Subgroups — C₃×C₄²×C₈³ #72201893

Order 24576.

1048860 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×C₄³×C₈²	C₂	1	7	yes	maximal
2	C₂×C₃×C₄×C₈³	C₂	1	24	yes	maximal
3	C₄²×C₈³	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×C₄⁴×C₈	C₂²	1	7	yes	—
5	C₂×C₃×C₄²×C₈²	C₄	1	112	yes	—
6	C₂×C₃×C₄²×C₈²	C₂²	1	84	yes	—
7	C₂²×C₃×C₈³	C₂²	1	64	yes	—
8	C₃×C₄×C₈³	C₄	1	384	yes	direct product factor
9	C₄³×C₈²	C₂×C₃	1	7	yes	—
10	C₂×C₄×C₈³	C₂×C₃	1	24	yes	—
11	C₃×C₄⁵	C₂³	1	1	yes	characteristic
12	C₂×C₃×C₄³×C₈	C₂×C₄	1	168	yes	—
13	C₂×C₃×C₄³×C₈	C₂³	1	42	yes	—
14	C₂²×C₃×C₄×C₈²	C₂×C₄	1	672	yes	—
15	C₂²×C₃×C₄×C₈²	C₂³	1	112	yes	—
16	C₃×C₄²×C₈²	C₈	1	1792	yes	direct product factor
17	C₃×C₄²×C₈²	C₂×C₄	1	1344	yes	—
18	C₂×C₃×C₈³	C₂×C₄	1	1536	yes	—
19	C₄⁴×C₈	C₂²×C₃	1	7	yes	—
20	C₂×C₄²×C₈²	C₃×C₄	1	112	yes	—
21	C₂×C₄²×C₈²	C₂²×C₃	1	84	yes	—
22	C₂²×C₈³	C₂²×C₃	1	64	yes	—
23	C₄×C₈³	C₃×C₄	1	384	yes	direct product factor
24	C₂×C₃×C₄⁴	C₂²×C₄	1	28	yes	—
25	C₂×C₃×C₄⁴	C₂⁴	1	3	yes	—
26	C₂²×C₃×C₄²×C₈	C₄²	1	448	yes	—
27	C₂²×C₃×C₄²×C₈	C₂²×C₄	1	504	yes	—
28	C₂²×C₃×C₄²×C₈	C₂⁴	1	28	yes	—
29	C₂³×C₃×C₈²	C₂²×C₄	1	448	yes	—
30	C₃×C₄³×C₈	C₂×C₈	1	2688	yes	—
31	C₃×C₄³×C₈	C₂²×C₄	1	672	yes	—
32	C₂×C₃×C₄×C₈²	C₄²	1	5376	yes	—
33	C₂×C₃×C₄×C₈²	C₂×C₈	1	10752	yes	—
34	C₂×C₃×C₄×C₈²	C₂²×C₄	1	2688	yes	—
35	C₃×C₈³	C₄²	1	4096	yes	direct product factor
36	C₄⁵	C₂³×C₃	1	1	yes	characteristic
37	C₂×C₄³×C₈	C₂×C₃×C₄	1	168	yes	—
38	C₂×C₄³×C₈	C₂³×C₃	1	42	yes	—
39	C₂²×C₄×C₈²	C₂×C₃×C₄	1	672	yes	—
40	C₂²×C₄×C₈²	C₂³×C₃	1	112	yes	—
41	C₄²×C₈²	C₃×C₈	1	1792	yes	direct product factor
42	C₄²×C₈²	C₂×C₃×C₄	1	1344	yes	—
43	C₂×C₈³	C₂×C₃×C₄	1	1536	yes	—
44	C₂³×C₃×C₄×C₈	C₂×C₄²	1	672	yes	—
45	C₂³×C₃×C₄×C₈	C₂³×C₄	1	168	yes	—
46	C₂²×C₃×C₄³	C₂×C₄²	1	112	yes	—
47	C₂²×C₃×C₄³	C₂³×C₄	1	42	yes	—
48	C₂²×C₃×C₄³	C₂⁵	1	1	yes	characteristic
49	C₂×C₃×C₄²×C₈	C₄×C₈	1	10752	yes	—
50	C₂×C₃×C₄²×C₈	C₂×C₄²	1	4032	yes	—
51	C₂×C₃×C₄²×C₈	C₂²×C₈	1	8064	yes	—
52	C₂×C₃×C₄²×C₈	C₂³×C₄	1	672	yes	—
