Subgroups — C₃×C₉×#380 #13144898

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₃×C₉×#95	C₂	1	2	yes	maximal, complemented
5	C₃²×#380	C₃	1	1	yes	maximal, characteristic
6	C₉×#380	C₃	1	3	yes	maximal, direct product factor
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	—	2	4	no	—
9	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	4	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	—	2	4	no	—
14	C₃×C₉×C₂²⋊C₄	—	2	4	no	—
15	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	2	yes	—
17	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
18	C₂³×C₄×C₉	C₂×C₃	1	3	yes	—
19	C₃²×#95	C₂×C₃	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	C₂×C₃	1	6	yes	complemented
21	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	6	yes	—
22	C₉×#95	C₂×C₃	1	6	yes	complemented
23	C₂³×C₃×C₉	—	2	2	no	—
24	C₂³×C₃×C₉	—	2	2	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
26	C₂³×C₃×C₉	D₄	1	2	yes	—
27	C₂³×C₃×C₉	Q₈	1	2	yes	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	8	no	—
31	C₂×C₃×C₄×C₉	—	2	4	no	—
32	C₂×C₃×C₄×C₉	—	2	4	no	—
33	C₂×C₃×C₄×C₉	—	2	4	no	—
34	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
37	C₃×#380	C₉	1	3	yes	direct product factor
38	C₃×#380	C₃²	1	1	yes	characteristic
39	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂⁴×C₉	C₂²×C₃	1	3	yes	—
46	C₃²×C₄⋊C₄	—	2	4	no	—
47	C₃²×C₂²⋊C₄	—	2	4	no	—
48	C₂²×C₄×C₉	—	2	12	no	—
49	C₂²×C₄×C₉	C₃×C₄	1	12	yes	complemented
50	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	—
51	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
52	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
53	C₉×C₄⋊C₄	—	2	12	no	—
54	C₉×C₂²⋊C₄	—	2	12	no	—
55	C₂²×C₃×C₉	—	2	2	no	—
56	C₂²×C₃×C₉	—	2	2	no	—
57	C₂²×C₃×C₉	—	2	4	no	—
58	C₂²×C₃×C₉	—	2	4	no	—
59	C₂²×C₃×C₉	C₄⋊C₄	1	4	yes	complemented
60	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃×C₉	#41	1	2	yes	—
65	C₃×C₄×C₉	—	2	2	no	—
66	C₃×C₄×C₉	—	4	4	no	—
67	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	complemented
68	C₂³×C₃×C₄	C₂×C₉	1	3	yes	—
69	C₂³×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
70	C₂×C₃×C₄⋊C₄	C₂×C₉	1	6	yes	complemented
71	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	2	yes	—
72	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	6	yes	—
73	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
74	C₃×#95	C₂×C₉	1	6	yes	complemented
75	C₃×#95	C₂×C₃²	1	2	yes	—
76	C₂³×C₃²	—	2	2	no	—
77	C₂³×C₃²	—	2	2	no	—
78	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
79	C₂³×C₃²	C₃×D₄	1	2	yes	—
80	C₂³×C₃²	C₃×Q₈	1	2	yes	—
81	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
90	C₂³×C₉	—	2	6	no	—
91	C₂³×C₉	—	2	6	no	—
92	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
93	C₂³×C₉	C₃×D₄	1	6	yes	—
94	C₂³×C₉	C₃×Q₈	1	6	yes	—
95	C₂³×C₉	C₂³×C₃	1	3	yes	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	—	2	24	no	—
98	C₂×C₄×C₉	—	2	12	no	—
99	C₂×C₄×C₉	—	2	12	no	—
100	C₂×C₄×C₉	—	2	12	no	—
101	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
102	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
103	C₂×C₄×C₉	C₃×D₄	1	12	yes	—
104	#380	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×C₃×C₉	—	2	4	no	—
106	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₉	#117	1	1	yes	characteristic
