Subgroups — C₃×C₉×#461 #13145141

Order 1728.

600 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×SD₁₆	C₂	1	2	yes	maximal
2	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#103	C₂	1	2	yes	maximal, complemented
6	C₃²×#461	C₃	1	1	yes	maximal, characteristic
7	C₉×#461	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
9	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	—	2	1	no	—
11	C₃×C₉×SD₁₆	—	2	4	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	—	2	1	no	—
14	C₃×C₉×C₄⋊C₄	—	2	1	no	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₃²×SD₁₆	C₂×C₃	1	2	yes	—
19	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×#103	C₂×C₃	1	2	yes	—
23	C₂×C₉×SD₁₆	C₂×C₃	1	6	yes	—
24	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
25	C₄×D₄×C₉	C₂×C₃	1	3	yes	—
26	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
27	C₉×#103	C₂×C₃	1	6	yes	complemented
28	C₂³×C₃×C₉	—	2	1	no	—
29	C₃×Q₈×C₉	—	2	2	no	—
30	C₃×Q₈×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	1	no	—
32	C₃×D₄×C₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	—	2	2	no	—
35	C₂×C₃×C₄×C₉	—	4	2	no	—
36	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃×#461	C₉	1	3	yes	direct product factor
41	C₃×#461	C₃²	1	1	yes	characteristic
42	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₃²×SD₁₆	—	2	4	no	—
46	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
47	C₃²×C₄⋊C₄	—	2	1	no	—
48	C₃²×C₄⋊C₄	—	2	1	no	—
49	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
50	C₃²×C₂²⋊C₄	—	2	1	no	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
53	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
54	C₂²×C₄×C₉	—	2	3	no	—
55	C₉×SD₁₆	—	2	12	no	—
56	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
57	C₉×C₄⋊C₄	—	2	3	no	—
58	C₉×C₄⋊C₄	—	2	3	no	—
59	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
60	C₉×C₂²⋊C₄	—	2	3	no	—
61	C₄²×C₉	C₂²×C₃	1	3	yes	—
62	C₂²×C₃×C₉	—	2	2	no	—
63	C₂²×C₃×C₉	—	4	1	no	—
64	C₂²×C₃×C₉	—	4	1	no	—
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	—	4	2	no	—
68	C₃×C₄×C₉	—	4	1	no	—
69	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
70	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₉	#41	1	1	yes	characteristic
72	C₂×C₃×SD₁₆	C₂×C₉	1	6	yes	—
73	C₂×C₃×SD₁₆	C₂×C₃²	1	2	yes	—
74	C₃×C₄⋊Q₈	C₂×C₉	1	3	yes	complemented
75	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
76	C₃×C₄×D₄	C₂×C₉	1	3	yes	—
77	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
78	C₃×C₄⋊C₈	C₂×C₉	1	3	yes	complemented
79	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
80	C₃×#103	C₂×C₉	1	6	yes	complemented
81	C₃×#103	C₂×C₃²	1	2	yes	—
82	C₂³×C₃²	—	2	1	no	—
83	C₃²×Q₈	—	2	2	no	—
84	C₃²×Q₈	—	2	2	no	—
85	C₃²×D₄	—	2	1	no	—
86	C₃²×D₄	C₃×D₄	1	2	yes	—
87	C₂×C₃²×C₄	—	2	1	no	—
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	4	2	no	—
90	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂³×C₉	—	2	3	no	—
94	C₃²×C₈	—	2	2	no	—
95	Q₈×C₉	—	2	6	no	—
96	Q₈×C₉	—	2	6	no	—
97	D₄×C₉	—	2	3	no	—
98	D₄×C₉	C₃×D₄	1	6	yes	—
99	C₂×C₄×C₉	—	2	3	no	—
100	C₂×C₄×C₉	—	2	6	no	—
101	C₂×C₄×C₉	—	4	6	no	—
102	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
103	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
104	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
105	C₈×C₉	—	2	6	no	—
106	#461	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₃×C₉	—	4	2	no	—
108	C₂×C₃×C₉	#121	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₃×C₉	#137	1	1	yes	characteristic
111	C₂×C₃×Q₈	C₂²×C₉	1	6	yes	—
112	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
113	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
114	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	2	3	no	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₃×SD₁₆	—	2	12	no	—
118	C₃×SD₁₆	—	2	4	no	—
119	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
120	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
121	C₃×C₄⋊C₄	—	2	3	no	—
122	C₃×C₄⋊C₄	—	2	3	no	—
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	C₂²×C₉	1	3	yes	—
126	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
127	C₃×C₂²⋊C₄	—	2	3	no	—
128	C₃×C₂²⋊C₄	—	2	1	no	—
129	C₃×C₄²	C₂²×C₉	1	3	yes	—
130	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	4	1	no	—
133	C₂²×C₃²	—	4	1	no	—
134	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
135	C₃²×C₄	—	2	1	no	—
136	C₃²×C₄	—	4	2	no	—
137	C₃²×C₄	—	4	1	no	—
138	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
139	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₃²×C₄	C₃×#41	1	1	yes	characteristic
