Subgroups — C₃×C₄³×C₈×C₉ #72192046

Order 13824.

33300 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⁴×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₈×C₉	C₂	1	14	yes	maximal
3	C₃²×C₄³×C₈	C₃	1	1	yes	maximal, characteristic
4	C₄³×C₈×C₉	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×C₄³×C₉	C₄	1	8	yes	—
6	C₂×C₃×C₄³×C₉	C₂²	1	7	yes	—
7	C₂²×C₃×C₄×C₈×C₉	C₂²	1	28	yes	—
8	C₃×C₄²×C₈×C₉	C₄	1	112	yes	direct product factor
9	C₄⁴×C₉	C₂×C₃	1	3	yes	—
10	C₃²×C₄⁴	C₂×C₃	1	1	yes	characteristic
11	C₂×C₃²×C₄²×C₈	C₂×C₃	1	14	yes	—
12	C₂×C₄²×C₈×C₉	C₂×C₃	1	42	yes	—
13	C₃×C₄³×C₈	C₉	1	3	yes	direct product factor
14	C₃×C₄³×C₈	C₃²	1	1	yes	characteristic
15	C₂³×C₃×C₈×C₉	C₂³	1	8	yes	—
16	C₂²×C₃×C₄²×C₉	C₂×C₄	1	28	yes	—
17	C₂²×C₃×C₄²×C₉	C₂³	1	7	yes	—
18	C₂×C₃×C₄×C₈×C₉	C₂×C₄	1	336	yes	—
19	C₃×C₄³×C₉	C₈	1	64	yes	direct product factor
20	C₃×C₄³×C₉	C₂×C₄	1	56	yes	—
21	C₂²×C₃²×C₄×C₈	C₂²×C₃	1	28	yes	—
22	C₂×C₃²×C₄³	C₃×C₄	1	8	yes	—
23	C₂×C₃²×C₄³	C₂²×C₃	1	7	yes	—
24	C₃²×C₄²×C₈	C₃×C₄	1	112	yes	—
25	C₂²×C₄×C₈×C₉	C₂²×C₃	1	84	yes	—
26	C₂×C₄³×C₉	C₃×C₄	1	24	yes	—
27	C₂×C₄³×C₉	C₂²×C₃	1	21	yes	—
28	C₄²×C₈×C₉	C₃×C₄	1	336	yes	direct product factor
29	C₂³×C₃×C₄×C₉	C₂²×C₄	1	14	yes	—
30	C₂³×C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
31	C₂²×C₃×C₈×C₉	C₂²×C₄	1	112	yes	—
32	C₂×C₃×C₄²×C₉	C₄²	1	112	yes	—
33	C₂×C₃×C₄²×C₉	C₂×C₈	1	224	yes	—
34	C₂×C₃×C₄²×C₉	C₂²×C₄	1	84	yes	—
35	C₃×C₄×C₈×C₉	C₄²	1	448	yes	direct product factor
36	C₂×C₃×C₄²×C₈	C₂×C₉	1	42	yes	—
37	C₂×C₃×C₄²×C₈	C₂×C₃²	1	14	yes	—
38	C₃×C₄⁴	C₂×C₉	1	3	yes	—
39	C₃×C₄⁴	C₂×C₃²	1	1	yes	characteristic
40	C₂³×C₃²×C₈	C₂³×C₃	1	8	yes	—
41	C₂²×C₃²×C₄²	C₂×C₃×C₄	1	28	yes	—
42	C₂²×C₃²×C₄²	C₂³×C₃	1	7	yes	—
43	C₂×C₃²×C₄×C₈	C₂×C₃×C₄	1	336	yes	—
44	C₃²×C₄³	C₃×C₈	1	64	yes	—
45	C₃²×C₄³	C₂×C₃×C₄	1	56	yes	—
46	C₂³×C₈×C₉	C₂³×C₃	1	24	yes	—
47	C₂²×C₄²×C₉	C₂×C₃×C₄	1	84	yes	—
48	C₂²×C₄²×C₉	C₂³×C₃	1	21	yes	—
49	C₂×C₄×C₈×C₉	C₂×C₃×C₄	1	1008	yes	—
50	C₄³×C₉	C₃×C₈	1	192	yes	direct product factor
51	C₄³×C₉	C₂×C₃×C₄	1	168	yes	—
