Subgroups — #72234404

Order 4096.

1131 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234399	C₂	1	2	yes	maximal
2	#72234396	C₄	1	4	yes	—
3	#72234385	C₂	1	4	yes	maximal
4	#72234375	C₄	1	16	yes	—
5	C₄×#7511	C₂²	1	4	yes	—
6	C₈×M₁₂₈	C₄	1	4	yes	—
7	#72234254	C₂	1	1	yes	maximal, characteristic
8	#72234239	C₄	1	1	yes	characteristic
9	#72234239	C₂²	1	2	yes	—
10	#72234233	C₄	1	2	yes	—
11	#72234233	C₂²	1	1	yes	characteristic
12	#72210602	C₄	1	1	yes	characteristic
13	C₂×#7511	C₂×C₄	1	16	yes	—
14	C₄×M₁₂₈	C₂×C₄	1	8	yes	—
15	C₄×#1397	C₂×C₄	1	2	yes	—
16	C₄×#1397	C₂³	1	1	yes	characteristic
17	C₄²×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂×#7330	C₈	1	4	yes	—
19	C₂×#7330	C₂×C₄	1	8	yes	—
20	C₂×#7329	C₈	1	4	yes	—
21	C₂×#7329	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₂	C₈	1	4	yes	—
23	C₂×C₈×C₃₂	C₂×C₄	1	2	yes	—
24	#128206	C₈	1	2	yes	—
25	#128206	C₈	1	1	yes	characteristic
26	#126519	C₂×C₄	1	1	yes	characteristic
27	#126519	C₂×C₄	1	2	yes	—
28	#96658	C₈	1	1	yes	characteristic
29	#94505	C₈	1	16	yes	complemented
30	#94504	C₈	1	8	yes	complemented
31	C₈×C₆₄	C₈	1	8	yes	complemented
32	C₂×M₁₂₈	C₄²	1	16	yes	—
33	C₂×#1397	C₄²	1	4	yes	—
34	C₂×#1397	C₂×C₈	1	4	yes	—
35	C₂×#1397	C₂²×C₄	1	4	yes	—
36	C₂×C₄×C₃₂	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₃₂	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₃₂	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₃₂	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₃₂	C₂×C₈	1	4	yes	—
41	C₂×C₄×C₃₂	C₂²×C₄	1	2	yes	—
42	C₄×#346	C₂×C₈	1	2	yes	—
43	C₄×#346	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₁₆	C₂²×C₄	1	1	yes	characteristic
45	C₂×#1311	C₄²	1	4	yes	—
46	C₂×#1311	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₆	C₄²	1	2	yes	—
48	C₂×C₈×C₁₆	C₂×C₈	1	4	yes	—
49	#7841	C₂×C₈	1	2	yes	—
50	#7841	C₂×C₈	1	1	yes	characteristic
51	#7511	—	2	32	no	—
52	#7510	C₂×C₈	1	16	yes	—
53	C₄×C₆₄	C₂×C₈	1	16	yes	—
54	#7330	—	2	16	no	—
55	#7330	—	2	8	no	—
56	#7329	C₁₆	1	16	yes	—
57	#7329	C₂×C₈	1	8	yes	—
58	C₈×C₃₂	C₁₆	1	16	yes	—
59	C₈×C₃₂	C₂×C₈	1	4	yes	—
60	C₈×C₃₂	C₂×C₈	1	4	yes	—
61	#7137	—	2	2	no	—
62	#7137	—	2	4	no	—
63	#7014	—	2	2	no	—
64	C₂²×C₃₂	C₄×C₈	1	8	yes	—
65	C₂²×C₃₂	C₄×C₈	1	1	yes	characteristic
66	C₂²×C₃₂	C₄×C₈	1	2	yes	—
67	C₂²×C₃₂	C₄×C₈	1	1	yes	characteristic
68	C₂²×C₃₂	C₂×C₄²	1	4	yes	—
69	C₂×#346	—	2	2	no	—
70	C₂×#346	—	2	2	no	—
71	C₂×#346	M₃₂	1	4	yes	—
72	C₂×C₄×C₁₆	C₄×C₈	1	4	yes	—
73	C₂×C₄×C₁₆	C₄×C₈	1	2	yes	—
74	C₂×C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₆	C₂×C₄²	1	2	yes	—
76	C₂×C₄×C₁₆	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
