Subgroups — C₃×C₉×#410 #13144988

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₃×C₉×#117	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#99	C₂	1	4	yes	maximal, complemented
4	C₃²×#410	C₃	1	1	yes	maximal, characteristic
5	C₉×#410	C₃	1	3	yes	maximal, direct product factor
6	C₃×C₉×#41	—	2	2	no	—
7	C₃×C₉×#41	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₄	1	2	yes	complemented
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	—	2	2	no	—
14	C₃×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₄²×C₉	—	2	2	no	—
16	C₂×C₃²×#41	C₂×C₃	1	1	yes	characteristic
17	C₃²×#117	C₂×C₃	1	2	yes	—
18	C₂×C₉×#41	C₂×C₃	1	3	yes	—
19	C₃²×#99	C₂×C₃	1	4	yes	—
20	C₉×#117	C₂×C₃	1	6	yes	complemented
21	C₉×#99	C₂×C₃	1	12	yes	complemented
22	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₃×Q₈×C₉	—	2	2	no	—
25	C₃×D₄×C₉	—	2	4	no	—
26	C₃×D₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	4	no	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₃×#410	C₉	1	3	yes	direct product factor
36	C₃×#410	C₃²	1	1	yes	characteristic
37	C₃²×#41	—	2	2	no	—
38	C₃²×#41	—	2	2	no	—
39	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
40	C₂×C₃²×D₄	C₃×C₄	1	1	yes	characteristic
41	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄⋊C₄	—	2	2	no	—
45	C₃²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
46	C₃²×C₄²	—	2	2	no	—
47	C₉×#41	—	2	6	no	—
48	C₉×#41	—	2	6	no	—
49	C₂×Q₈×C₉	C₃×C₄	1	3	yes	complemented
50	C₂×D₄×C₉	C₃×C₄	1	3	yes	complemented
51	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
52	C₂²×C₄×C₉	C₃×C₄	1	6	yes	complemented
53	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
54	C₉×C₄⋊C₄	—	2	6	no	—
55	C₉×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
56	C₄²×C₉	—	2	6	no	—
57	C₂²×C₃×C₉	—	2	2	no	—
58	C₂²×C₃×C₉	—	4	1	no	—
59	C₂²×C₃×C₉	—	4	2	no	—
60	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	—	4	4	no	—
64	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×#41	C₂×C₉	1	3	yes	—
68	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic
69	C₃×#117	C₂×C₉	1	6	yes	complemented
70	C₃×#117	C₂×C₃²	1	2	yes	—
71	C₃×#99	C₂×C₉	1	12	yes	complemented
72	C₃×#99	C₂×C₃²	1	4	yes	—
73	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
74	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
75	C₃²×Q₈	—	2	2	no	—
76	C₃²×D₄	—	2	4	no	—
77	C₃²×D₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
85	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
86	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
87	C₂³×C₉	C₂³×C₃	1	3	yes	—
88	Q₈×C₉	—	2	6	no	—
89	D₄×C₉	—	2	12	no	—
90	D₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	—	2	6	no	—
92	C₂×C₄×C₉	—	2	6	no	—
93	C₂×C₄×C₉	—	2	12	no	—
94	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
95	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
96	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
97	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
98	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
99	#410	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₃×C₉	—	2	2	no	—
101	C₂×C₃×C₉	—	2	1	no	—
102	C₂×C₃×C₉	—	4	2	no	—
103	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₃×#41	—	2	6	no	—
105	C₃×#41	—	2	6	no	—
106	C₃×#41	—	2	2	no	—
107	C₃×#41	—	2	2	no	—
108	C₂×C₃×Q₈	C₄×C₉	1	3	yes	complemented
109	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
110	C₂×C₃×D₄	C₄×C₉	1	3	yes	complemented
111	C₂×C₃×D₄	C₂²×C₉	1	6	yes	—
112	C₂×C₃×D₄	C₃²×C₄	1	1	yes	characteristic
113	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
114	C₂²×C₃×C₄	C₄×C₉	1	6	yes	complemented
115	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
116	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
117	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	2	6	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₂²⋊C₄	C₂²×C₉	1	12	yes	—
121	C₃×C₂²⋊C₄	C₂²×C₃²	1	4	yes	—
122	C₃×C₄²	—	2	6	no	—
123	C₃×C₄²	—	2	2	no	—
124	C₂²×C₃²	—	2	2	no	—
125	C₂²×C₃²	—	4	1	no	—
126	C₂²×C₃²	—	4	2	no	—
127	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	—	4	4	no	—
131	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
132	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
133	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₉	—	2	6	no	—
135	C₂²×C₉	—	4	3	no	—
136	C₂²×C₉	—	4	6	no	—
137	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
138	C₂²×C₉	C₂×C₃×D₄	1	6	yes	—
