Subgroups — #72225378

Order 2048.

423 conjugacy classes of subgroups.

The subgroup lattice has 236 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	#72225352	C₂	1	1	yes	maximal, characteristic
3	#72225313	C₂	1	1	yes	maximal, characteristic
4	#72216535	C₂	1	2	yes	maximal
5	#72216226	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×#7052	C₂²	1	1	yes	characteristic
14	#94053	C₄	1	2	yes	—
15	#94041	—	2	2	no	—
16	#93882	C₄	1	4	yes	complemented
17	#93666	C₄	1	4	yes	complemented
18	#13638	C₈	1	2	yes	—
19	C₂×#1422	—	2	2	no	—
20	#13551	—	4	2	no	—
21	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
22	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
23	C₄×M₆₄	D₄	1	1	yes	characteristic
24	C₄×M₆₄	D₄	1	1	yes	characteristic
25	#12670	C₂×C₄	1	2	yes	—
26	#12664	—	2	2	no	—
27	C₂×#1373	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₆	—	2	1	no	—
29	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
30	#7476	C₈	1	1	yes	characteristic
31	#7475	C₈	1	1	yes	characteristic
32	#7469	—	4	2	no	—
33	#7464	C₈	1	4	yes	—
34	#7459	—	2	2	no	—
35	#7459	C₂×C₄	1	4	yes	—
36	#7329	C₂×C₄	1	2	yes	—
37	#7329	C₂×C₄	1	2	yes	—
38	C₈×C₃₂	C₂×C₄	1	2	yes	—
39	C₈×C₃₂	Q₈	1	2	yes	—
40	#7179	—	2	2	no	—
41	#7052	—	2	1	no	—
42	#7052	C₂×C₄	1	2	yes	—
43	C₂×M₆₄	—	2	1	no	—
44	C₂×M₆₄	—	2	1	no	—
45	C₂×M₆₄	—	4	2	no	—
46	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
47	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
48	C₂×M₆₄	D₈	1	2	yes	—
49	C₂×#368	C₂×C₈	1	2	yes	—
50	#2158	C₂×C₈	1	2	yes	—
51	C₂×#364	—	2	2	no	—
52	#2108	—	4	2	no	—
53	C₂×C₄×C₁₆	—	2	2	no	—
54	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
56	#1772	—	2	2	no	—
57	C₂×C₈²	#41	1	1	yes	characteristic
58	#1426	—	2	4	no	—
59	#1422	—	2	4	no	—
60	#1422	—	4	2	no	—
61	#1398	—	8	4	no	—
62	#1397	—	2	1	no	—
63	#1397	—	2	1	no	—
64	#1397	C₂×C₈	1	2	yes	—
65	#1397	Q₁₆	1	2	yes	—
66	C₄×C₃₂	C₄²	1	2	yes	—
67	C₄×C₃₂	C₄²	1	2	yes	—
68	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
69	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
70	#1382	—	2	2	no	—
71	#1379	—	4	2	no	—
72	#1373	—	2	2	no	—
73	#1373	C₂×C₈	1	4	yes	—
74	C₈×C₁₆	—	2	2	no	—
75	C₈×C₁₆	—	2	2	no	—
76	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
77	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
78	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
79	C₈×C₁₆	#41	1	1	yes	characteristic
80	C₂×M₃₂	—	4	2	no	—
81	C₂²×C₁₆	—	4	2	no	—
82	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
83	C₂²×C₁₆	#131	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
85	C₂²×C₁₆	#135	1	1	yes	characteristic
86	C₂×#108	—	2	2	no	—
87	#427	—	2	2	no	—
88	#405	—	4	2	no	—
89	C₂×C₄×C₈	—	2	1	no	—
90	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
91	M₆₄	—	2	4	no	—
92	M₆₄	—	2	2	no	—
93	M₆₄	—	2	2	no	—
94	M₆₄	—	2	2	no	—
95	M₆₄	—	4	2	no	—
96	M₆₄	—	4	4	no	—
97	M₆₄	—	8	2	no	—
98	M₆₄	M₃₂	1	4	yes	—
99	C₂×C₃₂	—	2	2	no	—
100	C₂×C₃₂	—	2	2	no	—
101	C₂×C₃₂	—	8	2	no	—
102	C₂×C₃₂	C₄×C₈	1	2	yes	—
103	C₂×C₃₂	C₄×C₈	1	2	yes	—
104	C₂×C₃₂	#107	1	4	yes	—
105	#368	—	2	2	no	—
106	#368	—	2	2	no	—
107	#367	—	2	2	no	—
108	#366	—	2	2	no	—
109	#364	—	2	4	no	—
110	#364	—	4	2	no	—
111	#346	—	8	2	no	—
112	C₄×C₁₆	—	2	4	no	—
113	C₄×C₁₆	—	4	2	no	—
114	C₄×C₁₆	—	8	2	no	—
115	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
116	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
118	C₄×C₁₆	C₄×Q₈	1	1	yes	characteristic
119	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
120	C₄×C₁₆	#131	1	1	yes	characteristic
121	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