78	#1744	—	2	2	no	—
79	#1744	M₃₂	1	2	yes	—
80	C₄²×C₈	C₂²×C₈	1	1	yes	characteristic
81	C₂×#322	—	2	2	no	—
82	C₂×#322	M₃₂	1	2	yes	—
83	C₂×C₈²	C₄×C₈	1	4	yes	—
84	C₂×C₈²	C₄×C₈	1	2	yes	—
85	M₁₂₈	—	2	32	no	—
86	C₂×C₆₄	C₄×C₈	1	32	yes	—
87	#1397	—	2	8	no	—
88	#1397	—	2	8	no	—
89	#1397	#131	1	16	yes	—
90	C₄×C₃₂	C₄×C₈	1	2	yes	—
91	C₄×C₃₂	C₄×C₈	1	4	yes	—
92	C₄×C₃₂	C₄×C₈	1	2	yes	—
93	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
94	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
95	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
96	C₄×C₃₂	C₂×C₁₆	1	8	yes	—
97	C₄×C₃₂	C₂²×C₈	1	4	yes	—
98	C₄×C₃₂	C₂²×C₈	1	4	yes	—
99	#1311	—	2	8	no	—
100	#1311	M₃₂	1	8	yes	—
101	C₈×C₁₆	C₄×C₈	1	4	yes	—
102	C₈×C₁₆	C₄×C₈	1	4	yes	—
103	C₈×C₁₆	C₂×C₁₆	1	8	yes	—
104	C₈×C₁₆	C₂×C₁₆	1	8	yes	—
105	C₂²×C₁₆	—	2	4	no	—
106	C₂²×C₁₆	#346	1	4	yes	—
107	C₂²×C₁₆	C₄³	1	1	yes	characteristic
108	C₂²×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
109	C₂²×C₁₆	C₂×C₄×C₈	1	2	yes	—
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	#346	1	4	yes	—
112	C₂×C₄×C₈	C₂×C₄×C₈	1	2	yes	—
113	C₂×C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×M₃₂	1	2	yes	—
116	C₄³	C₂×M₃₂	1	1	yes	characteristic
117	C₂×C₃₂	—	2	8	no	—
118	C₂×C₃₂	—	2	8	no	—
119	C₂×C₃₂	C₈²	1	4	yes	—
120	C₂×C₃₂	C₈²	1	1	yes	characteristic
121	C₂×C₃₂	C₈²	1	2	yes	—
122	C₂×C₃₂	C₈²	1	1	yes	characteristic
123	C₂×C₃₂	C₄×C₁₆	1	16	yes	—
124	C₂×C₃₂	C₄×C₁₆	1	4	yes	—
125	C₂×C₃₂	C₄×C₁₆	1	4	yes	—
126	C₂×C₃₂	C₄×C₁₆	1	4	yes	—
127	C₂×C₃₂	C₄×C₁₆	1	4	yes	—
128	C₂×C₃₂	C₂×C₄×C₈	1	8	yes	—
129	C₂×C₃₂	#405	1	16	yes	—
130	#346	—	4	4	no	—
131	#346	—	4	4	no	—
132	#346	—	4	4	no	—
133	C₄×C₁₆	—	2	4	no	—
134	C₄×C₁₆	C₈²	1	4	yes	—
135	C₄×C₁₆	C₄×C₁₆	1	8	yes	—
136	C₄×C₁₆	#346	1	4	yes	—
137	C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
138	C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
139	C₄×C₁₆	#405	1	8	yes	—
140	C₄×C₁₆	C₂²×C₁₆	1	4	yes	—
141	C₄×C₁₆	C₂²×C₁₆	1	4	yes	—
142	C₄×C₁₆	C₂×M₃₂	1	4	yes	—
143	#336	—	4	2	no	—
144	#336	—	4	4	no	—
145	#322	—	4	2	no	—
146	#322	—	4	4	no	—
147	C₈²	—	2	4	no	—
148	C₈²	C₈²	1	4	yes	—
149	C₈²	C₄×C₁₆	1	8	yes	—
150	C₈²	#346	1	4	yes	—
151	C₆₄	—	2	32	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	—	4	4	no	—
154	C₂²×C₈	C₄²×C₈	1	1	yes	characteristic
155	C₂²×C₈	C₂×#346	1	1	yes	characteristic
156	C₂²×C₈	C₄×M₃₂	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×#346	1	1	yes	characteristic
159	C₂×C₄²	C₄×M₃₂	1	2	yes	—
160	C₂×C₁₆	—	2	8	no	—
161	C₂×C₁₆	—	2	8	no	—
162	C₂×C₁₆	—	4	4	no	—
163	C₂×C₁₆	—	4	4	no	—
164	C₂×C₁₆	#1311	1	8	yes	—
165	C₂×C₁₆	C₂×C₈²	1	1	yes	characteristic
