Subgroups — #72225366

Order 2048.

451 conjugacy classes of subgroups.

The subgroup lattice has 245 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225363	C₂	1	2	yes	maximal
2	#72225362	C₂	1	1	yes	maximal, characteristic
3	#72225313	C₂	1	1	yes	maximal, characteristic
4	#72216534	C₂	1	2	yes	maximal
5	C₁₆×M₆₄	C₂	1	1	yes	maximal, characteristic
6	#151998	C₄	1	1	yes	characteristic
7	#151989	—	2	2	no	—
8	#151040	C₄	1	1	yes	characteristic
9	C₂×#7459	C₂²	1	2	yes	—
10	C₈×M₆₄	C₂²	1	1	yes	characteristic
11	C₈×M₆₄	C₂²	1	1	yes	characteristic
12	#129142	C₂²	1	2	yes	—
13	C₂×C₁₆²	C₂²	1	1	yes	characteristic
14	#94053	C₄	1	2	yes	—
15	#94041	—	2	2	no	—
16	#93885	C₄	1	4	yes	complemented
17	#93664	C₄	1	2	yes	complemented
18	C₁₆×C₃₂	C₄	1	2	yes	complemented
19	#13638	C₈	1	2	yes	—
20	C₂×#1422	—	2	2	no	—
21	#13551	—	4	2	no	—
22	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
23	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
24	C₄×M₆₄	D₄	1	1	yes	characteristic
25	C₄×M₆₄	D₄	1	1	yes	characteristic
26	#12670	C₂×C₄	1	2	yes	—
27	#12664	—	2	2	no	—
28	C₂×#1373	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₆	—	2	1	no	—
30	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
31	#7476	C₈	1	1	yes	characteristic
32	#7475	C₈	1	1	yes	characteristic
33	#7469	—	4	2	no	—
34	#7464	C₈	1	4	yes	—
35	#7459	—	2	2	no	—
36	#7459	C₂×C₄	1	4	yes	—
37	#7329	C₂×C₄	1	2	yes	—
38	#7329	Q₈	1	2	yes	—
39	C₈×C₃₂	C₂×C₄	1	2	yes	—
40	C₈×C₃₂	C₂×C₄	1	2	yes	—
41	#7179	—	2	2	no	—
42	C₁₆²	—	2	1	no	—
43	C₁₆²	C₂×C₄	1	1	yes	characteristic
44	C₁₆²	C₂×C₄	1	1	yes	characteristic
45	C₂×M₆₄	—	2	1	no	—
46	C₂×M₆₄	—	2	1	no	—
47	C₂×M₆₄	—	4	2	no	—
48	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
49	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
50	C₂×M₆₄	D₈	1	1	yes	characteristic
51	C₂×M₆₄	D₈	1	1	yes	characteristic
52	C₂×#368	C₂×C₈	1	2	yes	—
53	#2158	C₂×C₈	1	2	yes	—
54	C₂×#364	—	2	2	no	—
55	#2108	—	4	2	no	—
56	C₂×C₄×C₁₆	—	2	2	no	—
57	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
59	#1772	—	2	2	no	—
60	C₂×C₈²	#41	1	1	yes	characteristic
61	#1426	—	2	4	no	—
62	#1422	—	2	4	no	—
63	#1422	—	4	2	no	—
64	#1398	—	8	4	no	—
65	#1397	—	2	1	no	—
66	#1397	—	2	1	no	—
67	#1397	C₂×C₈	1	2	yes	—
68	#1397	Q₁₆	1	2	yes	—
69	C₄×C₃₂	C₄²	1	2	yes	—
70	C₄×C₃₂	C₄²	1	2	yes	—
71	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
72	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
73	#1382	—	2	2	no	—
74	#1379	—	4	2	no	—
75	#1373	—	2	2	no	—
76	#1373	C₂×C₈	1	4	yes	—
77	C₈×C₁₆	—	2	2	no	—
78	C₈×C₁₆	—	2	1	no	—
79	C₈×C₁₆	—	2	1	no	—
80	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
83	C₈×C₁₆	#41	1	1	yes	characteristic
84	C₂×M₃₂	—	4	2	no	—
85	C₂²×C₁₆	—	2	4	no	—
86	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
87	C₂²×C₁₆	#131	1	1	yes	characteristic
88	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
89	C₂²×C₁₆	#135	1	1	yes	characteristic
90	C₂×#108	—	2	2	no	—
91	#427	—	2	2	no	—
92	#405	—	4	2	no	—
93	C₂×C₄×C₈	—	2	1	no	—
94	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
95	M₆₄	—	2	4	no	—
96	M₆₄	—	2	4	no	—
97	M₆₄	—	2	2	no	—
98	M₆₄	—	2	2	no	—
99	M₆₄	—	4	4	no	—
100	M₆₄	—	8	2	no	—
101	M₆₄	M₃₂	1	4	yes	—
102	M₆₄	SD₃₂	1	4	yes	—
103	C₂×C₃₂	—	2	2	no	—
104	C₂×C₃₂	—	2	2	no	—
105	C₂×C₃₂	—	8	2	no	—
106	C₂×C₃₂	C₄×C₈	1	2	yes	—
107	C₂×C₃₂	C₄×C₈	1	2	yes	—
108	C₂×C₃₂	#107	1	2	yes	—
109	C₂×C₃₂	#107	1	2	yes	—
110	#368	—	2	2	no	—
111	#368	—	2	2	no	—
112	#367	—	2	2	no	—
113	#366	—	2	2	no	—
114	#364	—	2	4	no	—
115	#364	—	4	2	no	—
116	#346	—	8	2	no	—
117	C₄×C₁₆	—	2	2	no	—
118	C₄×C₁₆	—	2	2	no	—
119	C₄×C₁₆	—	2	4	no	—
120	C₄×C₁₆	—	8	2	no	—
121	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
122	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
