Subgroups — C3×C9×#408 #13144982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#408 #13144982