Subgroups — #72225150

Order 4096.

913 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225142	C₂	1	4	yes	maximal
2	#72225140	C₄	1	16	yes	complemented
3	#72225139	C₂	1	2	yes	maximal
4	#72225138	C₄	1	4	yes	—
5	#72225137	C₄	1	4	yes	—
6	#72225137	C₂²	1	4	yes	—
7	#72216231	C₂	1	1	yes	maximal, characteristic
8	#72216226	C₂²	1	2	yes	—
9	C₈×#1398	C₄	1	2	yes	—
10	#72216220	C₄	1	1	yes	characteristic
11	C₁₆×M₆₄	C₄	1	1	yes	characteristic
12	#72216182	C₂²	1	1	yes	characteristic
13	C₄×#1398	—	2	4	no	—
14	C₄×#1398	C₈	1	4	yes	—
15	#149502	—	2	4	no	—
16	#149499	C₂×C₄	1	8	yes	—
17	#149495	C₈	1	4	yes	—
18	C₈×M₆₄	C₂×C₄	1	2	yes	—
19	C₈×M₆₄	C₂×C₄	1	2	yes	—
20	C₈×#346	C₂×C₄	1	1	yes	characteristic
21	C₈×#346	C₂×C₄	1	2	yes	—
22	C₂×#7052	C₂×C₄	1	2	yes	—
23	C₂×#7052	C₂³	1	1	yes	characteristic
24	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
25	#93667	—	2	8	no	—
26	#93667	C₂×C₄	1	16	yes	—
27	#93666	C₂×C₄	1	8	yes	—
28	#93665	C₈	1	4	yes	—
29	#93664	C₈	1	2	yes	—
30	C₁₆×C₃₂	C₈	1	1	yes	characteristic
31	C₁₆×C₃₂	C₈	1	1	yes	characteristic
32	C₂×#1398	—	2	8	no	—
33	C₂×#1398	—	4	8	no	—
34	C₄×M₆₄	—	2	8	no	—
35	C₄×M₆₄	C₂×C₈	1	4	yes	—
36	C₄×M₆₄	C₂×C₈	1	4	yes	—
37	C₄×#346	C₂×C₈	1	2	yes	—
38	C₄×#346	M₁₆	1	4	yes	—
39	C₂×#1311	C₂×C₈	1	2	yes	—
40	C₂×#1311	M₁₆	1	4	yes	—
41	C₂×C₈×C₁₆	C₄²	1	2	yes	—
42	C₂×C₈×C₁₆	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₆	C₂²×C₄	1	2	yes	—
45	C₄×C₈²	C₄²	1	1	yes	characteristic
46	#7332	—	4	8	no	—
47	#7331	—	2	8	no	—
48	#7331	C₄²	1	16	yes	—
49	#7330	—	2	8	no	—
50	#7329	C₄²	1	4	yes	—
51	#7329	C₄²	1	4	yes	—
52	C₈×C₃₂	C₂×C₈	1	2	yes	—
53	C₈×C₃₂	C₂×C₈	1	2	yes	—
54	C₈×C₃₂	C₂×C₈	1	4	yes	—
55	#7052	M₁₆	1	8	yes	—
56	#7052	C₂²×C₄	1	4	yes	—
57	C₁₆²	C₂×C₈	1	1	yes	characteristic
58	C₁₆²	C₂×C₈	1	1	yes	characteristic
59	C₁₆²	C₂×C₈	1	2	yes	—
60	C₂×M₆₄	—	2	4	no	—
61	C₂×M₆₄	—	4	8	no	—
62	C₂×M₆₄	M₃₂	1	8	yes	—
63	C₂×#346	—	2	8	no	—
64	C₂×#346	—	2	4	no	—
65	C₂×C₄×C₁₆	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₁₆	#97	1	4	yes	—
67	C₂×C₄×C₁₆	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₁₆	C₂×M₁₆	1	4	yes	—
69	C₄²×C₈	C₄×C₈	1	1	yes	characteristic
70	C₄²×C₈	#97	1	2	yes	—
71	C₂×C₈²	C₄×C₈	1	1	yes	characteristic
72	C₂×C₈²	#97	1	2	yes	—
73	C₂×C₈²	C₂×C₄²	1	1	yes	characteristic
74	#1398	—	4	16	no	—
75	#1398	—	8	16	no	—
76	#1397	—	2	4	no	—
77	#1397	—	4	8	no	—
78	#1397	M₃₂	1	8	yes	—
79	C₄×C₃₂	—	2	16	no	—
80	C₄×C₃₂	C₄×C₈	1	8	yes	—
81	C₄×C₃₂	C₄×C₈	1	4	yes	—
82	C₄×C₃₂	C₄×C₈	1	4	yes	—
83	#1311	—	2	8	no	—
84	#1311	—	2	4	no	—
85	C₈×C₁₆	C₄×C₈	1	2	yes	—
86	C₈×C₁₆	C₄×C₈	1	2	yes	—
87	C₈×C₁₆	#97	1	4	yes	—
88	C₈×C₁₆	#97	1	4	yes	—
89	C₈×C₁₆	C₂×C₄²	1	2	yes	—
90	C₈×C₁₆	C₂×C₄²	1	4	yes	—
91	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
92	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
93	C₈×C₁₆	C₂×M₁₆	1	4	yes	—
94	C₂²×C₁₆	—	2	2	no	—
95	C₂²×C₁₆	—	2	4	no	—
96	C₂²×C₁₆	#347	1	8	yes	—
97	C₂²×C₁₆	C₂×M₃₂	1	4	yes	—
98	C₂×C₄×C₈	—	2	2	no	—
99	C₂×C₄×C₈	—	2	4	no	—
100	C₂×C₄×C₈	#322	1	2	yes	—
101	C₂×C₄×C₈	#322	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×#97	1	2	yes	—
104	C₄³	#322	1	1	yes	characteristic
105	M₆₄	—	2	16	no	—
106	M₆₄	—	4	8	no	—
107	M₆₄	—	8	16	no	—