53	C₂²×C₃×C₈²	C₂×C₄²	1	5376	yes	—
54	C₂²×C₃×C₈²	C₂²×C₈	1	7168	yes	—
55	C₃×C₄⁴	C₂²×C₈	1	448	yes	—
56	C₃×C₄⁴	C₂³×C₄	1	48	yes	—
57	C₃×C₄×C₈²	C₄×C₈	1	43008	yes	direct product factor
58	C₃×C₄×C₈²	C₂×C₄²	1	7168	yes	—
59	C₂²×C₄²×C₈	C₃×C₄²	1	448	yes	—
60	C₂²×C₄²×C₈	C₂²×C₃×C₄	1	504	yes	—
61	C₂²×C₄²×C₈	C₂⁴×C₃	1	28	yes	—
62	C₂³×C₈²	C₂²×C₃×C₄	1	448	yes	—
63	C₂×C₄⁴	C₂²×C₃×C₄	1	28	yes	—
64	C₂×C₄⁴	C₂⁴×C₃	1	3	yes	—
65	C₄³×C₈	C₂×C₃×C₈	1	2688	yes	—
66	C₄³×C₈	C₂²×C₃×C₄	1	672	yes	—
67	C₂×C₄×C₈²	C₃×C₄²	1	5376	yes	—
68	C₂×C₄×C₈²	C₂×C₃×C₈	1	10752	yes	—
69	C₂×C₄×C₈²	C₂²×C₃×C₄	1	2688	yes	—
70	C₈³	C₃×C₄²	1	4096	yes	direct product factor
71	C₂⁴×C₃×C₈	C₂²×C₄²	1	112	yes	—
72	C₂³×C₃×C₄²	C₄³	1	64	yes	—
73	C₂³×C₃×C₄²	C₂²×C₄²	1	84	yes	—
74	C₂³×C₃×C₄²	C₂⁴×C₄	1	7	yes	—
75	C₂²×C₃×C₄×C₈	C₄³	1	2688	yes	—
76	C₂²×C₃×C₄×C₈	C₂×C₄×C₈	1	16128	yes	—
77	C₂²×C₃×C₄×C₈	C₂²×C₄²	1	2016	yes	—
78	C₂²×C₃×C₄×C₈	C₂³×C₈	1	2688	yes	—
79	C₂×C₃×C₄³	C₂×C₄×C₈	1	2688	yes	—
80	C₂×C₃×C₄³	C₂²×C₄²	1	336	yes	—
81	C₂×C₃×C₄³	C₂³×C₈	1	672	yes	—
82	C₂×C₃×C₄³	C₂⁴×C₄	1	24	yes	—
83	C₃×C₄²×C₈	C₈²	1	28672	yes	direct product factor
84	C₃×C₄²×C₈	C₂×C₄×C₈	1	32256	yes	—
85	C₃×C₄²×C₈	C₂²×C₄²	1	1792	yes	—
86	C₂×C₃×C₈²	C₄³	1	7168	yes	—
87	C₂×C₃×C₈²	C₂×C₄×C₈	1	43008	yes	—
88	C₂³×C₄×C₈	C₂×C₃×C₄²	1	672	yes	—
89	C₂³×C₄×C₈	C₂³×C₃×C₄	1	168	yes	—
90	C₂²×C₄³	C₂×C₃×C₄²	1	112	yes	—
91	C₂²×C₄³	C₂³×C₃×C₄	1	42	yes	—
92	C₂²×C₄³	C₂⁵×C₃	1	1	yes	characteristic
93	C₂×C₄²×C₈	C₃×C₄×C₈	1	10752	yes	—
94	C₂×C₄²×C₈	C₂×C₃×C₄²	1	4032	yes	—
95	C₂×C₄²×C₈	C₂²×C₃×C₈	1	8064	yes	—
96	C₂×C₄²×C₈	C₂³×C₃×C₄	1	672	yes	—
97	C₂²×C₈²	C₂×C₃×C₄²	1	5376	yes	—
98	C₂²×C₈²	C₂²×C₃×C₈	1	7168	yes	—
99	C₄⁴	C₂²×C₃×C₈	1	448	yes	—
100	C₄⁴	C₂³×C₃×C₄	1	48	yes	—
101	C₄×C₈²	C₃×C₄×C₈	1	43008	yes	direct product factor
102	C₄×C₈²	C₂×C₃×C₄²	1	7168	yes	—
103	C₂⁴×C₃×C₄	C₂×C₄³	1	24	yes	—
104	C₂⁴×C₃×C₄	C₂³×C₄²	1	7	yes	—
105	C₂³×C₃×C₈	C₂×C₄³	1	672	yes	—
106	C₂³×C₃×C₈	C₂²×C₄×C₈	1	2688	yes	—
107	C₂²×C₃×C₄²	C₄²×C₈	1	1792	yes	—
