Subgroups — #126861

Order 512.

501 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#20816	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#7174	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	—	2	1	no	—
6	#2920	—	2	1	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2285	—	2	1	no	—
9	C₄×M₃₂	—	2	1	no	—
10	C₄×M₃₂	—	2	1	no	—
11	C₄×M₃₂	C₄	1	2	yes	complemented
12	C₄×M₃₂	C₂²	1	2	yes	—
13	C₂×#346	—	2	1	no	—
14	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
15	#1844	C₄	1	2	yes	—
16	#1834	—	2	1	no	—
17	#1807	—	2	1	no	—
18	#1807	C₄	1	2	yes	—
19	C₄×#106	—	2	1	no	—
20	C₄×#106	C₂²	1	2	yes	—
21	#1568	—	2	1	no	—
22	#1568	C₄	1	2	yes	complemented
23	#1567	—	2	1	no	—
24	#1566	C₄	1	2	yes	complemented
25	#1564	—	2	1	no	—
26	#1564	C₂²	1	2	yes	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂×#117	—	4	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	8	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂²×C₁₆	—	2	2	no	—
35	#432	—	2	2	no	—
36	#432	—	4	1	no	—
37	#431	—	2	2	no	—
38	#431	—	4	1	no	—
39	C₂×#106	—	2	2	no	—
40	C₂×#106	C₂×C₄	1	4	yes	—
41	#423	—	2	2	no	—
42	#407	—	4	1	no	—
43	C₄×M₁₆	—	4	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
54	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	#384	—	2	2	no	—
57	#382	—	4	1	no	—
58	#380	—	2	2	no	—
59	#380	—	4	1	no	—
60	C₄×C₄⋊C₄	—	2	2	no	—
61	C₄×C₄⋊C₄	—	4	1	no	—
62	#346	—	2	2	no	—
63	#346	—	2	2	no	—
64	C₄×C₁₆	—	2	2	no	—
65	C₄×C₁₆	C₂×C₄	1	4	yes	—
66	#340	—	2	2	no	—
67	#340	—	2	2	no	—
68	#335	—	2	2	no	—
69	#335	—	2	2	no	—
70	#334	—	2	2	no	—
71	#334	C₂×C₄	1	4	yes	—
72	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₄	D₈	1	1	yes	characteristic
74	C₂³×C₄	SD₁₆	1	1	yes	characteristic
75	C₂×M₁₆	—	4	2	no	—
76	C₂²×C₈	—	2	4	no	—
77	C₂²×C₈	—	2	4	no	—
78	C₂²×C₈	C₄²	1	1	yes	characteristic
79	C₂²×C₈	C₄²	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	#117	—	8	4	no	—
91	C₂×C₄⋊C₄	—	2	4	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₂²⋊C₄	—	4	2	no	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	D₈	1	1	yes	characteristic
102	C₂×C₄²	SD₁₆	1	1	yes	characteristic
103	C₂×C₄²	SD₁₆	1	1	yes	characteristic
104	C₂×C₄²	SD₁₆	1	1	yes	characteristic
105	C₂×C₄²	Q₁₆	1	1	yes	characteristic
106	C₂×C₄²	Q₁₆	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	M₃₂	—	2	8	no	—
109	M₃₂	—	2	8	no	—
110	C₂×C₁₆	—	2	8	no	—
111	C₂×C₁₆	—	2	4	no	—
112	#106	—	2	8	no	—
113	#106	—	4	4	no	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	2	no	—
117	C₂²⋊C₈	—	4	2	no	—
118	#97	—	4	2	no	—
119	C₄×C₈	—	4	2	no	—
120	C₄×C₈	C₄²	1	1	yes	characteristic
121	C₄×C₈	C₄²	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
136	#95	—	4	2	no	—
137	#95	—	4	2	no	—
138	C₂⁴	#102	1	1	yes	characteristic
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	#95	1	1	yes	characteristic
145	C₂²×C₄	#95	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
147	C₂²×C₄	#102	1	1	yes	characteristic
148	C₂²×C₄	#102	1	1	yes	characteristic
149	C₂²×C₄	#102	1	1	yes	characteristic
