Subgroups — C₃×C₉×#474 #13145180

Order 1728.

540 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#107	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#102	C₂	1	2	yes	maximal
6	C₃²×#474	C₃	1	1	yes	maximal, characteristic
7	C₉×#474	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	—	2	1	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
19	C₃²×#107	C₂×C₃	1	2	yes	—
20	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₃²×#102	C₂×C₃	1	2	yes	—
22	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
23	C₄×D₄×C₉	C₂×C₃	1	3	yes	—
24	C₉×#107	C₂×C₃	1	6	yes	complemented
25	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
26	C₉×#102	C₂×C₃	1	6	yes	—
27	C₂³×C₃×C₉	—	2	1	no	—
28	C₃×Q₈×C₉	—	4	2	no	—
29	C₃×D₄×C₉	—	2	1	no	—
30	C₃×D₄×C₉	Q₈	1	2	yes	complemented
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	—	4	2	no	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₃×C₈×C₉	—	2	2	no	—
38	C₃×#474	C₉	1	3	yes	direct product factor
39	C₃×#474	C₃²	1	1	yes	characteristic
40	C₂×C₃²×Q₈	—	2	1	no	—
41	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	—	2	1	no	—
43	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
44	C₃²×C₄⋊C₄	—	2	1	no	—
45	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₃²×C₂²⋊C₄	—	2	1	no	—
48	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₂×Q₈×C₉	—	2	3	no	—
50	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
51	C₂²×C₄×C₉	—	2	3	no	—
52	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
53	C₉×C₄⋊C₄	—	2	3	no	—
54	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
55	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
56	C₉×C₂²⋊C₄	—	2	3	no	—
57	C₄²×C₉	C₂²×C₃	1	3	yes	—
58	C₂²×C₃×C₉	—	2	2	no	—
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	—	4	1	no	—
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	—	4	1	no	—
64	C₃×C₄×C₉	—	4	2	no	—
65	C₃×C₄×C₉	D₈	1	2	yes	—
66	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
67	C₃×C₄×C₉	#41	1	1	yes	characteristic
68	C₃×C₄⋊Q₈	C₂×C₉	1	3	yes	complemented
69	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
70	C₃×C₄×D₄	C₂×C₉	1	3	yes	—
71	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
72	C₃×#107	C₂×C₉	1	6	yes	complemented
73	C₃×#107	C₂×C₃²	1	2	yes	—
74	C₃×C₄⋊C₈	C₂×C₉	1	3	yes	complemented
75	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
76	C₃×#102	C₂×C₉	1	6	yes	—
77	C₃×#102	C₂×C₃²	1	2	yes	—
78	C₂³×C₃²	—	2	1	no	—
79	C₃²×Q₈	—	4	2	no	—
80	C₃²×D₄	—	2	1	no	—
81	C₃²×D₄	C₃×Q₈	1	2	yes	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	4	2	no	—
85	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂³×C₉	—	2	3	no	—
89	C₃²×C₈	—	2	2	no	—
90	Q₈×C₉	—	4	6	no	—
91	D₄×C₉	—	2	3	no	—
92	D₄×C₉	C₃×Q₈	1	6	yes	complemented
93	C₂×C₄×C₉	—	2	3	no	—
94	C₂×C₄×C₉	—	2	6	no	—
95	C₂×C₄×C₉	—	4	6	no	—
96	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
97	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
98	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
99	C₈×C₉	—	2	6	no	—
100	#474	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×C₃×C₉	—	4	2	no	—
102	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₃×C₉	#137	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	2	3	no	—
106	C₂×C₃×Q₈	—	2	1	no	—
107	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
108	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	3	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
112	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
113	C₃×C₄⋊C₄	—	2	3	no	—
114	C₃×C₄⋊C₄	—	2	1	no	—
115	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
116	C₃×C₄⋊C₄	C₂²×C₉	1	3	yes	—
117	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
118	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
119	C₃×C₂²⋊C₄	—	2	3	no	—
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₄²	C₂²×C₉	1	3	yes	—
122	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	—	4	1	no	—
125	C₂²×C₃²	—	4	1	no	—
126	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	4	1	no	—
129	C₃²×C₄	—	4	2	no	—
130	C₃²×C₄	C₃×D₈	1	2	yes	—
131	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₃²×C₄	C₃×#41	1	1	yes	characteristic
133	C₂²×C₉	—	2	6	no	—
