Subgroups — #498518

Order 512.

452 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	#33476	C₂	1	2	yes	maximal
2	#21735	C₂	1	1	yes	maximal, characteristic
3	#13265	C₂	1	2	yes	maximal
4	#13162	C₂	1	1	yes	maximal, characteristic
5	#13161	C₂	1	1	yes	maximal, characteristic
6	#13106	C₂	1	2	yes	maximal
7	#13106	C₂	1	2	yes	maximal
8	#12665	C₂	1	1	yes	maximal, characteristic
9	#12664	C₂	1	1	yes	maximal, characteristic
10	#12656	C₂	1	1	yes	maximal, characteristic
11	#12654	C₂	1	1	yes	maximal, characteristic
12	#3162	C₄	1	1	yes	characteristic
13	#2960	C₂²	1	2	yes	—
14	#2183	—	2	2	no	—
15	#2183	—	2	2	no	—
16	#2183	—	2	2	no	—
17	Q₈×C₁₆	—	2	2	no	—
18	C₂×#364	C₂²	1	1	yes	characteristic
19	C₂×#364	C₂²	1	1	yes	characteristic
20	#2151	C₂²	1	2	yes	—
21	#2151	C₂²	1	2	yes	—
22	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	#2108	C₂²	1	2	yes	—
25	#2108	C₂²	1	2	yes	—
26	C₄×M₃₂	C₂²	1	2	yes	—
27	C₂×#346	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
29	#1995	C₄	1	2	yes	—
30	#1946	C₄	1	1	yes	characteristic
31	#1946	C₄	1	1	yes	characteristic
32	#1946	C₄	1	2	yes	—
33	#1849	C₄	1	1	yes	characteristic
34	#1772	C₂²	1	2	yes	—
35	#1772	C₂²	1	1	yes	characteristic
36	#1772	C₂²	1	1	yes	characteristic
37	#1726	C₂²	1	1	yes	characteristic
38	#1379	C₂²	1	2	yes	—
39	#1379	C₂²	1	2	yes	—
40	#1379	C₂²	1	2	yes	—
41	#1379	C₂²	1	2	yes	—
42	#1376	C₂²	1	2	yes	—
43	#1374	C₂²	1	1	yes	characteristic
44	#1374	C₂²	1	1	yes	characteristic
45	#533	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	C₂³	1	2	yes	—
47	C₂×M₃₂	C₂³	1	2	yes	—
48	C₂²×C₁₆	C₂³	1	1	yes	characteristic
49	C₂²×C₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆	C₂³	1	1	yes	characteristic
52	#501	C₈	1	2	yes	—
53	#501	C₈	1	2	yes	—
54	#501	C₈	1	2	yes	—
55	#501	C₈	1	2	yes	—
56	#446	—	2	2	no	—
57	#446	—	2	2	no	—
58	C₈×Q₈	—	2	2	no	—
59	C₈×Q₈	—	2	2	no	—
60	C₂×#108	C₂×C₄	1	1	yes	characteristic
61	C₂×#108	C₂×C₄	1	1	yes	characteristic
62	C₂×#108	C₂×C₄	1	2	yes	—
63	#427	C₂×C₄	1	2	yes	—
64	#427	C₂×C₄	1	1	yes	characteristic
65	#427	C₂×C₄	1	1	yes	characteristic
66	#424	C₂×C₄	1	2	yes	—
67	#424	C₂×C₄	1	2	yes	—
68	#423	C₂×C₄	1	2	yes	—
69	#405	C₂³	1	2	yes	—
70	#405	C₂³	1	2	yes	—
71	C₂×#97	C₂×C₄	1	1	yes	characteristic
72	C₂×#97	C₂×C₄	1	1	yes	characteristic
73	C₂×#97	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
77	#376	C₂×C₄	1	1	yes	characteristic
78	#364	—	2	2	no	—
79	#364	Q₈	1	4	yes	complemented
80	C₄⋊C₁₆	—	2	2	no	—
81	C₄⋊C₁₆	—	2	2	no	—
82	C₄⋊C₁₆	—	2	2	no	—
83	C₄⋊C₁₆	—	2	2	no	—
84	C₄⋊C₁₆	—	2	2	no	—
85	C₄⋊C₁₆	D₄	1	2	yes	—
86	C₄⋊C₁₆	D₄	1	2	yes	—
87	#346	—	2	2	no	—
88	#346	—	2	2	no	—
89	#346	—	2	2	no	—
90	#346	—	2	2	no	—
91	C₄×C₁₆	—	2	2	no	—
92	C₄×C₁₆	—	2	2	no	—
93	C₄×C₁₆	—	2	2	no	—
94	C₄×C₁₆	—	2	2	no	—
95	#339	C₂×C₄	1	2	yes	—
96	#339	C₂×C₄	1	2	yes	—
97	#339	C₂×C₄	1	2	yes	—
98	#339	C₂×C₄	1	2	yes	—
99	C₂×M₁₆	C₂²×C₄	1	2	yes	—
100	C₂×M₁₆	C₂²×C₄	1	2	yes	—
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂⁴	1	1	yes	characteristic
105	C₄⋊Q₈	C₂×C₈	1	2	yes	—
106	C₄⋊Q₈	C₂×C₈	1	2	yes	—
107	#125	C₂×C₈	1	2	yes	—
108	#125	C₂×C₈	1	2	yes	—
109	C₂²⋊Q₈	—	2	2	no	—
110	C₂²⋊Q₈	—	2	2	no	—
111	C₄×Q₈	—	2	2	no	—
112	C₄×Q₈	—	2	2	no	—