139	C₄×C₉	—	2	6	no	—
140	C₄×C₉	—	4	12	no	—
141	C₄×C₉	C₃×C₂²⋊C₄	1	3	yes	—
142	C₄×C₉	C₃×C₂²⋊C₄	1	6	yes	—
143	C₄×C₉	C₃×C₂²⋊C₄	1	3	yes	—
144	C₂×#41	C₂×C₃×C₉	1	1	yes	characteristic
145	#117	C₂×C₃×C₉	1	2	yes	complemented
146	#99	C₂×C₃×C₉	1	4	yes	complemented
147	C₃×C₉	#410	1	1	yes	characteristic, direct product factor, Sylow
148	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
149	C₂³×C₃	C₂³×C₉	1	3	yes	—
150	C₂³×C₃	C₂×C₃²×C₄	1	2	yes	—
151	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
152	C₃×Q₈	—	2	6	no	—
153	C₃×Q₈	—	2	2	no	—
154	C₃×D₄	—	2	12	no	—
155	C₃×D₄	—	2	4	no	—
156	C₃×D₄	—	2	6	no	—
157	C₃×D₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	6	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	12	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
165	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
166	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
167	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
168	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
169	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
173	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
174	C₂×C₃²	—	2	2	no	—
175	C₂×C₃²	—	2	1	no	—
176	C₂×C₃²	—	4	2	no	—
177	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₉	—	2	6	no	—
179	C₂×C₉	—	2	3	no	—
180	C₂×C₉	—	4	6	no	—
181	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	C₂×Q₈	C₃×C₄×C₉	1	1	yes	characteristic, complemented
185	C₂×D₄	C₃×C₄×C₉	1	1	yes	characteristic, complemented
186	C₂×D₄	C₂²×C₃×C₉	1	2	yes	—
187	C₂²×C₄	C₃×C₄×C₉	1	2	yes	complemented
188	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	C₂²×C₃×C₉	1	4	yes	—
191	C₄²	—	2	2	no	—
192	C₂²×C₃	—	2	6	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	4	3	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	6	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
199	C₂²×C₃	C₂×D₄×C₉	1	6	yes	—
200	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
201	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
202	C₃×C₄	—	2	6	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	—	4	12	no	—
206	C₃×C₄	C₉×C₂²⋊C₄	1	3	yes	—
207	C₃×C₄	C₉×C₂²⋊C₄	1	3	yes	—
208	C₃×C₄	C₉×C₂²⋊C₄	1	6	yes	—
209	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
210	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
211	C₃×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
212	C₃²	C₃×#410	1	1	yes	characteristic
213	C₉	C₃×#410	1	3	yes	direct product factor
214	C₂³	C₂×C₃×C₄×C₉	1	2	yes	—
215	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
223	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
225	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
226	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
227	C₂×C₃	—	2	6	no	—
228	C₂×C₃	—	2	3	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	6	no	—
232	C₂×C₃	—	4	2	no	—
233	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
234	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
239	C₂²	C₂×C₃×D₄×C₉	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	4	4	no	—
242	C₄	C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
243	C₄	C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
244	C₄	C₃×C₉×C₂²⋊C₄	1	2	yes	—
245	C₃	C₉×#410	1	3	yes	direct product factor
246	C₃	C₃²×#410	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	2	1	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#117	C₂	semidirect
3. C₃×C₉×#99	C₂	semidirect
5. C₉×#410	C₃	direct (N × H)
8. C₂×C₃×Q₈×C₉	C₄	semidirect
9. C₂×C₃×D₄×C₉	C₄	semidirect
11. C₂²×C₃×C₄×C₉	C₄	semidirect
20. C₉×#117	C₂×C₃	semidirect
21. C₉×#99	C₂×C₃	semidirect
35. C₃×#410	C₉	direct (N × H)
49. C₂×Q₈×C₉	C₃×C₄	semidirect
50. C₂×D₄×C₉	C₃×C₄	semidirect
52. C₂²×C₄×C₉	C₃×C₄	semidirect
69. C₃×#117	C₂×C₉	semidirect
71. C₃×#99	C₂×C₉	semidirect
99. #410	C₃×C₉	direct (N × H)
108. C₂×C₃×Q₈	C₄×C₉	semidirect
110. C₂×C₃×D₄	C₄×C₉	semidirect
114. C₂²×C₃×C₄	C₄×C₉	semidirect
145. #117	C₂×C₃×C₉	semidirect
146. #99	C₂×C₃×C₉	semidirect
147. C₃×C₉	#410	direct (N × H)
184. C₂×Q₈	C₃×C₄×C₉	semidirect
185. C₂×D₄	C₃×C₄×C₉	semidirect
187. C₂²×C₄	C₃×C₄×C₉	semidirect
213. C₉	C₃×#410	direct (N × H)
245. C₃	C₉×#410	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
C₃×C₉×#117	2
C₃×C₉×#99	2
C₃²×#410	3
C₉×#410	3

Subgroups — C3×C9×#410 #13144988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#410 #13144988