52	C₄³×C₈	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂⁴×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄×C₉	C₂×C₄²	1	84	yes	—
55	C₂²×C₃×C₄×C₉	C₂²×C₈	1	112	yes	—
56	C₂²×C₃×C₄×C₉	C₂³×C₄	1	14	yes	—
57	C₂×C₃×C₈×C₉	C₂×C₄²	1	224	yes	—
58	C₃×C₄²×C₉	C₄×C₈	1	448	yes	direct product factor
59	C₃×C₄²×C₉	C₂×C₄²	1	112	yes	—
60	C₂²×C₃×C₄×C₈	C₂²×C₉	1	84	yes	—
61	C₂²×C₃×C₄×C₈	C₂²×C₃²	1	28	yes	—
62	C₂×C₃×C₄³	C₄×C₉	1	24	yes	—
63	C₂×C₃×C₄³	C₂²×C₉	1	21	yes	—
64	C₂×C₃×C₄³	C₃²×C₄	1	8	yes	—
65	C₂×C₃×C₄³	C₂²×C₃²	1	7	yes	—
66	C₃×C₄²×C₈	C₄×C₉	1	336	yes	direct product factor
67	C₃×C₄²×C₈	C₃²×C₄	1	112	yes	—
68	C₂³×C₃²×C₄	C₂²×C₃×C₄	1	14	yes	—
69	C₂³×C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
70	C₂²×C₃²×C₈	C₂²×C₃×C₄	1	112	yes	—
71	C₂×C₃²×C₄²	C₃×C₄²	1	112	yes	—
72	C₂×C₃²×C₄²	C₂×C₃×C₈	1	224	yes	—
73	C₂×C₃²×C₄²	C₂²×C₃×C₄	1	84	yes	—
74	C₂³×C₄×C₉	C₂²×C₃×C₄	1	42	yes	—
75	C₂³×C₄×C₉	C₂⁴×C₃	1	3	yes	—
76	C₃²×C₄×C₈	C₃×C₄²	1	448	yes	—
77	C₂²×C₈×C₉	C₂²×C₃×C₄	1	336	yes	—
78	C₂×C₄²×C₉	C₃×C₄²	1	336	yes	—
79	C₂×C₄²×C₉	C₂×C₃×C₈	1	672	yes	—
80	C₂×C₄²×C₉	C₂²×C₃×C₄	1	252	yes	—
81	C₄×C₈×C₉	C₃×C₄²	1	1344	yes	direct product factor
82	C₂×C₄²×C₈	C₂×C₃×C₉	1	14	yes	—
83	C₄⁴	C₂×C₃×C₉	1	1	yes	characteristic
84	C₂³×C₃×C₉	C₂²×C₄²	1	7	yes	—
85	C₂³×C₃×C₉	C₂³×C₈	1	8	yes	—
86	C₂×C₃×C₄×C₉	C₄³	1	56	yes	—
87	C₂×C₃×C₄×C₉	C₂×C₄×C₈	1	336	yes	—
88	C₂×C₃×C₄×C₉	C₂²×C₄²	1	28	yes	—
89	C₃×C₈×C₉	C₄³	1	64	yes	direct product factor
90	C₂³×C₃×C₈	C₂³×C₉	1	24	yes	—
91	C₂³×C₃×C₈	C₂³×C₃²	1	8	yes	—
92	C₂²×C₃×C₄²	C₂×C₄×C₉	1	84	yes	—
93	C₂²×C₃×C₄²	C₂³×C₉	1	21	yes	—
94	C₂²×C₃×C₄²	C₂×C₃²×C₄	1	28	yes	—
95	C₂²×C₃×C₄²	C₂³×C₃²	1	7	yes	—
96	C₂×C₃×C₄×C₈	C₂×C₄×C₉	1	1008	yes	—
97	C₂×C₃×C₄×C₈	C₂×C₃²×C₄	1	336	yes	—
98	C₃×C₄³	C₈×C₉	1	192	yes	direct product factor
99	C₃×C₄³	C₂×C₄×C₉	1	168	yes	—
100	C₃×C₄³	C₃²×C₈	1	64	yes	—
101	C₃×C₄³	C₂×C₃²×C₄	1	56	yes	—
102	C₂⁴×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
103	C₂²×C₃²×C₄	C₂×C₃×C₄²	1	84	yes	—
104	C₂²×C₃²×C₄	C₂²×C₃×C₈	1	112	yes	—
105	C₂²×C₃²×C₄	C₂³×C₃×C₄	1	14	yes	—
106	C₂⁴×C₉	C₂³×C₃×C₄	1	3	yes	—
107	C₂×C₃²×C₈	C₂×C₃×C₄²	1	224	yes	—
108	C₃²×C₄²	C₃×C₄×C₈	1	448	yes	—