166	C₂×C₁₆	C₂×C₈²	1	1	yes	characteristic
167	C₂×C₁₆	#1450	1	4	yes	—
168	C₂×C₁₆	C₄²×C₈	1	2	yes	—
169	C₂×C₁₆	#1726	1	4	yes	—
170	C₂×C₁₆	C₂×C₄×C₁₆	1	2	yes	—
171	C₂×C₁₆	C₂×C₄×C₁₆	1	2	yes	—
172	C₂×C₁₆	C₂×C₄×C₁₆	1	4	yes	—
173	C₂×C₁₆	C₄×M₃₂	1	4	yes	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	2	8	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	—	4	4	no	—
179	C₄×C₈	#1311	1	8	yes	—
180	C₄×C₈	C₂×C₈²	1	2	yes	—
181	C₄×C₈	#1450	1	4	yes	—
182	C₄×C₈	C₂×C₄×C₁₆	1	4	yes	—
183	C₄×C₈	C₂×#346	1	2	yes	—
184	C₃₂	—	2	16	no	—
185	C₃₂	C₈×C₁₆	1	8	yes	—
186	C₃₂	C₈×C₁₆	1	4	yes	—
187	C₃₂	C₈×C₁₆	1	2	yes	—
188	C₃₂	C₈×C₁₆	1	2	yes	—
189	C₃₂	#1450	1	16	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄×#346	1	1	yes	characteristic
192	C₂²×C₄	#13551	1	2	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	—	8	4	no	—
198	C₂×C₈	—	8	8	no	—
199	C₂×C₈	C₄×C₈²	1	1	yes	characteristic
200	C₂×C₈	#7841	1	2	yes	—
201	C₂×C₈	C₂×#1311	1	2	yes	—
202	C₂×C₈	#11410	1	4	yes	—
203	C₂×C₈	C₄²×C₁₆	1	2	yes	—
204	C₂×C₈	C₄×#346	1	1	yes	characteristic
205	C₂×C₈	#13551	1	4	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	8	2	no	—
208	C₄²	C₂×#1311	1	2	yes	—
209	C₄²	#11410	1	4	yes	—
210	C₄²	#13551	1	2	yes	—
211	C₁₆	—	2	8	no	—
212	C₁₆	—	4	8	no	—
213	C₁₆	#7841	1	4	yes	—
214	C₁₆	C₂×C₈×C₁₆	1	2	yes	—
215	C₁₆	C₂×C₈×C₁₆	1	1	yes	characteristic
216	C₁₆	C₂×C₈×C₁₆	1	1	yes	characteristic
217	C₁₆	#11410	1	8	yes	—
218	C₂³	#151750	1	1	yes	characteristic
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×#1311	1	1	yes	characteristic
223	C₂×C₄	#126530	1	2	yes	—
224	C₂×C₄	#149496	1	4	yes	—
225	C₂×C₄	#151750	1	1	yes	characteristic
226	C₂×C₄	#151750	1	2	yes	—
227	C₈	—	2	4	no	—
228	C₈	—	4	8	no	—
229	C₈	—	8	8	no	—
230	C₈	#94852	1	1	yes	characteristic
231	C₈	C₄×C₈×C₁₆	1	1	yes	characteristic
232	C₈	#126530	1	2	yes	—
233	C₈	#149496	1	4	yes	—
234	C₂²	—	2	2	no	—
235	C₂²	#72210680	1	2	yes	—
236	C₂²	#72234239	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	#72210680	1	2	yes	—
240	C₄	#72234228	1	1	yes	characteristic
241	C₄	#72234239	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	#72234241	1	1	yes	characteristic
244	C₂	#72234403	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₈²
commutator	C₈	C₄×C₈×C₁₆
Frattini, cosocle	C₄×#1397	C₂³
socle	C₂²	#72234239
prime radical	C₂³	#151750
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₄×#1397 ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
29. #94505	C₈	semidirect
30. #94504	C₈	semidirect
31. C₈×C₆₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72234399	2
#72234385	2
#72234254	2