124	C₄×C₁₆	C₄×Q₈	1	1	yes	characteristic
125	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
126	C₄×C₁₆	#131	1	1	yes	characteristic
127	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
128	C₄×C₁₆	#135	1	1	yes	characteristic
129	#339	—	4	2	no	—
130	C₈²	—	2	1	no	—
131	C₈²	#117	1	1	yes	characteristic
132	C₈²	C₄×Q₈	1	1	yes	characteristic
133	C₂×M₁₆	—	4	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	C₈×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₈	1	1	yes	characteristic
137	#117	—	4	2	no	—
138	C₂×C₄²	#443	1	1	yes	characteristic
139	M₃₂	—	4	4	no	—
140	M₃₂	—	8	2	no	—
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	—	4	8	no	—
146	C₂×C₁₆	—	8	2	no	—
147	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
148	C₂×C₁₆	#405	1	1	yes	characteristic
149	C₂×C₁₆	C₂×#107	1	1	yes	characteristic
150	C₂×C₁₆	#427	1	1	yes	characteristic
151	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
152	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
153	C₂×C₁₆	C₈×Q₈	1	1	yes	characteristic
154	C₂×C₁₆	C₈×Q₈	1	1	yes	characteristic
155	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
156	C₂×C₁₆	#504	1	2	yes	—
157	C₂×C₁₆	C₂×SD₃₂	1	2	yes	—
158	C₂×C₁₆	#508	1	2	yes	—
159	#108	—	4	2	no	—
160	#108	—	4	2	no	—
161	#107	—	4	2	no	—
162	#106	—	4	2	no	—
163	#97	—	8	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	8	2	no	—
168	C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₈	C₈×D₄	1	1	yes	characteristic
170	C₄×C₈	C₄×Q₁₆	1	1	yes	characteristic
171	C₄×C₈	#443	1	1	yes	characteristic
172	C₃₂	—	2	4	no	—
173	C₃₂	—	2	4	no	—
174	C₃₂	—	2	2	no	—
175	C₃₂	—	2	2	no	—
176	C₃₂	—	8	4	no	—
177	C₃₂	#346	1	4	yes	—
178	C₃₂	#367	1	4	yes	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₈×D₈	1	1	yes	characteristic
181	M₁₆	—	8	2	no	—
182	M₁₆	—	8	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₈×D₈	1	1	yes	characteristic
188	C₂×C₈	C₈×Q₁₆	1	1	yes	characteristic
189	C₂×C₈	C₈×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₄×#107	1	1	yes	characteristic
191	C₂×C₈	#2169	1	2	yes	—
192	C₂×C₈	#2170	1	2	yes	—
193	C₂×C₈	C₄×SD₃₂	1	2	yes	—
194	C₂×C₈	#2178	1	2	yes	—
195	C₄⋊C₄	—	8	2	no	—
196	C₂²⋊C₄	—	8	2	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	8	2	no	—
199	C₄²	C₈×Q₁₆	1	1	yes	characteristic
200	C₄²	#1772	1	1	yes	characteristic
201	C₁₆	—	2	8	no	—
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	4	4	no	—
207	C₁₆	—	8	4	no	—
208	C₁₆	C₂×#346	1	1	yes	characteristic
209	C₁₆	#2108	1	1	yes	characteristic
210	C₁₆	C₂×#367	1	1	yes	characteristic
211	C₁₆	#2158	1	1	yes	characteristic
212	C₁₆	C₄×SD₃₂	1	2	yes	—
213	C₁₆	#2183	1	2	yes	—
214	C₂³	#12046	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	16	2	no	—
219	C₂×C₄	C₈×#107	1	1	yes	characteristic
220	C₂×C₄	#12039	1	2	yes	—
221	C₂×C₄	C₈×SD₃₂	1	2	yes	—
222	C₂×C₄	#12046	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	—	4	2	no	—
225	C₈	—	16	2	no	—
226	C₈	C₈×SD₃₂	1	2	yes	—
227	C₈	#12054	1	2	yes	—
228	C₈	#12662	1	1	yes	characteristic
229	C₈	#12664	1	1	yes	characteristic
230	C₈	#12670	1	1	yes	characteristic
231	C₈	C₄×#367	1	1	yes	characteristic
232	C₂²	—	4	1	no	—
233	C₂²	#122948	1	2	yes	—
234	C₂²	#129142	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	32	2	no	—
238	C₄	#122948	1	2	yes	—
239	C₄	#129140	1	1	yes	characteristic
240	C₄	C₈×#367	1	1	yes	characteristic
241	C₂	—	2	1	no	—
242	C₂	—	2	2	no	—
243	C₂	#72225365	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#2158
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225365
prime radical	C₂³	#12046
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #93885	C₄	semidirect
17. #93664	C₄	semidirect
18. C₁₆×C₃₂	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225363	2
#72225362	2
#72225313	2
#72216534	2
C₁₆×M₆₄	2