108	C₂×C₃₂	—	2	8	no	—
109	C₂×C₃₂	—	4	16	no	—
110	C₂×C₃₂	#346	1	8	yes	—
111	C₂×C₃₂	#346	1	8	yes	—
112	#346	—	4	8	no	—
113	#346	—	4	16	no	—
114	C₄×C₁₆	—	2	2	no	—
115	C₄×C₁₆	—	2	4	no	—
116	C₄×C₁₆	#322	1	4	yes	—
117	C₄×C₁₆	#322	1	4	yes	—
118	C₄×C₁₆	#322	1	4	yes	—
119	C₄×C₁₆	#347	1	8	yes	—
120	C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
121	C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
122	C₄×C₁₆	C₄×M₁₆	1	8	yes	—
123	C₄×C₁₆	C₂×M₃₂	1	4	yes	—
124	C₈²	—	2	2	no	—
125	C₈²	—	2	4	no	—
126	C₈²	C₄³	1	1	yes	characteristic
127	C₈²	C₂×C₄×C₈	1	1	yes	characteristic
128	C₈²	C₂×#97	1	2	yes	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	C₄×M₃₂	1	2	yes	—
132	C₂²×C₈	C₂×#347	1	4	yes	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	C₂×#322	1	1	yes	characteristic
136	C₂×C₁₆	—	2	16	no	—
137	C₂×C₁₆	—	2	8	no	—
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	—	2	8	no	—
140	C₂×C₁₆	—	4	8	no	—
141	C₂×C₁₆	—	4	4	no	—
142	C₂×C₁₆	#1313	1	8	yes	—
143	C₂×C₁₆	#1313	1	8	yes	—
144	C₂×C₁₆	C₂×#346	1	4	yes	—
145	C₂×C₁₆	C₄×M₃₂	1	4	yes	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	—	4	8	no	—
151	C₄×C₈	C₂×#322	1	1	yes	characteristic
152	C₄×C₈	C₂×#322	1	2	yes	—
153	C₄×C₈	C₄²×C₈	1	1	yes	characteristic
154	C₄×C₈	C₄×#97	1	2	yes	—
155	C₄×C₈	C₄×M₃₂	1	2	yes	—
156	C₄×C₈	C₂×#347	1	4	yes	—
157	C₃₂	—	2	8	no	—
158	C₃₂	—	4	8	no	—
159	C₃₂	—	8	16	no	—
160	C₃₂	#1398	1	16	yes	complemented
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	#11402	1	1	yes	characteristic
163	C₂²×C₄	#11404	1	2	yes	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	#11402	1	2	yes	—
171	C₂×C₈	C₂×#1313	1	4	yes	—
172	C₂×C₈	C₄×#346	1	2	yes	—
173	C₂×C₈	C₄×#347	1	4	yes	—
174	C₄²	—	4	2	no	—
175	C₄²	—	4	4	no	—
176	C₄²	C₄×#322	1	1	yes	characteristic
177	C₄²	#11402	1	1	yes	characteristic
178	C₄²	#11404	1	2	yes	—
179	C₁₆	—	2	4	no	—
180	C₁₆	—	2	8	no	—
181	C₁₆	—	4	8	no	—
182	C₁₆	—	4	4	no	—
183	C₁₆	#7333	1	16	yes	—
184	C₁₆	C₂×#1398	1	8	yes	—
185	C₂³	#122220	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	8	1	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	#122220	1	2	yes	—
191	C₂×C₄	#122220	1	1	yes	characteristic
192	C₂×C₄	#126522	1	1	yes	characteristic
193	C₂×C₄	C₄×#1313	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	4	no	—
196	C₈	—	8	1	no	—
197	C₈	—	8	2	no	—
198	C₈	C₂×#7333	1	8	yes	—
199	C₈	C₄×#1398	1	4	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	2	no	—
202	C₂²	#72216178	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	16	1	no	—
206	C₄	C₄×#7333	1	4	yes	—
207	C₄	#72225138	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	#72225139	1	1	yes	characteristic
210	C₂	#72225149	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×#322
commutator	C₄×C₈	C₄²×C₈
Frattini, cosocle	C₂×#7052	C₂³
socle	C₂²	#72216178
prime radical	C₂³	#122220
LCS	G ▷ C₄×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ #72216182 ◁ G
LECS	G ▷ C₂×#7052 ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72225140	C₄	semidirect
160. C₃₂	#1398	semidirect

Maximal subgroups

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

subgroup	index
#72225142	2
#72225139	2
#72216231	2