Subgroups — C₂⁷×C₃×C₈² #72201900

Order 24576.

229062428 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×C₃×C₄×C₈	C₂	1	3	yes	maximal
2	C₂⁶×C₃×C₈²	C₂	1	508	yes	maximal, direct product factor
3	C₂⁷×C₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁷×C₃×C₄²	C₂²	1	1	yes	characteristic
5	C₂⁶×C₃×C₄×C₈	C₄	1	762	yes	—
6	C₂⁶×C₃×C₄×C₈	C₂²	1	762	yes	—
7	C₂⁸×C₃×C₈	C₄	1	6	yes	—
8	C₂⁵×C₃×C₈²	C₂²	1	42672	yes	direct product factor
9	C₂⁷×C₄×C₈	C₂×C₃	1	3	yes	—
10	C₂⁶×C₈²	C₂×C₃	1	508	yes	direct product factor
11	C₂⁸×C₃×C₄	C₂×C₄	1	3	yes	—
12	C₂⁶×C₃×C₄²	C₂×C₄	1	381	yes	—
13	C₂⁶×C₃×C₄²	C₂³	1	127	yes	—
14	C₂⁷×C₃×C₈	C₈	1	1536	yes	direct product factor
15	C₂⁷×C₃×C₈	C₂×C₄	1	1524	yes	—
16	C₂⁵×C₃×C₄×C₈	C₂×C₄	1	96012	yes	—
17	C₂⁵×C₃×C₄×C₈	C₂³	1	32004	yes	—
18	C₂⁴×C₃×C₈²	C₂³	1	755904	yes	direct product factor
19	C₂⁷×C₄²	C₂²×C₃	1	1	yes	characteristic
20	C₂⁸×C₈	C₃×C₄	1	6	yes	—
21	C₂⁶×C₄×C₈	C₃×C₄	1	762	yes	—
22	C₂⁶×C₄×C₈	C₂²×C₃	1	762	yes	—
23	C₂⁵×C₈²	C₂²×C₃	1	42672	yes	direct product factor
24	C₂⁷×C₃×C₄	C₄²	1	381	yes	—
25	C₂⁷×C₃×C₄	C₂×C₈	1	768	yes	—
26	C₂⁷×C₃×C₄	C₂²×C₄	1	381	yes	—
27	C₂⁹×C₃	C₄²	1	1	yes	characteristic
28	C₂⁵×C₃×C₄²	C₄²	1	16002	yes	—
29	C₂⁵×C₃×C₄²	C₂²×C₄	1	24003	yes	—
30	C₂⁵×C₃×C₄²	C₂⁴	1	2667	yes	—
31	C₂⁶×C₃×C₈	C₂×C₈	1	195072	yes	direct product factor
32	C₂⁶×C₃×C₈	C₂²×C₄	1	64008	yes	—
33	C₂⁴×C₃×C₄×C₈	C₂²×C₄	1	1984248	yes	—
34	C₂⁴×C₃×C₄×C₈	C₂⁴	1	283464	yes	—
35	C₂³×C₃×C₈²	C₂⁴	1	3023616	yes	direct product factor
36	C₂⁶×C₄²	C₂×C₃×C₄	1	381	yes	—
37	C₂⁶×C₄²	C₂³×C₃	1	127	yes	—
38	C₂⁸×C₄	C₂×C₃×C₄	1	3	yes	—
39	C₂⁵×C₄×C₈	C₂×C₃×C₄	1	96012	yes	—
40	C₂⁵×C₄×C₈	C₂³×C₃	1	32004	yes	—
41	C₂⁷×C₈	C₃×C₈	1	1536	yes	direct product factor
42	C₂⁷×C₈	C₂×C₃×C₄	1	1524	yes	—
43	C₂⁴×C₈²	C₂³×C₃	1	755904	yes	direct product factor
44	C₂⁸×C₃	C₄×C₈	1	384	yes	—