150	C₂²×C₄	#102	1	1	yes	characteristic
151	C₂²×C₄	#103	1	1	yes	characteristic
152	C₂²×C₄	#103	1	1	yes	characteristic
153	C₂²×C₄	#103	1	1	yes	characteristic
154	C₂²×C₄	#103	1	1	yes	characteristic
155	C₂²×C₄	#106	1	1	yes	characteristic
156	C₂²×C₄	#106	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	#107	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
162	M₁₆	—	8	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	#95	1	1	yes	characteristic
175	C₂×C₈	#95	1	1	yes	characteristic
176	C₂×C₈	#95	1	1	yes	characteristic
177	C₂×C₈	#95	1	1	yes	characteristic
178	C₂×C₈	#104	1	4	yes	—
179	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₈	#131	1	4	yes	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	8	4	no	—
187	C₄⋊C₄	—	8	4	no	—
188	C₂²⋊C₄	—	8	4	no	—
189	C₄²	—	8	4	no	—
190	C₄²	#95	1	1	yes	characteristic
191	C₄²	#95	1	1	yes	characteristic
192	C₄²	#102	1	1	yes	characteristic
193	C₄²	#102	1	1	yes	characteristic
194	C₄²	#103	1	1	yes	characteristic
195	C₄²	#103	1	1	yes	characteristic
196	C₄²	#103	1	1	yes	characteristic
197	C₄²	#103	1	1	yes	characteristic
198	C₄²	#106	1	1	yes	characteristic
199	C₄²	#106	1	1	yes	characteristic
200	C₄²	#107	1	1	yes	characteristic
201	C₄²	#107	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×SD₁₆	1	1	yes	characteristic
205	C₄²	C₂×Q₁₆	1	1	yes	characteristic
206	C₁₆	—	2	8	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	4	1	no	—
210	C₂³	#340	1	1	yes	characteristic
211	C₂³	#340	1	1	yes	characteristic
212	C₂³	C₂×#102	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	#340	1	1	yes	characteristic
235	C₂×C₄	#340	1	1	yes	characteristic
236	C₂×C₄	#340	1	1	yes	characteristic
237	C₂×C₄	#340	1	1	yes	characteristic
238	C₂×C₄	#340	1	1	yes	characteristic
239	C₂×C₄	#340	1	1	yes	characteristic
240	C₂×C₄	#365	1	4	yes	—
241	C₂×C₄	C₂×#95	1	1	yes	characteristic
242	C₂×C₄	#408	1	2	yes	—
243	C₂×C₄	C₂×#102	1	1	yes	characteristic
244	C₂×C₄	C₂×#103	1	1	yes	characteristic
245	C₂×C₄	C₂×#103	1	1	yes	characteristic
246	C₂×C₄	C₂×#104	1	2	yes	—
247	C₂×C₄	C₂×#106	1	1	yes	characteristic
248	C₂×C₄	C₂×#107	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	8	4	no	—
254	C₈	#339	1	2	yes	—
255	C₈	#339	1	2	yes	—
256	C₈	#405	1	2	yes	—
257	C₈	#424	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	4	4	no	—
266	C₂²	#1383	1	2	yes	—
267	C₂²	#1492	1	2	yes	—
268	C₂²	C₂×#340	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	16	4	no	—
277	C₄	#1383	1	2	yes	—
278	C₄	#1384	1	2	yes	—
279	C₄	#1493	1	2	yes	—
280	C₄	#1729	1	1	yes	characteristic
281	C₄	C₂×#339	1	1	yes	characteristic
282	C₄	C₂×#365	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	1	no	—
286	C₂	#7897	1	1	yes	characteristic
287	C₂	C₂×#1383	1	1	yes	characteristic
288	C₂	C₂×#1384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7174	C₂	semidirect
11. C₄×M₃₂	C₄	semidirect
14. C₂×C₄×C₁₆	C₄	semidirect
22. #1568	C₄	semidirect
24. #1566	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
#20816	2
#17793	2
#7174	2