108	C₂²×C₃×C₄²	C₂×C₄³	1	336	yes	—
109	C₂²×C₃×C₄²	C₂²×C₄×C₈	1	2016	yes	—
110	C₂²×C₃×C₄²	C₂³×C₄²	1	84	yes	—
111	C₂²×C₃×C₄²	C₂⁴×C₈	1	112	yes	—
112	C₂×C₃×C₄×C₈	C₂×C₈²	1	43008	yes	—
113	C₂×C₃×C₄×C₈	C₄²×C₈	1	32256	yes	—
114	C₂×C₃×C₄×C₈	C₂×C₄³	1	2688	yes	—
115	C₂×C₃×C₄×C₈	C₂²×C₄×C₈	1	16128	yes	—
116	C₃×C₄³	C₂×C₈²	1	7168	yes	—
117	C₃×C₄³	C₂²×C₄×C₈	1	2688	yes	—
118	C₃×C₄³	C₂³×C₄²	1	64	yes	—
119	C₃×C₈²	C₄²×C₈	1	28672	yes	direct product factor
120	C₂⁴×C₈	C₂²×C₃×C₄²	1	112	yes	—
121	C₂³×C₄²	C₃×C₄³	1	64	yes	—
122	C₂³×C₄²	C₂²×C₃×C₄²	1	84	yes	—
123	C₂³×C₄²	C₂⁴×C₃×C₄	1	7	yes	—
124	C₂²×C₄×C₈	C₃×C₄³	1	2688	yes	—
125	C₂²×C₄×C₈	C₂×C₃×C₄×C₈	1	16128	yes	—
126	C₂²×C₄×C₈	C₂²×C₃×C₄²	1	2016	yes	—
127	C₂²×C₄×C₈	C₂³×C₃×C₈	1	2688	yes	—
128	C₂×C₄³	C₂×C₃×C₄×C₈	1	2688	yes	—
129	C₂×C₄³	C₂²×C₃×C₄²	1	336	yes	—
130	C₂×C₄³	C₂³×C₃×C₈	1	672	yes	—
131	C₂×C₄³	C₂⁴×C₃×C₄	1	24	yes	—
132	C₄²×C₈	C₃×C₈²	1	28672	yes	direct product factor
133	C₄²×C₈	C₂×C₃×C₄×C₈	1	32256	yes	—
134	C₄²×C₈	C₂²×C₃×C₄²	1	1792	yes	—
135	C₂×C₈²	C₃×C₄³	1	7168	yes	—
136	C₂×C₈²	C₂×C₃×C₄×C₈	1	43008	yes	—
137	C₂⁵×C₃	C₂²×C₄³	1	1	yes	characteristic
138	C₂³×C₃×C₄	C₄⁴	1	48	yes	—
139	C₂³×C₃×C₄	C₂×C₄²×C₈	1	672	yes	—
140	C₂³×C₃×C₄	C₂²×C₄³	1	42	yes	—
141	C₂³×C₃×C₄	C₂³×C₄×C₈	1	168	yes	—
142	C₂²×C₃×C₈	C₄⁴	1	448	yes	—
143	C₂²×C₃×C₈	C₂²×C₈²	1	7168	yes	—
144	C₂²×C₃×C₈	C₂×C₄²×C₈	1	8064	yes	—
145	C₂×C₃×C₄²	C₄×C₈²	1	7168	yes	—
146	C₂×C₃×C₄²	C₂²×C₈²	1	5376	yes	—
147	C₂×C₃×C₄²	C₂×C₄²×C₈	1	4032	yes	—
148	C₂×C₃×C₄²	C₂²×C₄³	1	112	yes	—
149	C₂×C₃×C₄²	C₂³×C₄×C₈	1	672	yes	—
150	C₃×C₄×C₈	C₄×C₈²	1	43008	yes	direct product factor
151	C₃×C₄×C₈	C₂×C₄²×C₈	1	10752	yes	—
152	C₂⁴×C₄	C₂×C₃×C₄³	1	24	yes	—
153	C₂⁴×C₄	C₂³×C₃×C₄²	1	7	yes	—
154	C₂³×C₈	C₂×C₃×C₄³	1	672	yes	—
155	C₂³×C₈	C₂²×C₃×C₄×C₈	1	2688	yes	—
156	C₂²×C₄²	C₃×C₄²×C₈	1	1792	yes	—
157	C₂²×C₄²	C₂×C₃×C₄³	1	336	yes	—
158	C₂²×C₄²	C₂²×C₃×C₄×C₈	1	2016	yes	—
159	C₂²×C₄²	C₂³×C₃×C₄²	1	84	yes	—
160	C₂²×C₄²	C₂⁴×C₃×C₈	1	112	yes	—
161	C₂×C₄×C₈	C₂×C₃×C₈²	1	43008	yes	—