45	C₂⁸×C₃	C₂×C₄²	1	127	yes	—
46	C₂⁶×C₃×C₄	C₄×C₈	1	48768	yes	—
47	C₂⁶×C₃×C₄	C₂×C₄²	1	24003	yes	—
48	C₂⁶×C₃×C₄	C₂²×C₈	1	48768	yes	—
49	C₂⁶×C₃×C₄	C₂³×C₄	1	8001	yes	—
50	C₂⁵×C₃×C₈	C₂²×C₈	1	4096512	yes	direct product factor
51	C₂⁵×C₃×C₈	C₂³×C₄	1	566928	yes	—
52	C₂⁴×C₃×C₄²	C₂×C₄²	1	496062	yes	—
53	C₂⁴×C₃×C₄²	C₂³×C₄	1	248031	yes	—
54	C₂⁴×C₃×C₄²	C₂⁵	1	11811	yes	—
55	C₂³×C₃×C₄×C₈	C₂³×C₄	1	8503920	yes	—
56	C₂³×C₃×C₄×C₈	C₂⁵	1	566928	yes	—
57	C₂²×C₃×C₈²	C₂⁵	1	2731008	yes	direct product factor
58	C₂⁹	C₃×C₄²	1	1	yes	characteristic
59	C₂⁷×C₄	C₃×C₄²	1	381	yes	—
60	C₂⁷×C₄	C₂×C₃×C₈	1	768	yes	—
61	C₂⁷×C₄	C₂²×C₃×C₄	1	381	yes	—
62	C₂⁶×C₈	C₂×C₃×C₈	1	195072	yes	direct product factor
63	C₂⁶×C₈	C₂²×C₃×C₄	1	64008	yes	—
64	C₂⁵×C₄²	C₃×C₄²	1	16002	yes	—
65	C₂⁵×C₄²	C₂²×C₃×C₄	1	24003	yes	—
66	C₂⁵×C₄²	C₂⁴×C₃	1	2667	yes	—
67	C₂⁴×C₄×C₈	C₂²×C₃×C₄	1	1984248	yes	—
68	C₂⁴×C₄×C₈	C₂⁴×C₃	1	283464	yes	—
69	C₂³×C₈²	C₂⁴×C₃	1	3023616	yes	direct product factor
70	C₂⁷×C₃	C₈²	1	16384	yes	direct product factor
71	C₂⁷×C₃	C₂×C₄×C₈	1	24384	yes	—
72	C₂⁷×C₃	C₂²×C₄²	1	2667	yes	—
73	C₂⁵×C₃×C₄	C₂×C₄×C₈	1	1536192	yes	—
74	C₂⁵×C₃×C₄	C₂²×C₄²	1	248031	yes	—
75	C₂⁵×C₃×C₄	C₂³×C₈	1	512064	yes	—
76	C₂⁵×C₃×C₄	C₂⁴×C₄	1	35433	yes	—
77	C₂⁴×C₃×C₈	C₂³×C₈	1	18141696	yes	direct product factor
78	C₂⁴×C₃×C₈	C₂⁴×C₄	1	1133856	yes	—
79	C₂³×C₃×C₄²	C₂²×C₄²	1	2480310	yes	—
80	C₂³×C₃×C₄²	C₂⁴×C₄	1	531495	yes	—
81	C₂³×C₃×C₄²	C₂⁶	1	11811	yes	—
82	C₂²×C₃×C₄×C₈	C₂⁴×C₄	1	7936992	yes	—
83	C₂²×C₃×C₄×C₈	C₂⁶	1	256032	yes	—
84	C₂×C₃×C₈²	C₂⁶	1	520192	yes	direct product factor
85	C₂⁸	C₃×C₄×C₈	1	384	yes	—
86	C₂⁸	C₂×C₃×C₄²	1	127	yes	—
87	C₂⁶×C₄	C₃×C₄×C₈	1	48768	yes	—
88	C₂⁶×C₄	C₂×C₃×C₄²	1	24003	yes	—
89	C₂⁶×C₄	C₂²×C₃×C₈	1	48768	yes	—
90	C₂⁶×C₄	C₂³×C₃×C₄	1	8001	yes	—
91	C₂⁵×C₈	C₂²×C₃×C₈	1	4096512	yes	direct product factor
92	C₂⁵×C₈	C₂³×C₃×C₄	1	566928	yes	—