122	C₄×C₁₆	#135	1	1	yes	characteristic
123	#339	—	4	2	no	—
124	C₈²	—	2	1	no	—
125	C₈²	#117	1	1	yes	characteristic
126	C₈²	C₄×Q₈	1	1	yes	characteristic
127	C₂×M₁₆	—	4	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	C₈×D₄	1	1	yes	characteristic
130	C₂²×C₈	C₄×D₈	1	1	yes	characteristic
131	#117	—	4	2	no	—
132	C₂×C₄²	#443	1	1	yes	characteristic
133	M₃₂	—	4	4	no	—
134	M₃₂	—	8	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	4	4	no	—
138	C₂×C₁₆	—	8	4	no	—
139	C₂×C₁₆	—	8	2	no	—
140	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
141	C₂×C₁₆	#405	1	1	yes	characteristic
142	C₂×C₁₆	C₂×#107	1	1	yes	characteristic
143	C₂×C₁₆	#427	1	1	yes	characteristic
144	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
145	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₁₆	C₈×Q₈	1	1	yes	characteristic
147	C₂×C₁₆	C₈×Q₈	1	1	yes	characteristic
148	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
149	C₂×C₁₆	#504	1	2	yes	—
150	C₂×C₁₆	#509	1	2	yes	—
151	C₂×C₁₆	#510	1	2	yes	—
152	#108	—	4	2	no	—
153	#108	—	4	2	no	—
154	#107	—	4	2	no	—
155	#106	—	4	2	no	—
156	#97	—	8	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	8	2	no	—
161	C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₈	C₈×D₄	1	1	yes	characteristic
163	C₄×C₈	C₄×Q₁₆	1	1	yes	characteristic
164	C₄×C₈	#443	1	1	yes	characteristic
165	C₃₂	—	2	4	no	—
166	C₃₂	—	2	2	no	—
167	C₃₂	—	2	4	no	—
168	C₃₂	—	4	2	no	—
169	C₃₂	—	8	4	no	—
170	C₃₂	#346	1	4	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₈×D₈	1	1	yes	characteristic
173	M₁₆	—	8	2	no	—
174	M₁₆	—	8	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	8	2	no	—
179	C₂×C₈	C₈×D₈	1	1	yes	characteristic
180	C₂×C₈	C₈×Q₁₆	1	1	yes	characteristic
181	C₂×C₈	C₈×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₄×#107	1	1	yes	characteristic
183	C₂×C₈	#2169	1	2	yes	—
184	C₂×C₈	#2170	1	2	yes	—
185	C₂×C₈	#2175	1	2	yes	—
186	C₂×C₈	#2179	1	2	yes	—
187	C₄⋊C₄	—	8	2	no	—
188	C₂²⋊C₄	—	8	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	8	2	no	—
191	C₄²	C₈×Q₁₆	1	1	yes	characteristic
192	C₄²	#1772	1	1	yes	characteristic
193	C₁₆	—	2	1	no	—
194	C₁₆	—	2	1	no	—
195	C₁₆	—	2	1	no	—
196	C₁₆	—	2	1	no	—
197	C₁₆	—	8	4	no	—
198	C₁₆	—	8	4	no	—
199	C₁₆	C₂×#346	1	1	yes	characteristic
200	C₁₆	#2108	1	1	yes	characteristic
201	C₁₆	#2159	1	2	yes	—
202	C₁₆	#2176	1	1	yes	characteristic
203	C₁₆	#2177	1	1	yes	characteristic
204	C₁₆	#2183	1	2	yes	—
205	C₂³	#12047	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	16	2	no	—
210	C₂×C₄	C₈×#107	1	1	yes	characteristic
211	C₂×C₄	#12039	1	2	yes	—
212	C₂×C₄	#12043	1	2	yes	—
213	C₂×C₄	#12047	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	4	2	no	—
216	C₈	—	16	2	no	—
217	C₈	#12043	1	2	yes	—
218	C₈	#12054	1	2	yes	—
219	C₈	#12662	1	1	yes	characteristic
220	C₈	#12664	1	1	yes	characteristic
221	C₈	#12671	1	1	yes	characteristic
222	C₈	#12674	1	1	yes	characteristic
223	C₂²	—	4	1	no	—
224	C₂²	#122950	1	2	yes	—
225	C₂²	#129143	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	4	1	no	—
228	C₄	—	32	2	no	—
229	C₄	#122950	1	2	yes	—
230	C₄	#129140	1	1	yes	characteristic
231	C₄	#129146	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	2	no	—
234	C₂	#72225377	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#12674
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225377
prime radical	C₂³	#12047
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ 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. #93882	C₄	semidirect
17. #93666	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
#72225352	2
#72225313	2
#72216535	2
#72216226	2