162	C₂×C₄×C₈	C₃×C₄²×C₈	1	32256	yes	—
163	C₂×C₄×C₈	C₂×C₃×C₄³	1	2688	yes	—
164	C₂×C₄×C₈	C₂²×C₃×C₄×C₈	1	16128	yes	—
165	C₄³	C₂×C₃×C₈²	1	7168	yes	—
166	C₄³	C₂²×C₃×C₄×C₈	1	2688	yes	—
167	C₄³	C₂³×C₃×C₄²	1	64	yes	—
168	C₈²	C₃×C₄²×C₈	1	28672	yes	direct product factor
169	C₂⁴×C₃	C₂×C₄⁴	1	3	yes	—
170	C₂⁴×C₃	C₂²×C₄²×C₈	1	28	yes	—
171	C₂²×C₃×C₄	C₂×C₄×C₈²	1	2688	yes	—
172	C₂²×C₃×C₄	C₄³×C₈	1	672	yes	—
173	C₂²×C₃×C₄	C₂×C₄⁴	1	28	yes	—
174	C₂²×C₃×C₄	C₂³×C₈²	1	448	yes	—
175	C₂²×C₃×C₄	C₂²×C₄²×C₈	1	504	yes	—
176	C₂×C₃×C₈	C₂×C₄×C₈²	1	10752	yes	—
177	C₂×C₃×C₈	C₄³×C₈	1	2688	yes	—
178	C₃×C₄²	C₈³	1	4096	yes	direct product factor
179	C₃×C₄²	C₂×C₄×C₈²	1	5376	yes	—
180	C₃×C₄²	C₂²×C₄²×C₈	1	448	yes	—
181	C₂⁵	C₂²×C₃×C₄³	1	1	yes	characteristic
182	C₂³×C₄	C₃×C₄⁴	1	48	yes	—
183	C₂³×C₄	C₂×C₃×C₄²×C₈	1	672	yes	—
184	C₂³×C₄	C₂²×C₃×C₄³	1	42	yes	—
185	C₂³×C₄	C₂³×C₃×C₄×C₈	1	168	yes	—
186	C₂²×C₈	C₃×C₄⁴	1	448	yes	—
187	C₂²×C₈	C₂²×C₃×C₈²	1	7168	yes	—
188	C₂²×C₈	C₂×C₃×C₄²×C₈	1	8064	yes	—
189	C₂×C₄²	C₃×C₄×C₈²	1	7168	yes	—
190	C₂×C₄²	C₂²×C₃×C₈²	1	5376	yes	—
191	C₂×C₄²	C₂×C₃×C₄²×C₈	1	4032	yes	—
192	C₂×C₄²	C₂²×C₃×C₄³	1	112	yes	—
193	C₂×C₄²	C₂³×C₃×C₄×C₈	1	672	yes	—
194	C₄×C₈	C₃×C₄×C₈²	1	43008	yes	direct product factor
195	C₄×C₈	C₂×C₃×C₄²×C₈	1	10752	yes	—
196	C₂³×C₃	C₂²×C₄×C₈²	1	112	yes	—
197	C₂³×C₃	C₂×C₄³×C₈	1	42	yes	—
198	C₂³×C₃	C₄⁵	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₈³	1	1536	yes	—
200	C₂×C₃×C₄	C₄²×C₈²	1	1344	yes	—
201	C₂×C₃×C₄	C₂²×C₄×C₈²	1	672	yes	—
202	C₂×C₃×C₄	C₂×C₄³×C₈	1	168	yes	—
203	C₃×C₈	C₄²×C₈²	1	1792	yes	direct product factor
204	C₂⁴	C₂²×C₃×C₄²×C₈	1	28	yes	—
205	C₂⁴	C₂×C₃×C₄⁴	1	3	yes	—
206	C₂²×C₄	C₂×C₃×C₄×C₈²	1	2688	yes	—
207	C₂²×C₄	C₃×C₄³×C₈	1	672	yes	—
208	C₂²×C₄	C₂³×C₃×C₈²	1	448	yes	—
209	C₂²×C₄	C₂²×C₃×C₄²×C₈	1	504	yes	—
210	C₂²×C₄	C₂×C₃×C₄⁴	1	28	yes	—
211	C₂×C₈	C₂×C₃×C₄×C₈²	1	10752	yes	—
212	C₂×C₈	C₃×C₄³×C₈	1	2688	yes	—
213	C₄²	C₃×C₈³	1	4096	yes	direct product factor
214	C₄²	C₂×C₃×C₄×C₈²	1	5376	yes	—
