Subgroups — #72234414

Order 4096.

849 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234335	C₂	1	1	yes	maximal, characteristic
2	#72234311	—	2	2	no	—
3	#72234303	C₂	1	2	yes	maximal
4	#72234299	—	2	2	no	—
5	#72234299	—	2	2	no	—
6	#72234280	C₄	1	4	yes	—
7	C₄×#7052	C₂²	1	1	yes	characteristic
8	C₄×#7329	C₄	1	2	yes	—
9	#151876	—	2	4	no	—
10	C₄²×C₃₂	C₈	1	4	yes	—
11	C₂×#7331	—	2	4	no	—
12	C₂×#7329	—	2	2	no	—
13	C₂×#7329	—	2	2	no	—
14	C₂×C₈×C₃₂	C₈	1	8	yes	—
15	C₄×C₈×C₁₆	C₂×C₄	1	2	yes	—
16	C₄×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×#7052	D₄	1	1	yes	characteristic
18	C₂×#7052	D₄	1	2	yes	—
19	C₂×#7052	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₃₂	—	2	8	no	—
21	C₂×C₄×C₃₂	—	2	4	no	—
22	C₂×C₄×C₃₂	—	2	4	no	—
23	C₂×C₄×C₃₂	C₁₆	1	8	yes	—
24	C₄²×C₁₆	C₂×C₈	1	2	yes	—
25	C₄²×C₁₆	M₁₆	1	4	yes	—
26	C₂×C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₆	C₂²⋊C₄	1	2	yes	—
28	C₂×C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₆	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₆	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₁₆	M₁₆	1	2	yes	—
32	C₄×C₈²	C₄²	1	1	yes	characteristic
33	#7465	—	4	8	no	—
34	#7331	—	4	4	no	—
35	#7331	—	4	4	no	—
36	#7329	—	4	4	no	—
37	#7329	—	4	4	no	—
38	C₈×C₃₂	—	2	8	no	—
39	C₈×C₃₂	C₁₆	1	16	yes	—
40	#7052	—	4	2	no	—
41	#7052	—	4	1	no	—
42	#7052	—	4	1	no	—
43	C₂²×C₃₂	—	2	8	no	—
44	C₂²×C₃₂	—	4	8	no	—
45	C₂×C₄×C₁₆	—	2	8	no	—
46	C₂×C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
47	C₂×C₄×C₁₆	C₂²⋊C₈	1	4	yes	—
48	C₂×C₄×C₁₆	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₁₆	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₁₆	C₂×C₁₆	1	4	yes	—
51	C₂×C₄×C₁₆	M₃₂	1	4	yes	—
52	C₄²×C₈	C₄×C₈	1	1	yes	characteristic
53	C₄²×C₈	#97	1	2	yes	—
54	C₂×C₈²	#95	1	1	yes	characteristic
55	C₂×C₈²	C₄×C₈	1	2	yes	—
56	C₂×C₈²	#97	1	1	yes	characteristic
57	C₄×C₃₂	—	2	16	no	—
58	C₄×C₃₂	—	2	8	no	—
59	C₄×C₃₂	—	4	4	no	—
60	C₄×C₃₂	—	4	4	no	—
61	C₄×C₃₂	—	4	8	no	—
62	C₄×C₃₂	—	4	8	no	—
63	C₄×C₃₂	C₃₂	1	16	yes	complemented
64	C₈×C₁₆	—	2	2	no	—
65	C₈×C₁₆	—	2	2	no	—
66	C₈×C₁₆	—	2	2	no	—
67	C₈×C₁₆	—	2	2	no	—
68	C₈×C₁₆	—	2	1	no	—
69	C₈×C₁₆	—	2	1	no	—
70	C₈×C₁₆	—	2	2	no	—
71	C₈×C₁₆	—	2	4	no	—
72	C₈×C₁₆	C₂×C₁₆	1	8	yes	—
73	C₈×C₁₆	M₃₂	1	8	yes	—
74	C₂²×C₁₆	—	2	4	no	—
75	C₂²×C₁₆	—	2	4	no	—
76	C₂²×C₁₆	C₂²⋊C₁₆	1	4	yes	—
77	C₂²×C₁₆	C₄⋊C₁₆	1	4	yes	—
78	C₂×C₄×C₈	C₈²	1	1	yes	characteristic
79	C₂×C₄×C₈	#322	1	2	yes	—
80	C₂×C₄×C₈	#336	1	2	yes	—
81	C₂×C₄×C₈	#336	1	1	yes	characteristic
82	C₂×C₄×C₈	C₄×C₁₆	1	2	yes	—
83	C₂×C₄×C₈	#346	1	2	yes	—
84	C₂×C₄×C₈	#347	1	8	yes	—
85	C₄³	#322	1	1	yes	characteristic
86	C₂×C₃₂	—	2	16	no	—
87	C₂×C₃₂	—	4	8	no	—
88	C₂×C₃₂	—	4	8	no	—
89	C₂×C₃₂	—	4	16	no	—
90	C₂×C₃₂	—	8	16	no	—
91	C₄×C₁₆	—	2	2	no	—
92	C₄×C₁₆	—	2	2	no	—
93	C₄×C₁₆	—	2	8	no	—
94	C₄×C₁₆	—	2	2	no	—
95	C₄×C₁₆	—	2	4	no	—
96	C₄×C₁₆	—	2	2	no	—
97	C₄×C₁₆	—	2	4	no	—
98	C₄×C₁₆	—	8	8	no	—
99	C₄×C₁₆	C₂²⋊C₁₆	1	8	yes	—
100	C₄×C₁₆	C₄⋊C₁₆	1	8	yes	—
101	C₄×C₁₆	C₂×C₃₂	1	8	yes	—
102	C₄×C₁₆	M₆₄	1	8	yes	—
103	C₈²	H(3,ℤ/4)	1	2	yes	—
104	C₈²	H(3,ℤ/4)	1	1	yes	characteristic
105	C₈²	#338	1	1	yes	characteristic