141	C₂²×C₉	—	2	6	no	—
142	C₂²×C₉	—	4	3	no	—
143	C₂²×C₉	—	4	3	no	—
144	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
145	C₄×C₉	—	2	3	no	—
146	C₄×C₉	—	4	6	no	—
147	C₄×C₉	—	4	3	no	—
148	C₄×C₉	C₃×SD₁₆	1	6	yes	complemented
149	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
150	C₄×C₉	C₃×#41	1	3	yes	—
151	C₂×SD₁₆	C₂×C₃×C₉	1	2	yes	—
152	C₄⋊Q₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
153	C₄×D₄	C₂×C₃×C₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
155	#103	C₂×C₃×C₉	1	2	yes	complemented
156	C₃×C₉	#461	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³×C₃	—	2	3	no	—
158	C₂³×C₃	—	2	1	no	—
159	C₃×Q₈	—	2	6	no	—
160	C₃×Q₈	—	2	6	no	—
161	C₃×Q₈	—	2	2	no	—
162	C₃×Q₈	—	2	2	no	—
163	C₃×D₄	—	2	3	no	—
164	C₃×D₄	—	2	1	no	—
165	C₃×D₄	D₄×C₉	1	6	yes	—
166	C₃×D₄	C₃²×D₄	1	2	yes	—
167	C₂×C₃×C₄	—	2	3	no	—
168	C₂×C₃×C₄	—	2	6	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	4	6	no	—
172	C₂×C₃×C₄	—	4	2	no	—
173	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
174	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
175	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
176	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
179	C₃×C₈	—	2	6	no	—
180	C₃×C₈	—	2	2	no	—
181	C₂×C₃²	—	4	2	no	—
182	C₂×C₃²	C₃×#121	1	1	yes	characteristic
183	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
184	C₂×C₃²	C₃×#137	1	1	yes	characteristic
185	C₂×C₉	—	4	6	no	—
186	C₂×C₉	C₃×#121	1	3	yes	—
187	C₂×C₉	C₂×C₃×SD₁₆	1	3	yes	—
188	C₂×C₉	C₃×#137	1	3	yes	—
189	C₂×Q₈	C₂²×C₃×C₉	1	2	yes	—
190	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	SD₁₆	—	2	4	no	—
193	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₂²×C₃×C₉	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	1	no	—
198	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
199	C₂²×C₃	—	2	6	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	4	3	no	—
202	C₂²×C₃	—	4	3	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
206	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
207	C₃×C₄	—	2	3	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	4	6	no	—
210	C₃×C₄	—	4	3	no	—
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	C₉×SD₁₆	1	6	yes	complemented
214	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
215	C₃×C₄	C₉×#41	1	3	yes	—
216	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
217	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
218	C₃×C₄	C₃²×#41	1	1	yes	characteristic
219	C₃²	C₃×#461	1	1	yes	characteristic
220	C₉	C₃×#461	1	3	yes	direct product factor
221	C₂³	—	2	1	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	C₃×D₄×C₉	1	2	yes	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
230	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
231	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₂×C₃	—	4	2	no	—
234	C₂×C₃	—	4	6	no	—
235	C₂×C₃	C₉×#121	1	3	yes	—
236	C₂×C₃	C₂×C₉×SD₁₆	1	3	yes	—
237	C₂×C₃	C₉×#137	1	3	yes	—
238	C₂×C₃	C₃²×#121	1	1	yes	characteristic
239	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
240	C₂×C₃	C₃²×#137	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	1	no	—
248	C₄	C₃×C₉×SD₁₆	1	2	yes	complemented
249	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
250	C₄	C₃×C₉×#41	1	1	yes	characteristic
251	C₃	C₉×#461	1	3	yes	direct product factor
252	C₃	C₃²×#461	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	C₃×C₉×#121	1	1	yes	characteristic
255	C₂	C₂×C₃×C₉×SD₁₆	1	1	yes	characteristic
256	C₂	C₃×C₉×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃²×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₄⋊Q₈	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₃×C₉×#103	C₂	semidirect
7. C₉×#461	C₃	direct (N × H)
24. C₉×C₄⋊Q₈	C₂×C₃	semidirect
26. C₉×C₄⋊C₈	C₂×C₃	semidirect
27. C₉×#103	C₂×C₃	semidirect
40. C₃×#461	C₉	direct (N × H)
69. C₃×C₄×C₉	SD₁₆	semidirect
74. C₃×C₄⋊Q₈	C₂×C₉	semidirect
78. C₃×C₄⋊C₈	C₂×C₉	semidirect
80. C₃×#103	C₂×C₉	semidirect
106. #461	C₃×C₉	direct (N × H)
148. C₄×C₉	C₃×SD₁₆	semidirect
152. C₄⋊Q₈	C₂×C₃×C₉	semidirect
154. C₄⋊C₈	C₂×C₃×C₉	semidirect
155. #103	C₂×C₃×C₉	semidirect
156. C₃×C₉	#461	direct (N × H)
213. C₃×C₄	C₉×SD₁₆	semidirect
220. C₉	C₃×#461	direct (N × H)
248. C₄	C₃×C₉×SD₁₆	semidirect
251. C₃	C₉×#461	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×SD₁₆	2
C₃×C₉×C₄⋊Q₈	2
C₃×C₄×D₄×C₉	2
C₃×C₉×C₄⋊C₈	2
C₃×C₉×#103	2
C₃²×#461	3
C₉×#461	3

Subgroups — C3×C9×#461 #13145141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#461 #13145141