Subgroups — #498512

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2183	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#13263	C₂	1	1	yes	maximal, characteristic
5	#13263	C₂	1	1	yes	maximal, characteristic
6	#13160	C₂	1	2	yes	maximal
7	#13105	C₂	1	2	yes	maximal
8	#13104	C₂	1	2	yes	maximal
9	#12663	C₂	1	2	yes	maximal
10	#12653	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
15	#2183	—	2	4	no	—
16	Q₈×C₁₆	—	2	4	no	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₂×#346	C₂²	1	2	yes	—
24	C₂×#346	C₂²	1	2	yes	—
25	C₂×C₄×C₁₆	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	#1984	C₄	1	1	yes	characteristic
29	#1984	C₄	1	1	yes	characteristic
30	#1939	C₄	1	2	yes	—
31	#1939	C₄	1	2	yes	—
32	#1842	C₄	1	1	yes	characteristic
33	C₈×C₄⋊C₄	C₂²	1	2	yes	—
34	C₈×C₄⋊C₄	C₂²	1	2	yes	—
35	C₄²×C₈	C₂²	1	1	yes	characteristic
36	#1374	C₂²	1	1	yes	characteristic
37	#1374	C₂²	1	2	yes	—
38	#1374	C₂²	1	2	yes	—
39	#1374	C₂²	1	2	yes	—
40	#1374	C₂²	1	1	yes	characteristic
41	#1374	C₂²	1	2	yes	—
42	#1374	C₂²	1	2	yes	—
43	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
44	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₁₆	C₂³	1	2	yes	—
49	C₂²×C₁₆	C₂³	1	2	yes	—
50	C₂²×C₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆	C₂³	1	1	yes	characteristic
52	C₈×Q₈	—	2	2	no	—
53	C₈×Q₈	—	2	2	no	—
54	C₈×Q₈	—	2	2	no	—
55	C₈×Q₈	—	2	2	no	—
56	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	2	yes	—
66	C₂×C₄×C₈	C₂³	1	2	yes	—
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	#391	C₈	1	1	yes	characteristic
71	#391	C₈	1	1	yes	characteristic
72	#389	C₈	1	2	yes	—
73	#389	C₈	1	2	yes	—
74	#384	C₈	1	1	yes	characteristic
75	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
76	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
77	C₄³	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₁₆	—	2	4	no	—
79	C₄⋊C₁₆	—	2	4	no	—
80	C₄⋊C₁₆	—	2	2	no	—
81	C₄⋊C₁₆	—	2	2	no	—
82	C₄⋊C₁₆	D₄	1	4	yes	—
83	C₄⋊C₁₆	Q₈	1	4	yes	complemented
84	#346	—	2	4	no	—
85	#346	—	2	4	no	—
86	C₄×C₁₆	—	2	4	no	—
87	C₄×C₁₆	—	2	2	no	—
88	C₄×C₁₆	—	2	2	no	—
89	#336	C₂×C₄	1	2	yes	—
90	#336	C₂×C₄	1	2	yes	—
91	#336	C₂×C₄	1	2	yes	—
92	#336	C₂×C₄	1	2	yes	—
93	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
94	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₄	1	2	yes	—
97	C₂²×C₈	C₂²×C₄	1	2	yes	—
98	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂⁴	1	1	yes	characteristic
101	C₄⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
109	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
112	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
114	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	2	yes	—
119	C₂×C₄²	C₂²×C₄	1	2	yes	—
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	C₂×D₄	1	4	yes	—
139	C₂×C₁₆	C₂×Q₈	1	2	yes	—
140	C₂×C₁₆	C₂×Q₈	1	2	yes	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	4	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	4	1	no	—
156	C₄×C₈	C₂×D₄	1	2	yes	—
157	C₄×C₈	C₂×Q₈	1	2	yes	—
158	C₄×C₈	#41	1	2	yes	—
159	C₄×C₈	#41	1	2	yes	—
160	#95	C₂×C₈	1	2	yes	—
161	#95	C₂×C₈	1	2	yes	—
162	#95	C₂×C₈	1	2	yes	—
163	#95	C₂×C₈	1	2	yes	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	M₃₂	1	4	yes	—
170	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₈	1	2	yes	—
172	C₂²×C₄	C₂²×C₈	1	2	yes	—
173	C₂²×C₄	C₂²×C₈	1	2	yes	—
174	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₈	1	2	yes	—
178	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₄×D₄	1	2	yes	—
195	C₂×C₈	C₄×Q₈	1	2	yes	—
196	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	1	yes	characteristic
199	C₂×C₈	C₂×#41	1	1	yes	characteristic
200	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
201	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	C₄×D₄	1	2	yes	—
221	C₄²	C₄×Q₈	1	2	yes	—
222	C₄²	#131	1	2	yes	—
223	C₄²	#131	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	2	2	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	4	1	no	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	C₂³×C₈	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	4	4	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	C₈×D₄	1	2	yes	—
256	C₂×C₄	C₈×D₄	1	2	yes	—
257	C₂×C₄	C₈×Q₈	1	2	yes	—
258	C₂×C₄	C₈×Q₈	1	2	yes	—
259	C₂×C₄	C₂×M₃₂	1	4	yes	—
260	C₂×C₄	C₂×M₃₂	1	2	yes	—
261	C₂×C₄	#504	1	2	yes	—
262	C₂×C₄	#504	1	2	yes	—
263	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
265	C₂×C₄	#519	1	1	yes	characteristic
266	C₂×C₄	C₂×#131	1	1	yes	characteristic
267	C₂×C₄	C₂×#131	1	1	yes	characteristic
268	C₂×C₄	#568	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	1	no	—
274	C₈	#547	1	1	yes	characteristic
275	C₈	D₄×Q₈	1	1	yes	characteristic
276	C₈	#556	1	1	yes	characteristic
277	C₈	#557	1	1	yes	characteristic
278	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
279	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
280	C₂²	#2982	1	1	yes	characteristic
281	C₂²	C₂²×M₃₂	1	1	yes	characteristic
282	C₂²	C₂×#504	1	1	yes	characteristic
283	C₂²	C₂×#504	1	1	yes	characteristic
284	C₂²	#3407	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	1	no	—
294	C₄	#2168	1	4	yes	—
295	C₄	#2365	1	1	yes	characteristic
296	C₄	#2935	1	1	yes	characteristic
297	C₄	#2962	1	1	yes	characteristic
298	C₄	#2986	1	1	yes	characteristic
299	C₂	#23149	1	1	yes	characteristic
300	C₂	C₂×#2168	1	1	yes	characteristic
301	C₂	#33438	1	1	yes	characteristic
302	C₂	#33474	1	1	yes	characteristic
303	C₂	Q₈×M₃₂	1	1	yes	characteristic
304	C₂	#33498	1	1	yes	characteristic
305	C₂	#33499	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
83. C₄⋊C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2183	2
C₂×Q₈×C₁₆	2
C₈×C₄⋊Q₈	2
#13263	2
#13160	2
#13105	2
#13104	2
#12663	2
#12653	2
C₄×C₄⋊C₁₆	2