93	C₂⁴×C₄²	C₂×C₃×C₄²	1	496062	yes	—
94	C₂⁴×C₄²	C₂³×C₃×C₄	1	248031	yes	—
95	C₂⁴×C₄²	C₂⁵×C₃	1	11811	yes	—
96	C₂³×C₄×C₈	C₂³×C₃×C₄	1	8503920	yes	—
97	C₂³×C₄×C₈	C₂⁵×C₃	1	566928	yes	—
98	C₂²×C₈²	C₂⁵×C₃	1	2731008	yes	direct product factor
99	C₂⁶×C₃	C₂×C₈²	1	520192	yes	direct product factor
100	C₂⁶×C₃	C₂²×C₄×C₈	1	256032	yes	—
101	C₂⁶×C₃	C₂³×C₄²	1	11811	yes	—
102	C₂⁴×C₃×C₄	C₂²×C₄×C₈	1	7936992	yes	—
103	C₂⁴×C₃×C₄	C₂³×C₄²	1	531495	yes	—
104	C₂⁴×C₃×C₄	C₂⁴×C₈	1	1133856	yes	—
105	C₂⁴×C₃×C₄	C₂⁵×C₄	1	35433	yes	—
106	C₂³×C₃×C₈	C₂⁴×C₈	1	18141696	yes	direct product factor
107	C₂³×C₃×C₈	C₂⁵×C₄	1	512064	yes	—
108	C₂²×C₃×C₄²	C₂³×C₄²	1	2480310	yes	—
109	C₂²×C₃×C₄²	C₂⁵×C₄	1	248031	yes	—
110	C₂²×C₃×C₄²	C₂⁷	1	2667	yes	—
111	C₂×C₃×C₄×C₈	C₂⁵×C₄	1	1536192	yes	—
112	C₂×C₃×C₄×C₈	C₂⁷	1	24384	yes	—
113	C₃×C₈²	C₂⁷	1	16384	yes	direct product factor
114	C₂⁷	C₃×C₈²	1	16384	yes	direct product factor
115	C₂⁷	C₂×C₃×C₄×C₈	1	24384	yes	—
116	C₂⁷	C₂²×C₃×C₄²	1	2667	yes	—
117	C₂⁵×C₄	C₂×C₃×C₄×C₈	1	1536192	yes	—
118	C₂⁵×C₄	C₂²×C₃×C₄²	1	248031	yes	—
119	C₂⁵×C₄	C₂³×C₃×C₈	1	512064	yes	—
120	C₂⁵×C₄	C₂⁴×C₃×C₄	1	35433	yes	—
121	C₂⁴×C₈	C₂³×C₃×C₈	1	18141696	yes	direct product factor
122	C₂⁴×C₈	C₂⁴×C₃×C₄	1	1133856	yes	—
123	C₂³×C₄²	C₂²×C₃×C₄²	1	2480310	yes	—
124	C₂³×C₄²	C₂⁴×C₃×C₄	1	531495	yes	—
125	C₂³×C₄²	C₂⁶×C₃	1	11811	yes	—
126	C₂²×C₄×C₈	C₂⁴×C₃×C₄	1	7936992	yes	—
127	C₂²×C₄×C₈	C₂⁶×C₃	1	256032	yes	—
128	C₂×C₈²	C₂⁶×C₃	1	520192	yes	direct product factor
129	C₂⁵×C₃	C₂²×C₈²	1	2731008	yes	direct product factor
130	C₂⁵×C₃	C₂³×C₄×C₈	1	566928	yes	—
131	C₂⁵×C₃	C₂⁴×C₄²	1	11811	yes	—
132	C₂³×C₃×C₄	C₂³×C₄×C₈	1	8503920	yes	—
133	C₂³×C₃×C₄	C₂⁴×C₄²	1	248031	yes	—
134	C₂³×C₃×C₄	C₂⁵×C₈	1	566928	yes	—
135	C₂³×C₃×C₄	C₂⁶×C₄	1	8001	yes	—
136	C₂²×C₃×C₈	C₂⁵×C₈	1	4096512	yes	direct product factor
