Subgroups — #72234366

Order 1024.

587 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512761	C₂	1	1	yes	maximal, characteristic
2	#483823	C₂	1	2	yes	maximal, complemented
3	#151129	C₂	1	2	yes	maximal, complemented
4	#151128	C₂	1	2	yes	maximal, complemented
5	#151127	C₂	1	4	yes	maximal, complemented
6	#151037	C₂	1	1	yes	maximal, characteristic, complemented
7	#150062	C₂	1	2	yes	maximal, complemented
8	C₂×#7330	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#2262	C₄	1	1	yes	characteristic
10	C₄×#508	C₂²	1	2	yes	—
11	#17886	—	2	2	no	—
12	#13654	C₄	1	1	yes	characteristic
13	#13654	C₂²	1	1	yes	characteristic
14	#13653	C₄	1	1	yes	characteristic
15	#13653	C₂²	1	1	yes	characteristic
16	#13652	C₄	1	2	yes	—
17	#13652	C₂²	1	2	yes	—
18	#13633	C₄	1	1	yes	characteristic
19	#13615	C₄	1	2	yes	—
20	C₂×#1397	C₂²	1	1	yes	characteristic
21	C₂×#1397	C₂²	1	1	yes	characteristic
22	#12086	C₂²	1	4	yes	—
23	#12085	C₂²	1	4	yes	—
24	#12084	—	2	4	no	—
25	#12008	C₂²	1	2	yes	—
26	#11591	—	2	2	no	—
27	C₂×#1311	C₂²	1	1	yes	characteristic
28	#7463	C₂²	1	2	yes	—
29	#7463	C₂²	1	2	yes	—
30	#7386	C₂²	1	4	yes	—
31	#7385	C₂²	1	4	yes	—
32	#7330	C₂²	1	2	yes	—
33	#7330	C₂²	1	2	yes	—
34	C₂×#508	C₂×C₄	1	2	yes	—
35	#2965	—	4	2	no	—
36	C₄×#135	—	2	2	no	—
37	#2262	—	2	4	no	—
38	C₂×Q₆₄	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₆₄	C₂×C₄	1	2	yes	—
40	C₂×D₃₂	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃₂	—	2	1	no	—
42	C₂²×C₃₂	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃₂	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₃₂	C₂×C₄	1	2	yes	—
45	C₄×Q₃₂	C₂³	1	2	yes	—
46	C₄×SD₃₂	—	2	2	no	—
47	C₄×SD₃₂	—	2	2	no	—
48	C₄×D₁₆	C₂×C₄	1	2	yes	—
49	C₄×D₁₆	C₂³	1	2	yes	—
50	#2158	C₂×C₄	1	2	yes	—
51	#2138	—	2	2	no	—
52	C₂×C₄×C₁₆	D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
54	#1591	—	2	4	no	—
55	#1590	—	4	4	no	—
56	#1589	—	2	4	no	—
57	#1564	—	2	2	no	—
58	#1476	—	4	2	no	—
59	C₂×C₈²	D₄	1	1	yes	characteristic
60	#1424	C₂×C₄	1	1	yes	characteristic
61	#1424	C₂×C₄	1	1	yes	characteristic
62	#1423	C₂×C₄	1	2	yes	—
63	#1416	C₂×C₄	1	2	yes	—
64	#1416	C₂×C₄	1	2	yes	—
65	#1415	C₂×C₄	1	2	yes	—
66	#1415	C₂×C₄	1	2	yes	—
67	#1397	—	2	1	no	—
68	#1397	C₂×C₄	1	1	yes	characteristic
69	#1397	C₂×C₄	1	1	yes	characteristic
70	#1397	C₂³	1	2	yes	—
71	#1397	C₂³	1	1	yes	characteristic
72	#1397	C₂³	1	1	yes	characteristic
73	#1372	C₂³	1	4	yes	—
74	#1371	—	2	2	no	—
75	#1332	—	2	4	no	—
76	#1331	—	2	4	no	—
77	#1311	D₄	1	2	yes	—
78	#1311	C₂³	1	2	yes	—
79	C₂×#135	—	2	2	no	—
80	C₄×#41	—	4	2	no	—
81	#508	—	2	4	no	—
82	#508	—	2	4	no	—
83	C₂×Q₃₂	C₂²×C₄	1	2	yes	—
84	C₂×SD₃₂	—	2	2	no	—
85	C₂×D₁₆	C₂²×C₄	1	2	yes	—
86	C₂²×C₁₆	—	2	1	no	—
