Subgroups — #495471

Order 512.

553 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2163	C₂	1	1	yes	maximal, characteristic
2	#20826	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#108	C₂	1	1	yes	maximal, characteristic
4	#13179	C₂	1	4	yes	maximal, complemented
5	#12922	C₂	1	4	yes	maximal
6	#12676	C₂	1	4	yes	maximal, complemented
7	#2920	—	2	1	no	—
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂×C₄×M₁₆	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2163	C₄	1	2	yes	—
12	#2163	C₄	1	2	yes	—
13	#2163	C₄	1	2	yes	—
14	#2163	C₄	1	2	yes	—
15	C₂×#368	C₂²	1	8	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₂×#346	C₂²	1	2	yes	—
19	#1848	C₂²	1	2	yes	—
20	#1837	—	2	1	no	—
21	#1810	—	2	1	no	—
22	#1810	C₂²	1	2	yes	—
23	C₄×#108	—	2	1	no	—
24	C₄×#108	—	2	1	no	—
25	C₄×#108	—	2	1	no	—
26	C₄×#108	C₂²	1	2	yes	—
27	C₄×#108	C₂²	1	2	yes	—
28	C₄×#108	C₂²	1	2	yes	—
29	C₂×#341	—	2	1	no	—
30	#1381	C₂²	1	8	yes	—
31	C₂²×M₁₆	—	2	2	no	—
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂³×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄²	D₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂³	1	4	yes	—
36	#432	—	4	1	no	—
37	#432	—	4	1	no	—
38	#431	—	4	1	no	—
39	#431	—	4	1	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	8	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	C₂³	1	4	yes	—
44	C₂×#108	C₂³	1	4	yes	—
45	#423	—	2	2	no	—
46	#407	—	2	2	no	—
47	C₄×M₁₆	—	2	2	no	—
48	C₄×M₁₆	—	2	2	no	—
49	C₄×M₁₆	—	2	2	no	—
50	C₄×M₁₆	—	4	1	no	—
51	C₄×M₁₆	—	4	1	no	—
52	C₂×#97	—	4	1	no	—
53	C₂×C₄×C₈	—	4	1	no	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	#368	—	2	16	no	—
67	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
68	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
69	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
70	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
71	#346	C₂×C₄	1	4	yes	—
72	#346	C₂×C₄	1	4	yes	—
73	#341	—	2	2	no	—
74	#341	—	2	2	no	—
75	#341	—	2	2	no	—
76	#341	—	2	2	no	—
77	C₂³×C₄	Q₁₆	1	1	yes	characteristic
78	C₂³×C₄	Q₁₆	1	1	yes	characteristic
79	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	8	no	—
81	C₂×M₁₆	—	2	4	no	—
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	—	4	4	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂⁴	1	1	yes	characteristic
100	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	D₈	1	1	yes	characteristic
105	C₂×C₄²	D₈	1	1	yes	characteristic
106	C₂×C₄²	D₈	1	1	yes	characteristic
107	C₂×C₄²	D₈	1	1	yes	characteristic
108	C₂×C₄²	Q₁₆	1	1	yes	characteristic
109	C₂×C₄²	Q₁₆	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₁₆	—	2	8	no	—
113	C₂×C₁₆	C₂²×C₄	1	8	yes	—
114	#108	—	2	8	no	—
115	#108	—	2	4	no	—
116	#108	—	2	8	no	—
117	#108	—	4	4	no	—
118	#108	#41	1	8	yes	—
119	C₄⋊C₈	—	4	2	no	—
120	C₄⋊C₈	—	4	2	no	—
121	C₂²⋊C₈	—	4	2	no	—
122	C₂²⋊C₈	—	4	2	no	—
123	#97	—	4	2	no	—
124	#97	—	4	2	no	—
125	#97	—	4	2	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
142	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	#107	1	1	yes	characteristic
151	C₂²×C₄	#107	1	1	yes	characteristic
152	C₂²×C₄	#107	1	1	yes	characteristic
153	C₂²×C₄	#107	1	1	yes	characteristic
154	C₂²×C₄	#107	1	1	yes	characteristic
155	C₂²×C₄	#107	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
162	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	M₁₆	—	4	8	no	—
170	M₁₆	—	4	4	no	—
171	M₁₆	—	4	4	no	—
172	M₁₆	—	4	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	8	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₈	C₂×#41	1	4	yes	—
196	C₄²	#107	1	1	yes	characteristic
197	C₄²	#107	1	1	yes	characteristic
198	C₄²	#107	1	1	yes	characteristic
199	C₄²	#107	1	1	yes	characteristic
200	C₄²	#107	1	1	yes	characteristic
201	C₄²	#107	1	1	yes	characteristic
202	C₄²	#107	1	1	yes	characteristic
203	C₄²	#107	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×D₈	1	1	yes	characteristic
209	C₄²	C₂×D₈	1	1	yes	characteristic
210	C₄²	C₂×D₈	1	1	yes	characteristic
211	C₄²	C₂×D₈	1	1	yes	characteristic
212	C₁₆	—	2	8	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	C₂×#107	1	1	yes	characteristic
217	C₂³	C₂×#107	1	1	yes	characteristic
218	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	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	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	C₂×#107	1	1	yes	characteristic
239	C₂×C₄	C₂×#107	1	1	yes	characteristic
240	C₂×C₄	C₂×#107	1	1	yes	characteristic
241	C₂×C₄	C₂×#107	1	1	yes	characteristic
242	C₂×C₄	C₂×#107	1	1	yes	characteristic
243	C₂×C₄	C₂×#107	1	1	yes	characteristic
244	C₂×C₄	C₂×#107	1	1	yes	characteristic
245	C₂×C₄	C₂×#107	1	1	yes	characteristic
246	C₂×C₄	C₂×#107	1	1	yes	characteristic
247	C₂×C₄	C₂×#107	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₄	#525	1	2	yes	—
250	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
251	C₂×C₄	#575	1	2	yes	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	8	no	—
259	C₈	—	8	4	no	—
260	C₈	C₄×#41	1	2	yes	—
261	C₈	#533	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	#2161	1	2	yes	—
271	C₂²	C₂²×#107	1	1	yes	characteristic
272	C₂²	#3104	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	#2161	1	2	yes	—
283	C₄	#2296	1	1	yes	characteristic
284	C₄	#2911	1	1	yes	characteristic
285	C₄	#3100	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	2	2	no	—
288	C₂	—	4	1	no	—
289	C₂	#21358	1	1	yes	characteristic
290	C₂	C₂×#2161	1	1	yes	characteristic
291	C₂	C₂×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×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
4. #13179	C₂	semidirect
6. #12676	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2163	2
#20826	2
C₂×C₄×#108	2
#13179	2
#12922	2
#12676	2