113	#117	C₂²×C₄	1	2	yes	—
114	#117	C₂²×C₄	1	2	yes	—
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	—	2	2	no	—
120	M₃₂	—	2	2	no	—
121	M₃₂	C₂×Q₈	1	4	yes	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
135	C₂×C₁₆	C₂×D₄	1	2	yes	—
136	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
137	#108	—	2	2	no	—
138	#108	—	2	2	no	—
139	#108	—	4	2	no	—
140	#107	C₂×C₈	1	2	yes	—
141	#107	C₂×C₈	1	2	yes	—
142	#107	C₂×C₈	1	1	yes	characteristic
143	#107	C₂×C₈	1	1	yes	characteristic
144	#107	C₂×C₈	1	1	yes	characteristic
145	#107	C₂×C₈	1	1	yes	characteristic
146	#106	C₂×C₈	1	2	yes	—
147	#106	C₂×C₈	1	2	yes	—
148	#106	C₂×C₈	1	2	yes	—
149	#106	C₂×C₈	1	2	yes	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	#97	—	4	1	no	—
160	#97	C₂×C₈	1	2	yes	—
161	#97	C₂×C₈	1	2	yes	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	C₂×D₄	1	2	yes	—
167	C₄×C₈	C₂×Q₈	1	2	yes	—
168	C₄×C₈	#41	1	2	yes	—
169	C₄×C₈	#41	1	2	yes	—
170	#95	—	2	1	no	—
171	#95	—	2	1	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	#131	1	1	yes	characteristic
177	C₂²×C₄	#131	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	2	no	—
182	M₁₆	—	2	2	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	C₄×D₄	1	2	yes	—
190	C₂×C₈	C₄×Q₈	1	2	yes	—
191	C₂×C₈	C₂²×C₈	1	2	yes	—
192	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
194	C₂×C₈	#131	1	2	yes	—
195	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
197	C₂×C₈	C₂×#41	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	1	yes	characteristic
199	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
200	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₂²×C₈	1	2	yes	—
208	C₄⋊C₄	C₂²×C₈	1	2	yes	—
209	C₄⋊C₄	C₂²×C₈	1	2	yes	—
210	C₄⋊C₄	C₂²×C₈	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₄×D₄	1	2	yes	—
219	C₄²	C₄×Q₈	1	2	yes	—
220	C₄²	C₂²×C₈	1	2	yes	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	4	2	no	—
226	C₁₆	—	4	1	no	—
227	C₁₆	—	4	1	no	—
228	C₂³	C₂×#131	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	8	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	C₈×D₄	1	2	yes	—
244	C₂×C₄	C₈×D₄	1	2	yes	—
245	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
247	C₂×C₄	#519	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×#131	1	1	yes	characteristic
250	C₂×C₄	#568	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	4	2	no	—
255	C₈	C₈×Q₈	1	2	yes	—
256	C₈	C₈×Q₈	1	2	yes	—
257	C₈	#547	1	1	yes	characteristic
258	C₈	D₄×Q₈	1	1	yes	characteristic
259	C₈	#556	1	1	yes	characteristic
260	C₈	#557	1	1	yes	characteristic
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
264	C₂²	#2962	1	1	yes	characteristic
265	C₂²	#2986	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	1	no	—
274	C₄	#2365	1	1	yes	characteristic
275	C₄	#2935	1	1	yes	characteristic
276	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
277	C₄	#2982	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	#23149	1	1	yes	characteristic
281	C₂	#33441	1	1	yes	characteristic
282	C₂	#33442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
79. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33476	2
#21735	2
#13265	2
#13162	2
#13161	2
#13106	2
#12665	2
#12664	2
#12656	2
#12654	2