87	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₁₆	#41	1	1	yes	characteristic
89	#446	—	4	4	no	—
90	C₄×Q₁₆	—	2	2	no	—
91	C₄×Q₁₆	—	2	2	no	—
92	C₄×SD₁₆	—	4	2	no	—
93	C₄×SD₁₆	—	4	2	no	—
94	C₄×D₈	—	2	2	no	—
95	C₄×D₈	—	2	2	no	—
96	#436	—	4	4	no	—
97	#435	—	8	4	no	—
98	#431	—	4	2	no	—
99	#431	—	8	2	no	—
100	#427	—	2	2	no	—
101	#416	—	4	2	no	—
102	C₄×M₁₆	—	8	2	no	—
103	C₂×C₄×C₈	D₈	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
105	Q₆₄	—	2	2	no	—
106	SD₆₄	—	2	4	no	—
107	D₃₂	—	2	2	no	—
108	C₂×C₃₂	—	2	4	no	—
109	C₂×C₃₂	—	2	2	no	—
110	C₂×C₃₂	—	2	2	no	—
111	C₂×C₃₂	M₁₆	1	4	yes	complemented
112	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
116	#367	—	2	2	no	—
117	#366	C₂²×C₄	1	2	yes	—
118	#366	C₂²×C₄	1	2	yes	—
119	#358	—	2	2	no	—
120	#358	—	2	2	no	—
121	#357	—	2	2	no	—
122	#357	—	2	2	no	—
123	C₄×C₁₆	—	2	1	no	—
124	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₁₆	C₂×D₄	1	2	yes	—
126	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₄×C₁₆	#41	1	1	yes	characteristic
128	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
129	#335	—	2	4	no	—
130	#334	—	4	2	no	—
131	Q₈⋊C₈	—	4	4	no	—
132	D₄⋊C₈	—	4	4	no	—
133	C₈²	D₈	1	2	yes	—
134	C₈²	C₂×D₄	1	2	yes	—
135	C₂×#41	—	4	2	no	—
136	#135	—	4	4	no	—
137	#135	—	4	4	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×SD₁₆	—	4	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×M₁₆	—	8	4	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₈	#135	1	1	yes	characteristic
145	C₄×Q₈	—	4	2	no	—
146	C₄×Q₈	—	4	2	no	—
147	C₄×D₄	—	4	2	no	—
148	C₄×D₄	—	4	2	no	—
149	C₄×D₄	—	8	2	no	—
150	C₄×D₄	—	8	2	no	—
151	#117	—	4	2	no	—
152	#117	—	8	2	no	—
153	C₂×C₄²	—	8	2	no	—
154	C₂×C₄²	C₂×D₈	1	1	yes	characteristic
155	Q₃₂	—	2	2	no	—
156	Q₃₂	—	2	2	no	—
157	SD₃₂	—	4	4	no	—
158	D₁₆	—	2	2	no	—
159	D₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	4	2	no	—
163	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
164	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
165	C₂×C₁₆	C₂×M₁₆	1	2	yes	—
166	C₂×C₁₆	#131	1	2	yes	—
167	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₁₆	C₂×#41	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#106	—	4	2	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₄⋊C₈	—	8	4	no	—
174	#103	—	4	2	no	—
175	#103	—	4	2	no	—
176	#102	—	4	2	no	—
177	#102	—	4	2	no	—
178	C₂²⋊C₈	—	8	4	no	—
179	C₂²⋊C₈	—	8	2	no	—
180	#97	—	8	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	8	2	no	—
183	C₄×C₈	C₄×D₄	1	1	yes	characteristic
184	C₄×C₈	C₂×D₈	1	4	yes	—
185	C₄×C₈	C₂×D₈	1	1	yes	characteristic
186	C₄×C₈	#135	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
188	C₃₂	—	2	4	no	—
189	C₃₂	—	2	1	no	—
190	C₃₂	—	2	1	no	—
191	C₃₂	C₂×M₁₆	1	2	yes	—