134	C₂²×C₉	—	4	3	no	—
135	C₂²×C₉	—	4	3	no	—
136	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
137	C₄×C₉	—	2	3	no	—
138	C₄×C₉	—	4	3	no	—
139	C₄×C₉	—	4	6	no	—
140	C₄×C₉	C₃×D₈	1	6	yes	—
141	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
142	C₄×C₉	C₃×#41	1	3	yes	—
143	C₄⋊Q₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
144	C₄×D₄	C₂×C₃×C₉	1	1	yes	characteristic
145	#107	C₂×C₃×C₉	1	2	yes	complemented
146	C₄⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
147	#102	C₂×C₃×C₉	1	2	yes	—
148	C₃×C₉	#474	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×C₃	—	2	3	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₃×Q₈	—	4	6	no	—
152	C₃×Q₈	—	4	2	no	—
153	C₃×D₄	—	2	3	no	—
154	C₃×D₄	—	2	1	no	—
155	C₃×D₄	Q₈×C₉	1	6	yes	complemented
156	C₃×D₄	C₃²×Q₈	1	2	yes	—
157	C₂×C₃×C₄	—	2	3	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	4	6	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
164	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
165	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
166	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
169	C₃×C₈	—	2	6	no	—
170	C₃×C₈	—	2	2	no	—
171	C₂×C₃²	—	4	2	no	—
172	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
174	C₂×C₃²	C₃×#137	1	1	yes	characteristic
175	C₂×C₉	—	4	6	no	—
176	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
177	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
178	C₂×C₉	C₃×#137	1	3	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
185	C₄⋊C₄	C₂²×C₃×C₉	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	1	no	—
187	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
188	C₂²×C₃	—	2	6	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	4	3	no	—
191	C₂²×C₃	—	4	3	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
195	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
196	C₃×C₄	—	2	3	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	3	no	—
199	C₃×C₄	—	4	6	no	—
200	C₃×C₄	—	4	1	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	C₉×D₈	1	6	yes	—
203	C₃×C₄	C₂×Q₈×C₉	1	3	yes	—
204	C₃×C₄	C₉×#41	1	3	yes	—
205	C₃×C₄	C₃²×D₈	1	2	yes	—
206	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
207	C₃×C₄	C₃²×#41	1	1	yes	characteristic
208	C₃²	C₃×#474	1	1	yes	characteristic
209	C₉	C₃×#474	1	3	yes	direct product factor
210	C₂³	—	2	1	no	—
211	Q₈	—	4	2	no	—
212	D₄	—	2	1	no	—
213	D₄	C₃×Q₈×C₉	1	2	yes	complemented
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
218	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
219	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₂×C₃	—	4	2	no	—
222	C₂×C₃	—	4	6	no	—
223	C₂×C₃	C₉×C₂²⋊Q₈	1	3	yes	—
224	C₂×C₃	C₂×C₉×D₈	1	3	yes	—
225	C₂×C₃	C₉×#137	1	3	yes	—
226	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
228	C₂×C₃	C₃²×#137	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	C₃×C₉×D₈	1	2	yes	—
237	C₄	C₂×C₃×Q₈×C₉	1	1	yes	characteristic
238	C₄	C₃×C₉×#41	1	1	yes	characteristic
239	C₃	C₉×#474	1	3	yes	direct product factor
240	C₃	C₃²×#474	1	1	yes	characteristic
241	C₂	—	4	2	no	—
242	C₂	C₃×C₉×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂	C₂×C₃×C₉×D₈	1	1	yes	characteristic
244	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
1. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. C₃×C₉×#107	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
7. C₉×#474	C₃	direct (N × H)
22. C₉×C₄⋊Q₈	C₂×C₃	semidirect
24. C₉×#107	C₂×C₃	semidirect
25. C₉×C₄⋊C₈	C₂×C₃	semidirect
30. C₃×D₄×C₉	Q₈	semidirect
38. C₃×#474	C₉	direct (N × H)
68. C₃×C₄⋊Q₈	C₂×C₉	semidirect
72. C₃×#107	C₂×C₉	semidirect
74. C₃×C₄⋊C₈	C₂×C₉	semidirect
92. D₄×C₉	C₃×Q₈	semidirect
100. #474	C₃×C₉	direct (N × H)
143. C₄⋊Q₈	C₂×C₃×C₉	semidirect
145. #107	C₂×C₃×C₉	semidirect
146. C₄⋊C₈	C₂×C₃×C₉	semidirect
148. C₃×C₉	#474	direct (N × H)
155. C₃×D₄	Q₈×C₉	semidirect
209. C₉	C₃×#474	direct (N × H)
213. D₄	C₃×Q₈×C₉	semidirect
239. C₃	C₉×#474	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₃×C₄×D₄×C₉	2
C₃×C₉×#107	2
C₃×C₉×C₄⋊C₈	2
C₃×C₉×#102	2
C₃²×#474	3
C₉×#474	3

Subgroups — C3×C9×#474 #13145180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#474 #13145180