137	C₂²×C₃×C₈	C₂⁶×C₄	1	48768	yes	—
138	C₂×C₃×C₄²	C₂⁴×C₄²	1	496062	yes	—
139	C₂×C₃×C₄²	C₂⁶×C₄	1	24003	yes	—
140	C₂×C₃×C₄²	C₂⁸	1	127	yes	—
141	C₃×C₄×C₈	C₂⁶×C₄	1	48768	yes	—
142	C₃×C₄×C₈	C₂⁸	1	384	yes	—
143	C₂⁶	C₂×C₃×C₈²	1	520192	yes	direct product factor
144	C₂⁶	C₂²×C₃×C₄×C₈	1	256032	yes	—
145	C₂⁶	C₂³×C₃×C₄²	1	11811	yes	—
146	C₂⁴×C₄	C₂²×C₃×C₄×C₈	1	7936992	yes	—
147	C₂⁴×C₄	C₂³×C₃×C₄²	1	531495	yes	—
148	C₂⁴×C₄	C₂⁴×C₃×C₈	1	1133856	yes	—
149	C₂⁴×C₄	C₂⁵×C₃×C₄	1	35433	yes	—
150	C₂³×C₈	C₂⁴×C₃×C₈	1	18141696	yes	direct product factor
151	C₂³×C₈	C₂⁵×C₃×C₄	1	512064	yes	—
152	C₂²×C₄²	C₂³×C₃×C₄²	1	2480310	yes	—
153	C₂²×C₄²	C₂⁵×C₃×C₄	1	248031	yes	—
154	C₂²×C₄²	C₂⁷×C₃	1	2667	yes	—
155	C₂×C₄×C₈	C₂⁵×C₃×C₄	1	1536192	yes	—
156	C₂×C₄×C₈	C₂⁷×C₃	1	24384	yes	—
157	C₈²	C₂⁷×C₃	1	16384	yes	direct product factor
158	C₂⁴×C₃	C₂³×C₈²	1	3023616	yes	direct product factor
159	C₂⁴×C₃	C₂⁴×C₄×C₈	1	283464	yes	—
160	C₂⁴×C₃	C₂⁵×C₄²	1	2667	yes	—
161	C₂²×C₃×C₄	C₂⁴×C₄×C₈	1	1984248	yes	—
162	C₂²×C₃×C₄	C₂⁵×C₄²	1	24003	yes	—
163	C₂²×C₃×C₄	C₂⁶×C₈	1	64008	yes	—
164	C₂²×C₃×C₄	C₂⁷×C₄	1	381	yes	—
165	C₂×C₃×C₈	C₂⁶×C₈	1	195072	yes	direct product factor
166	C₂×C₃×C₈	C₂⁷×C₄	1	768	yes	—
167	C₃×C₄²	C₂⁵×C₄²	1	16002	yes	—
168	C₃×C₄²	C₂⁷×C₄	1	381	yes	—
169	C₃×C₄²	C₂⁹	1	1	yes	characteristic
170	C₂⁵	C₂²×C₃×C₈²	1	2731008	yes	direct product factor
171	C₂⁵	C₂³×C₃×C₄×C₈	1	566928	yes	—
172	C₂⁵	C₂⁴×C₃×C₄²	1	11811	yes	—
173	C₂³×C₄	C₂³×C₃×C₄×C₈	1	8503920	yes	—
174	C₂³×C₄	C₂⁴×C₃×C₄²	1	248031	yes	—
175	C₂³×C₄	C₂⁵×C₃×C₈	1	566928	yes	—
176	C₂³×C₄	C₂⁶×C₃×C₄	1	8001	yes	—
177	C₂²×C₈	C₂⁵×C₃×C₈	1	4096512	yes	direct product factor
178	C₂²×C₈	C₂⁶×C₃×C₄	1	48768	yes	—
179	C₂×C₄²	C₂⁴×C₃×C₄²	1	496062	yes	—
180	C₂×C₄²	C₂⁶×C₃×C₄	1	24003	yes	—
181	C₂×C₄²	C₂⁸×C₃	1	127	yes	—
182	C₄×C₈	C₂⁶×C₃×C₄	1	48768	yes	—
183	C₄×C₈	C₂⁸×C₃	1	384	yes	—