109	C₂×C₃×C₉	#121	1	2	yes	—
110	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
111	C₂⁴×C₃	C₂²×C₉	1	3	yes	—
112	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	12	no	—
114	C₂²×C₃×C₄	—	2	4	no	—
115	C₂²×C₃×C₄	C₄×C₉	1	12	yes	complemented
116	C₂²×C₃×C₄	C₂²×C₉	1	12	yes	—
117	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
118	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
119	C₂²×C₃×C₄	C₃²×C₄	1	4	yes	—
120	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
121	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
123	C₃×C₄⋊C₄	—	2	12	no	—
124	C₃×C₄⋊C₄	—	2	4	no	—
125	C₃×C₂²⋊C₄	—	2	12	no	—
126	C₃×C₂²⋊C₄	—	2	4	no	—
127	C₂²×C₃²	—	2	2	no	—
128	C₂²×C₃²	—	2	2	no	—
129	C₂²×C₃²	—	2	4	no	—
130	C₂²×C₃²	—	2	4	no	—
131	C₂²×C₃²	C₃×C₄⋊C₄	1	4	yes	—
132	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
134	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
136	C₂²×C₃²	C₃×#41	1	2	yes	—
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	—	4	4	no	—
139	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₉	—	2	6	no	—
141	C₂²×C₉	—	2	6	no	—
142	C₂²×C₉	—	2	12	no	—
143	C₂²×C₉	—	2	12	no	—
144	C₂²×C₉	C₃×C₄⋊C₄	1	12	yes	complemented
145	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
146	C₂²×C₉	C₂×C₃×D₄	1	6	yes	—
147	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
148	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
149	C₂²×C₉	C₃×#41	1	6	yes	—
150	C₄×C₉	—	2	6	no	—
151	C₄×C₉	—	4	12	no	—
152	C₄×C₉	C₃×C₂²⋊C₄	1	12	yes	complemented
153	C₂³×C₄	C₂×C₃×C₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×C₃×C₉	1	2	yes	complemented
155	C₂×C₂²⋊C₄	C₂×C₃×C₉	1	2	yes	—
156	#95	C₂×C₃×C₉	1	2	yes	complemented
157	C₃×C₉	#380	1	1	yes	characteristic, direct product factor, Sylow
158	C₂³×C₃	—	2	6	no	—
159	C₂³×C₃	—	2	6	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
163	C₂³×C₃	D₄×C₉	1	6	yes	—
164	C₂³×C₃	Q₈×C₉	1	6	yes	—
165	C₂³×C₃	C₂³×C₉	1	3	yes	—
166	C₂³×C₃	C₂×C₃²×C₄	1	2	yes	—
167	C₂³×C₃	C₃²×D₄	1	2	yes	—
168	C₂³×C₃	C₃²×Q₈	1	2	yes	—
169	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	24	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	12	no	—
175	C₂×C₃×C₄	—	2	12	no	—
176	C₂×C₃×C₄	—	2	12	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
181	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
182	C₂×C₃×C₄	D₄×C₉	1	12	yes	—
183	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
184	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
185	C₂×C₃×C₄	C₃²×D₄	1	4	yes	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₃²	C₃×#117	1	1	yes	characteristic
190	C₂×C₃²	C₃×#121	1	2	yes	—
191	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
192	C₂×C₉	—	2	12	no	—
193	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
194	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
195	C₂×C₉	C₃×#117	1	3	yes	—
196	C₂×C₉	C₃×#121	1	6	yes	—
197	C₂×C₉	C₃×C₂²⋊Q₈	1	6	yes	—
198	C₂⁴	C₂²×C₃×C₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₃×C₄×C₉	1	4	yes	complemented
201	C₂²×C₄	C₂²×C₃×C₉	1	4	yes	—
202	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²×C₃	—	2	6	no	—
207	C₂²×C₃	—	2	6	no	—
208	C₂²×C₃	—	2	12	no	—