109	C₃²×C₄²	C₂×C₃×C₄²	1	112	yes	—
110	C₂²×C₄×C₉	C₂×C₃×C₄²	1	252	yes	—
111	C₂²×C₄×C₉	C₂²×C₃×C₈	1	336	yes	—
112	C₂²×C₄×C₉	C₂³×C₃×C₄	1	42	yes	—
113	C₂×C₈×C₉	C₂×C₃×C₄²	1	672	yes	—
114	C₄²×C₉	C₃×C₄×C₈	1	1344	yes	direct product factor
115	C₄²×C₉	C₂×C₃×C₄²	1	336	yes	—
116	C₂²×C₄×C₈	C₂²×C₃×C₉	1	28	yes	—
117	C₂×C₄³	C₃×C₄×C₉	1	8	yes	—
118	C₂×C₄³	C₂²×C₃×C₉	1	7	yes	—
119	C₄²×C₈	C₃×C₄×C₉	1	112	yes	direct product factor
120	C₂²×C₃×C₉	C₂×C₄³	1	7	yes	—
121	C₂²×C₃×C₉	C₂²×C₄×C₈	1	28	yes	—
122	C₃×C₄×C₉	C₄²×C₈	1	112	yes	direct product factor
123	C₃×C₄×C₉	C₂×C₄³	1	8	yes	—
124	C₂³×C₃×C₄	C₂²×C₄×C₉	1	42	yes	—
125	C₂³×C₃×C₄	C₂⁴×C₉	1	3	yes	—
126	C₂³×C₃×C₄	C₂²×C₃²×C₄	1	14	yes	—
127	C₂³×C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
128	C₂²×C₃×C₈	C₂²×C₄×C₉	1	336	yes	—
129	C₂²×C₃×C₈	C₂²×C₃²×C₄	1	112	yes	—
130	C₂×C₃×C₄²	C₄²×C₉	1	336	yes	—
131	C₂×C₃×C₄²	C₂×C₈×C₉	1	672	yes	—
132	C₂×C₃×C₄²	C₂²×C₄×C₉	1	252	yes	—
133	C₂×C₃×C₄²	C₃²×C₄²	1	112	yes	—
134	C₂×C₃×C₄²	C₂×C₃²×C₈	1	224	yes	—
135	C₂×C₃×C₄²	C₂²×C₃²×C₄	1	84	yes	—
136	C₃×C₄×C₈	C₄²×C₉	1	1344	yes	direct product factor
137	C₃×C₄×C₈	C₃²×C₄²	1	448	yes	—
138	C₂³×C₃²	C₂²×C₃×C₄²	1	7	yes	—
139	C₂³×C₃²	C₂³×C₃×C₈	1	8	yes	—
140	C₂×C₃²×C₄	C₃×C₄³	1	56	yes	—
141	C₂×C₃²×C₄	C₂×C₃×C₄×C₈	1	336	yes	—
142	C₂×C₃²×C₄	C₂²×C₃×C₄²	1	28	yes	—
143	C₂³×C₉	C₂²×C₃×C₄²	1	21	yes	—
144	C₂³×C₉	C₂³×C₃×C₈	1	24	yes	—
145	C₃²×C₈	C₃×C₄³	1	64	yes	—
146	C₂×C₄×C₉	C₃×C₄³	1	168	yes	—
147	C₂×C₄×C₉	C₂×C₃×C₄×C₈	1	1008	yes	—
148	C₂×C₄×C₉	C₂²×C₃×C₄²	1	84	yes	—
149	C₈×C₉	C₃×C₄³	1	192	yes	direct product factor
150	C₂³×C₈	C₂³×C₃×C₉	1	8	yes	—
151	C₂²×C₄²	C₂×C₃×C₄×C₉	1	28	yes	—
152	C₂²×C₄²	C₂³×C₃×C₉	1	7	yes	—
153	C₂×C₄×C₈	C₂×C₃×C₄×C₉	1	336	yes	—
154	C₄³	C₃×C₈×C₉	1	64	yes	direct product factor
155	C₄³	C₂×C₃×C₄×C₉	1	56	yes	—
156	C₂×C₃×C₉	C₄⁴	1	1	yes	characteristic
157	C₂×C₃×C₉	C₂×C₄²×C₈	1	14	yes	—
158	C₂⁴×C₃	C₂³×C₄×C₉	1	3	yes	—
159	C₂⁴×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₂×C₄²×C₉	1	252	yes	—
161	C₂²×C₃×C₄	C₂²×C₈×C₉	1	336	yes	—
162	C₂²×C₃×C₄	C₂³×C₄×C₉	1	42	yes	—
163	C₂²×C₃×C₄	C₂×C₃²×C₄²	1	84	yes	—