215	C₄²	C₂²×C₃×C₄²×C₈	1	448	yes	—
216	C₂²×C₃	C₂²×C₈³	1	64	yes	—
217	C₂²×C₃	C₂×C₄²×C₈²	1	84	yes	—
218	C₂²×C₃	C₄⁴×C₈	1	7	yes	—
219	C₃×C₄	C₄×C₈³	1	384	yes	direct product factor
220	C₃×C₄	C₂×C₄²×C₈²	1	112	yes	—
221	C₂³	C₂²×C₃×C₄×C₈²	1	112	yes	—
222	C₂³	C₂×C₃×C₄³×C₈	1	42	yes	—
223	C₂³	C₃×C₄⁵	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×C₈³	1	1536	yes	—
225	C₂×C₄	C₃×C₄²×C₈²	1	1344	yes	—
226	C₂×C₄	C₂²×C₃×C₄×C₈²	1	672	yes	—
227	C₂×C₄	C₂×C₃×C₄³×C₈	1	168	yes	—
228	C₈	C₃×C₄²×C₈²	1	1792	yes	direct product factor
229	C₂×C₃	C₂×C₄×C₈³	1	24	yes	—
230	C₂×C₃	C₄³×C₈²	1	7	yes	—
231	C₂²	C₂²×C₃×C₈³	1	64	yes	—
232	C₂²	C₂×C₃×C₄²×C₈²	1	84	yes	—
233	C₂²	C₃×C₄⁴×C₈	1	7	yes	—
234	C₄	C₃×C₄×C₈³	1	384	yes	direct product factor
235	C₄	C₂×C₃×C₄²×C₈²	1	112	yes	—
236	C₃	C₄²×C₈³	1	1	yes	characteristic, direct product factor, Sylow
237	C₂	C₂×C₃×C₄×C₈³	1	24	yes	—
238	C₂	C₃×C₄³×C₈²	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄³	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂²×C₄³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄²×C₈³	C₃	direct (N × H)
8. C₃×C₄×C₈³	C₄	direct (N × H)
16. C₃×C₄²×C₈²	C₈	direct (N × H)
23. C₄×C₈³	C₃×C₄	direct (N × H)
35. C₃×C₈³	C₄²	direct (N × H)
41. C₄²×C₈²	C₃×C₈	direct (N × H)
57. C₃×C₄×C₈²	C₄×C₈	direct (N × H)
70. C₈³	C₃×C₄²	direct (N × H)
83. C₃×C₄²×C₈	C₈²	direct (N × H)
101. C₄×C₈²	C₃×C₄×C₈	direct (N × H)
119. C₃×C₈²	C₄²×C₈	direct (N × H)
132. C₄²×C₈	C₃×C₈²	direct (N × H)
150. C₃×C₄×C₈	C₄×C₈²	direct (N × H)
168. C₈²	C₃×C₄²×C₈	direct (N × H)
178. C₃×C₄²	C₈³	direct (N × H)
194. C₄×C₈	C₃×C₄×C₈²	direct (N × H)
203. C₃×C₈	C₄²×C₈²	direct (N × H)
213. C₄²	C₃×C₈³	direct (N × H)
219. C₃×C₄	C₄×C₈³	direct (N × H)
228. C₈	C₃×C₄²×C₈²	direct (N × H)
234. C₄	C₃×C₄×C₈³	direct (N × H)
236. C₃	C₄²×C₈³	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×C₄³×C₈²	2
C₂×C₃×C₄×C₈³	2
C₄²×C₈³	3

Subgroups — C3×C42×C83 #72201893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₈³ #72201893