209	C₂²×C₃	—	2	12	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	C₉×C₄⋊C₄	1	12	yes	complemented
215	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
216	C₂²×C₃	C₂×D₄×C₉	1	6	yes	—
217	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
218	C₂²×C₃	C₂×Q₈×C₉	1	3	yes	—
219	C₂²×C₃	C₉×#41	1	6	yes	—
220	C₂²×C₃	C₃²×C₄⋊C₄	1	4	yes	—
221	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
222	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
223	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
225	C₂²×C₃	C₃²×#41	1	2	yes	—
226	C₃×C₄	—	2	6	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	4	12	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	C₉×C₂²⋊C₄	1	12	yes	complemented
231	C₃×C₄	C₃²×C₂²⋊C₄	1	4	yes	—
232	C₃²	C₃×#380	1	1	yes	characteristic
233	C₉	C₃×#380	1	3	yes	direct product factor
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₃×C₄×C₉	1	2	yes	—
237	C₂³	C₃×D₄×C₉	1	2	yes	—
238	C₂³	C₃×Q₈×C₉	1	2	yes	—
239	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
246	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
247	C₂×C₄	C₃×D₄×C₉	1	4	yes	—
248	C₂×C₃	—	2	12	no	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
251	C₂×C₃	C₂×C₉×C₄⋊C₄	1	3	yes	—
252	C₂×C₃	C₉×#117	1	3	yes	—
253	C₂×C₃	C₉×#121	1	6	yes	—
254	C₂×C₃	C₉×C₂²⋊Q₈	1	6	yes	—
255	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₃	C₃²×#117	1	1	yes	characteristic
258	C₂×C₃	C₃²×#121	1	2	yes	—
259	C₂×C₃	C₃²×C₂²⋊Q₈	1	2	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	C₃×C₉×C₄⋊C₄	1	4	yes	complemented
265	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
266	C₂²	C₂×C₃×D₄×C₉	1	2	yes	—
267	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
268	C₂²	C₂×C₃×Q₈×C₉	1	1	yes	characteristic
269	C₂²	C₃×C₉×#41	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	4	4	no	—
272	C₄	C₃×C₉×C₂²⋊C₄	1	4	yes	complemented
273	C₃	C₉×#380	1	3	yes	direct product factor
274	C₃	C₃²×#380	1	1	yes	characteristic
275	C₂	—	2	4	no	—
276	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
277	C₂	C₂×C₃×C₉×C₄⋊C₄	1	1	yes	characteristic
278	C₂	C₃×C₉×#117	1	1	yes	characteristic
279	C₂	C₃×C₉×#121	1	2	yes	—
280	C₂	C₃×C₉×C₂²⋊Q₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂⁴×C₃²	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
4. C₃×C₉×#95	C₂	semidirect
6. C₉×#380	C₃	direct (N × H)
9. C₂²×C₃×C₄×C₉	C₄	semidirect
20. C₂×C₉×C₄⋊C₄	C₂×C₃	semidirect
22. C₉×#95	C₂×C₃	semidirect
37. C₃×#380	C₉	direct (N × H)
49. C₂²×C₄×C₉	C₃×C₄	semidirect
59. C₂²×C₃×C₉	C₄⋊C₄	semidirect
67. C₃×C₄×C₉	C₂²⋊C₄	semidirect
70. C₂×C₃×C₄⋊C₄	C₂×C₉	semidirect
74. C₃×#95	C₂×C₉	semidirect
104. #380	C₃×C₉	direct (N × H)
115. C₂²×C₃×C₄	C₄×C₉	semidirect
144. C₂²×C₉	C₃×C₄⋊C₄	semidirect
152. C₄×C₉	C₃×C₂²⋊C₄	semidirect
154. C₂×C₄⋊C₄	C₂×C₃×C₉	semidirect
156. #95	C₂×C₃×C₉	semidirect
157. C₃×C₉	#380	direct (N × H)
200. C₂²×C₄	C₃×C₄×C₉	semidirect
214. C₂²×C₃	C₉×C₄⋊C₄	semidirect
230. C₃×C₄	C₉×C₂²⋊C₄	semidirect
233. C₉	C₃×#380	direct (N × H)
264. C₂²	C₃×C₉×C₄⋊C₄	semidirect
272. C₄	C₃×C₉×C₂²⋊C₄	semidirect
273. C₃	C₉×#380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₃×C₉×#95	2
C₃²×#380	3
C₉×#380	3

Subgroups — C3×C9×#380 #13144898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#380 #13144898