164	C₂²×C₃×C₄	C₂²×C₃²×C₈	1	112	yes	—
165	C₂²×C₃×C₄	C₂³×C₃²×C₄	1	14	yes	—
166	C₂×C₃×C₈	C₂×C₄²×C₉	1	672	yes	—
167	C₂×C₃×C₈	C₂×C₃²×C₄²	1	224	yes	—
168	C₃×C₄²	C₄×C₈×C₉	1	1344	yes	direct product factor
169	C₃×C₄²	C₂×C₄²×C₉	1	336	yes	—
170	C₃×C₄²	C₃²×C₄×C₈	1	448	yes	—
171	C₃×C₄²	C₂×C₃²×C₄²	1	112	yes	—
172	C₂²×C₃²	C₂×C₃×C₄³	1	7	yes	—
173	C₂²×C₃²	C₂²×C₃×C₄×C₈	1	28	yes	—
174	C₃²×C₄	C₃×C₄²×C₈	1	112	yes	—
175	C₃²×C₄	C₂×C₃×C₄³	1	8	yes	—
176	C₂²×C₉	C₂×C₃×C₄³	1	21	yes	—
177	C₂²×C₉	C₂²×C₃×C₄×C₈	1	84	yes	—
178	C₄×C₉	C₃×C₄²×C₈	1	336	yes	direct product factor
179	C₄×C₉	C₂×C₃×C₄³	1	24	yes	—
180	C₂³×C₄	C₂²×C₃×C₄×C₉	1	14	yes	—
181	C₂³×C₄	C₂⁴×C₃×C₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²×C₃×C₄×C₉	1	112	yes	—
183	C₂×C₄²	C₃×C₄²×C₉	1	112	yes	—
184	C₂×C₄²	C₂×C₃×C₈×C₉	1	224	yes	—
185	C₂×C₄²	C₂²×C₃×C₄×C₉	1	84	yes	—
186	C₄×C₈	C₃×C₄²×C₉	1	448	yes	direct product factor
187	C₃×C₉	C₄³×C₈	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³×C₃	C₂²×C₄²×C₉	1	21	yes	—
189	C₂³×C₃	C₂³×C₈×C₉	1	24	yes	—
190	C₂³×C₃	C₂²×C₃²×C₄²	1	7	yes	—
191	C₂³×C₃	C₂³×C₃²×C₈	1	8	yes	—
192	C₂×C₃×C₄	C₄³×C₉	1	168	yes	—
193	C₂×C₃×C₄	C₂×C₄×C₈×C₉	1	1008	yes	—
194	C₂×C₃×C₄	C₂²×C₄²×C₉	1	84	yes	—
195	C₂×C₃×C₄	C₃²×C₄³	1	56	yes	—
196	C₂×C₃×C₄	C₂×C₃²×C₄×C₈	1	336	yes	—
197	C₂×C₃×C₄	C₂²×C₃²×C₄²	1	28	yes	—
198	C₃×C₈	C₄³×C₉	1	192	yes	direct product factor
199	C₃×C₈	C₃²×C₄³	1	64	yes	—
200	C₂×C₃²	C₃×C₄⁴	1	1	yes	characteristic
201	C₂×C₃²	C₂×C₃×C₄²×C₈	1	14	yes	—
202	C₂×C₉	C₃×C₄⁴	1	3	yes	—
203	C₂×C₉	C₂×C₃×C₄²×C₈	1	42	yes	—
204	C₂⁴	C₂³×C₃×C₄×C₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₃×C₄²×C₉	1	84	yes	—
206	C₂²×C₄	C₂²×C₃×C₈×C₉	1	112	yes	—
207	C₂²×C₄	C₂³×C₃×C₄×C₉	1	14	yes	—
208	C₂×C₈	C₂×C₃×C₄²×C₉	1	224	yes	—
209	C₄²	C₃×C₄×C₈×C₉	1	448	yes	direct product factor
210	C₄²	C₂×C₃×C₄²×C₉	1	112	yes	—
211	C₂²×C₃	C₂×C₄³×C₉	1	21	yes	—
212	C₂²×C₃	C₂²×C₄×C₈×C₉	1	84	yes	—
213	C₂²×C₃	C₂×C₃²×C₄³	1	7	yes	—
214	C₂²×C₃	C₂²×C₃²×C₄×C₈	1	28	yes	—
215	C₃×C₄	C₄²×C₈×C₉	1	336	yes	direct product factor
216	C₃×C₄	C₂×C₄³×C₉	1	24	yes	—
217	C₃×C₄	C₃²×C₄²×C₈	1	112	yes	—