184	C₂³×C₃	C₂⁴×C₈²	1	755904	yes	direct product factor
185	C₂³×C₃	C₂⁵×C₄×C₈	1	32004	yes	—
186	C₂³×C₃	C₂⁶×C₄²	1	127	yes	—
187	C₂×C₃×C₄	C₂⁷×C₈	1	1524	yes	—
188	C₂×C₃×C₄	C₂⁵×C₄×C₈	1	96012	yes	—
189	C₂×C₃×C₄	C₂⁸×C₄	1	3	yes	—
190	C₂×C₃×C₄	C₂⁶×C₄²	1	381	yes	—
191	C₃×C₈	C₂⁷×C₈	1	1536	yes	direct product factor
192	C₂⁴	C₂³×C₃×C₈²	1	3023616	yes	direct product factor
193	C₂⁴	C₂⁴×C₃×C₄×C₈	1	283464	yes	—
194	C₂⁴	C₂⁵×C₃×C₄²	1	2667	yes	—
195	C₂²×C₄	C₂⁴×C₃×C₄×C₈	1	1984248	yes	—
196	C₂²×C₄	C₂⁶×C₃×C₈	1	64008	yes	—
197	C₂²×C₄	C₂⁵×C₃×C₄²	1	24003	yes	—
198	C₂²×C₄	C₂⁷×C₃×C₄	1	381	yes	—
199	C₂×C₈	C₂⁶×C₃×C₈	1	195072	yes	direct product factor
200	C₂×C₈	C₂⁷×C₃×C₄	1	768	yes	—
201	C₄²	C₂⁵×C₃×C₄²	1	16002	yes	—
202	C₄²	C₂⁹×C₃	1	1	yes	characteristic
203	C₄²	C₂⁷×C₃×C₄	1	381	yes	—
204	C₂²×C₃	C₂⁵×C₈²	1	42672	yes	direct product factor
205	C₂²×C₃	C₂⁶×C₄×C₈	1	762	yes	—
206	C₂²×C₃	C₂⁷×C₄²	1	1	yes	characteristic
207	C₃×C₄	C₂⁶×C₄×C₈	1	762	yes	—
208	C₃×C₄	C₂⁸×C₈	1	6	yes	—
209	C₂³	C₂⁴×C₃×C₈²	1	755904	yes	direct product factor
210	C₂³	C₂⁵×C₃×C₄×C₈	1	32004	yes	—
211	C₂³	C₂⁶×C₃×C₄²	1	127	yes	—
212	C₂×C₄	C₂⁵×C₃×C₄×C₈	1	96012	yes	—
213	C₂×C₄	C₂⁷×C₃×C₈	1	1524	yes	—
214	C₂×C₄	C₂⁶×C₃×C₄²	1	381	yes	—
215	C₂×C₄	C₂⁸×C₃×C₄	1	3	yes	—
216	C₈	C₂⁷×C₃×C₈	1	1536	yes	direct product factor
217	C₂×C₃	C₂⁶×C₈²	1	508	yes	direct product factor
218	C₂×C₃	C₂⁷×C₄×C₈	1	3	yes	—
219	C₂²	C₂⁵×C₃×C₈²	1	42672	yes	direct product factor
220	C₂²	C₂⁶×C₃×C₄×C₈	1	762	yes	—
221	C₂²	C₂⁷×C₃×C₄²	1	1	yes	characteristic
222	C₄	C₂⁸×C₃×C₈	1	6	yes	—
223	C₄	C₂⁶×C₃×C₄×C₈	1	762	yes	—
224	C₃	C₂⁷×C₈²	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	C₂⁶×C₃×C₈²	1	508	yes	direct product factor
226	C₂	C₂⁷×C₃×C₄×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁹×C₃