192	C₃₂	C₂×M₁₆	1	2	yes	—
193	#41	—	8	4	no	—
194	#41	—	8	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	8	2	no	—
198	C₂²×C₄	—	8	2	no	—
199	C₂²×C₄	—	8	2	no	—
200	C₂²×C₄	—	8	2	no	—
201	C₂²×C₄	C₄×D₈	1	1	yes	characteristic
202	C₂²×C₄	#509	1	2	yes	—
203	Q₁₆	—	4	2	no	—
204	Q₁₆	—	4	2	no	—
205	SD₁₆	—	8	4	no	—
206	D₈	—	4	2	no	—
207	D₈	—	4	2	no	—
208	M₁₆	—	16	8	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	8	2	no	—
212	C₂×C₈	—	8	4	no	—
213	C₂×C₈	#436	1	2	yes	—
214	C₂×C₈	C₄×D₈	1	1	yes	characteristic
215	C₂×C₈	C₄×D₈	1	1	yes	characteristic
216	C₂×C₈	#443	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×C₈	C₂×#135	1	1	yes	characteristic
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	8	2	no	—
222	C₄⋊C₄	—	8	2	no	—
223	C₂²⋊C₄	—	8	2	no	—
224	C₂²⋊C₄	—	8	2	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₄²	—	8	2	no	—
227	C₄²	—	8	2	no	—
228	C₄²	—	8	2	no	—
229	C₄²	C₄×D₈	1	1	yes	characteristic
230	C₄²	#509	1	2	yes	—
231	C₄²	C₂²×D₈	1	1	yes	characteristic
232	C₁₆	—	2	1	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	—	4	2	no	—
235	C₁₆	#436	1	2	yes	—
236	C₁₆	C₂²×M₁₆	1	1	yes	characteristic
237	C₁₆	C₂×#131	1	1	yes	characteristic
238	C₂³	—	8	2	no	—
239	C₂³	#2177	1	1	yes	characteristic
240	Q₈	—	8	2	no	—
241	Q₈	—	8	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	16	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	16	4	no	—
250	C₂×C₄	—	16	4	no	—
251	C₂×C₄	—	16	4	no	—
252	C₂×C₄	#1589	1	2	yes	—
253	C₂×C₄	#2177	1	2	yes	—
254	C₂×C₄	#2177	1	1	yes	characteristic
255	C₂×C₄	#2179	1	2	yes	—
256	C₂×C₄	C₂×C₄×D₈	1	1	yes	characteristic
257	C₂×C₄	C₂×#509	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	4	no	—
261	C₈	—	16	4	no	—
262	C₈	#1589	1	2	yes	—
263	C₈	C₂×#436	1	1	yes	characteristic
264	C₈	C₂×#443	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	16	2	no	—
267	C₂²	—	16	2	no	—
268	C₂²	—	16	2	no	—
269	C₂²	#12086	1	2	yes	—
270	C₂²	C₂×#2177	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	16	2	no	—
274	C₄	—	16	2	no	—
275	C₄	—	16	2	no	—
276	C₄	#12086	1	2	yes	—
277	C₄	C₂×#1589	1	1	yes	characteristic
278	C₄	C₂×#2179	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	32	2	no	—
281	C₂	C₂×#12086	1	1	yes	characteristic
282	C₂	#512778	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂×#2177
prime radical	C₂×#2262	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #483823	C₂	semidirect
3. #151129	C₂	semidirect
4. #151128	C₂	semidirect
5. #151127	C₂	semidirect
6. #151037	C₂	semidirect
7. #150062	C₂	semidirect
8. C₂×#7330	C₂	semidirect
111. C₂×C₃₂	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#512761	2
#483823	2
#151129	2
#151128	2
#151127	2
#151037	2
#150062	2
C₂×#7330	2