218	C₃×C₄	C₂×C₃²×C₄³	1	8	yes	—
219	C₃²	C₃×C₄³×C₈	1	1	yes	characteristic
220	C₉	C₃×C₄³×C₈	1	3	yes	direct product factor
221	C₂³	C₂²×C₃×C₄²×C₉	1	7	yes	—
222	C₂³	C₂³×C₃×C₈×C₉	1	8	yes	—
223	C₂×C₄	C₃×C₄³×C₉	1	56	yes	—
224	C₂×C₄	C₂×C₃×C₄×C₈×C₉	1	336	yes	—
225	C₂×C₄	C₂²×C₃×C₄²×C₉	1	28	yes	—
226	C₈	C₃×C₄³×C₉	1	64	yes	direct product factor
227	C₂×C₃	C₂×C₄²×C₈×C₉	1	42	yes	—
228	C₂×C₃	C₂×C₃²×C₄²×C₈	1	14	yes	—
229	C₂×C₃	C₃²×C₄⁴	1	1	yes	characteristic
230	C₂×C₃	C₄⁴×C₉	1	3	yes	—
231	C₂²	C₂²×C₃×C₄×C₈×C₉	1	28	yes	—
232	C₂²	C₂×C₃×C₄³×C₉	1	7	yes	—
233	C₄	C₃×C₄²×C₈×C₉	1	112	yes	direct product factor
234	C₄	C₂×C₃×C₄³×C₉	1	8	yes	—
235	C₃	C₄³×C₈×C₉	1	3	yes	direct product factor
236	C₃	C₃²×C₄³×C₈	1	1	yes	characteristic
237	C₂	C₂×C₃×C₄²×C₈×C₉	1	14	yes	—
238	C₂	C₃×C₄⁴×C₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₄³×C₈×C₉	C₃	direct (N × H)
8. C₃×C₄²×C₈×C₉	C₄	direct (N × H)
13. C₃×C₄³×C₈	C₉	direct (N × H)
19. C₃×C₄³×C₉	C₈	direct (N × H)
28. C₄²×C₈×C₉	C₃×C₄	direct (N × H)
35. C₃×C₄×C₈×C₉	C₄²	direct (N × H)
50. C₄³×C₉	C₃×C₈	direct (N × H)
52. C₄³×C₈	C₃×C₉	direct (N × H)
58. C₃×C₄²×C₉	C₄×C₈	direct (N × H)
66. C₃×C₄²×C₈	C₄×C₉	direct (N × H)
81. C₄×C₈×C₉	C₃×C₄²	direct (N × H)
89. C₃×C₈×C₉	C₄³	direct (N × H)
98. C₃×C₄³	C₈×C₉	direct (N × H)
114. C₄²×C₉	C₃×C₄×C₈	direct (N × H)
119. C₄²×C₈	C₃×C₄×C₉	direct (N × H)
122. C₃×C₄×C₉	C₄²×C₈	direct (N × H)
136. C₃×C₄×C₈	C₄²×C₉	direct (N × H)
149. C₈×C₉	C₃×C₄³	direct (N × H)
154. C₄³	C₃×C₈×C₉	direct (N × H)
168. C₃×C₄²	C₄×C₈×C₉	direct (N × H)
178. C₄×C₉	C₃×C₄²×C₈	direct (N × H)
186. C₄×C₈	C₃×C₄²×C₉	direct (N × H)
187. C₃×C₉	C₄³×C₈	direct (N × H)
198. C₃×C₈	C₄³×C₉	direct (N × H)
209. C₄²	C₃×C₄×C₈×C₉	direct (N × H)
215. C₃×C₄	C₄²×C₈×C₉	direct (N × H)
220. C₉	C₃×C₄³×C₈	direct (N × H)
226. C₈	C₃×C₄³×C₉	direct (N × H)
233. C₄	C₃×C₄²×C₈×C₉	direct (N × H)
235. C₃	C₄³×C₈×C₉	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×C₄⁴×C₉	2
C₂×C₃×C₄²×C₈×C₉	2
C₃²×C₄³×C₈	3
C₄³×C₈×C₉	3

Subgroups — C3×C43×C8×C9 #72192046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄³×C₈×C₉ #72192046