106	C₈²	C₄×C₁₆	1	4	yes	—
107	C₈²	#346	1	4	yes	—
108	C₈²	#347	1	4	yes	—
109	C₂²×C₈	C₈×C₁₆	1	1	yes	characteristic
110	C₂²×C₈	#1311	1	1	yes	characteristic
111	C₂²×C₈	#1313	1	4	yes	—
112	C₂²×C₈	#1374	1	2	yes	—
113	C₂²×C₈	#1375	1	4	yes	—
114	C₂×C₄²	#1273	1	1	yes	characteristic
115	C₂×C₄²	#1312	1	2	yes	—
116	C₂×C₄²	#1313	1	4	yes	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	2	4	no	—
119	C₂×C₁₆	—	2	4	no	—
120	C₂×C₁₆	—	2	4	no	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	—	2	8	no	—
123	C₂×C₁₆	—	4	8	no	—
124	C₂×C₁₆	—	4	8	no	—
125	C₂×C₁₆	C₂²⋊C₃₂	1	8	yes	—
126	C₂×C₁₆	C₄⋊C₃₂	1	8	yes	—
127	C₄×C₈	—	4	8	no	—
128	C₄×C₈	#1274	1	2	yes	—
129	C₄×C₈	#1274	1	1	yes	characteristic
130	C₄×C₈	#1275	1	2	yes	—
131	C₄×C₈	#1275	1	1	yes	characteristic
132	C₄×C₈	C₈×C₁₆	1	2	yes	—
133	C₄×C₈	#1311	1	2	yes	—
134	C₄×C₈	#1312	1	4	yes	—
135	C₄×C₈	#1313	1	4	yes	—
136	C₄×C₈	#1374	1	4	yes	—
137	C₄×C₈	#1375	1	2	yes	—
138	C₄×C₈	C₄×C₃₂	1	4	yes	—
139	C₄×C₈	#1397	1	4	yes	—
140	C₄×C₈	#1398	1	8	yes	—
141	C₃₂	—	4	16	no	—
142	C₃₂	—	8	16	no	—
143	C₂²×C₄	#7053	1	4	yes	—
144	C₂²×C₄	#7136	1	1	yes	characteristic
145	C₂²×C₄	#7137	1	2	yes	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	C₁₆²	1	1	yes	characteristic
149	C₂×C₈	#7052	1	1	yes	characteristic
150	C₂×C₈	#7052	1	2	yes	—
151	C₂×C₈	#7053	1	8	yes	—
152	C₂×C₈	#7142	1	4	yes	—
153	C₂×C₈	#7142	1	2	yes	—
154	C₂×C₈	#7145	1	4	yes	—
155	C₂×C₈	#7145	1	2	yes	—
156	C₂×C₈	C₈×C₃₂	1	2	yes	—
157	C₂×C₈	#7329	1	2	yes	—
158	C₂×C₈	#7330	1	4	yes	—
159	C₂×C₈	#7465	1	4	yes	—
160	C₂×C₈	#7466	1	4	yes	—
161	C₄²	—	2	8	no	—
162	C₄²	#7014	1	1	yes	characteristic
163	C₄²	#7136	1	2	yes	—
164	C₄²	#7137	1	1	yes	characteristic
165	C₄²	#7331	1	4	yes	—
166	C₄²	#7332	1	4	yes	—
167	C₁₆	—	2	8	no	—
168	C₁₆	—	2	8	no	—
169	C₁₆	—	4	8	no	—
170	C₁₆	—	4	8	no	—
171	C₂³	#92896	1	1	yes	characteristic
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	#92895	1	1	yes	characteristic
174	C₂×C₄	#92896	1	2	yes	—
175	C₂×C₄	#93249	1	1	yes	characteristic
176	C₂×C₄	#93249	1	2	yes	—
177	C₂×C₄	#93667	1	8	yes	—
178	C₂×C₄	#93989	1	2	yes	—
179	C₂×C₄	#93990	1	2	yes	—
180	C₂×C₄	#93995	1	8	yes	—
181	C₈	—	2	8	no	—
182	C₈	—	2	4	no	—
183	C₈	—	2	4	no	—
184	C₈	C₁₆×C₃₂	1	1	yes	characteristic
185	C₈	#93664	1	1	yes	characteristic
186	C₈	#93665	1	2	yes	—
187	C₈	#93666	1	4	yes	—
188	C₈	#93998	1	4	yes	—
189	C₈	#94005	1	4	yes	—
190	C₂²	#72210714	1	1	yes	characteristic
191	C₂²	#72234264	1	4	yes	—
192	C₂²	#72234299	1	2	yes	—
193	C₄	—	2	8	no	—
194	C₄	#72210682	1	4	yes	—
195	C₄	#72234264	1	4	yes	—
196	C₄	#72234282	1	2	yes	—
197	C₄	#72234310	1	1	yes	characteristic
198	C₄	#72234311	1	1	yes	characteristic
199	C₂	#72234265	1	2	yes	—
200	C₂	#72234301	1	4	yes	—
201	C₂	#72234312	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₈²
commutator	C₈	C₁₆×C₃₂
Frattini, cosocle	C₄×#7052	C₂²
socle, prime radical	C₂³	#92896
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×#7052 ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
63. C₄×C₃₂	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
#72234335	2
#72234303	2