socle, prime radical	C₂⁹×C₃	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁶×C₃×C₈²	C₂	direct (N × H)
3. C₂⁷×C₈²	C₃	direct (N × H)
8. C₂⁵×C₃×C₈²	C₂²	direct (N × H)
10. C₂⁶×C₈²	C₂×C₃	direct (N × H)
14. C₂⁷×C₃×C₈	C₈	direct (N × H)
18. C₂⁴×C₃×C₈²	C₂³	direct (N × H)
23. C₂⁵×C₈²	C₂²×C₃	direct (N × H)
31. C₂⁶×C₃×C₈	C₂×C₈	direct (N × H)
35. C₂³×C₃×C₈²	C₂⁴	direct (N × H)
41. C₂⁷×C₈	C₃×C₈	direct (N × H)
43. C₂⁴×C₈²	C₂³×C₃	direct (N × H)
50. C₂⁵×C₃×C₈	C₂²×C₈	direct (N × H)
57. C₂²×C₃×C₈²	C₂⁵	direct (N × H)
62. C₂⁶×C₈	C₂×C₃×C₈	direct (N × H)
69. C₂³×C₈²	C₂⁴×C₃	direct (N × H)
70. C₂⁷×C₃	C₈²	direct (N × H)
77. C₂⁴×C₃×C₈	C₂³×C₈	direct (N × H)
84. C₂×C₃×C₈²	C₂⁶	direct (N × H)
91. C₂⁵×C₈	C₂²×C₃×C₈	direct (N × H)
98. C₂²×C₈²	C₂⁵×C₃	direct (N × H)
99. C₂⁶×C₃	C₂×C₈²	direct (N × H)
106. C₂³×C₃×C₈	C₂⁴×C₈	direct (N × H)
113. C₃×C₈²	C₂⁷	direct (N × H)
114. C₂⁷	C₃×C₈²	direct (N × H)
121. C₂⁴×C₈	C₂³×C₃×C₈	direct (N × H)
128. C₂×C₈²	C₂⁶×C₃	direct (N × H)
129. C₂⁵×C₃	C₂²×C₈²	direct (N × H)
136. C₂²×C₃×C₈	C₂⁵×C₈	direct (N × H)
143. C₂⁶	C₂×C₃×C₈²	direct (N × H)
150. C₂³×C₈	C₂⁴×C₃×C₈	direct (N × H)
157. C₈²	C₂⁷×C₃	direct (N × H)
158. C₂⁴×C₃	C₂³×C₈²	direct (N × H)
165. C₂×C₃×C₈	C₂⁶×C₈	direct (N × H)
170. C₂⁵	C₂²×C₃×C₈²	direct (N × H)
177. C₂²×C₈	C₂⁵×C₃×C₈	direct (N × H)
184. C₂³×C₃	C₂⁴×C₈²	direct (N × H)
191. C₃×C₈	C₂⁷×C₈	direct (N × H)
192. C₂⁴	C₂³×C₃×C₈²	direct (N × H)
199. C₂×C₈	C₂⁶×C₃×C₈	direct (N × H)
204. C₂²×C₃	C₂⁵×C₈²	direct (N × H)
209. C₂³	C₂⁴×C₃×C₈²	direct (N × H)
216. C₈	C₂⁷×C₃×C₈	direct (N × H)
217. C₂×C₃	C₂⁶×C₈²	direct (N × H)
219. C₂²	C₂⁵×C₃×C₈²	direct (N × H)
224. C₃	C₂⁷×C₈²	direct (N × H)
225. C₂	C₂⁶×C₃×C₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷×C₃×C₄×C₈	2
C₂⁶×C₃×C₈²	2
C₂⁷×C₈²	3

Subgroups — C27×C3×C82 #72201900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁷×C₃×C₈² #72201900