Subgroups — #109380

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17793	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#8084	C₂	1	2	yes	maximal
5	#7884	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	2	yes	—
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#1568	—	2	1	no	—
12	#1568	—	2	1	no	—
13	#1567	C₄	1	2	yes	—
14	#1566	C₄	1	2	yes	—
15	#1564	—	2	1	no	—
16	#1564	—	2	1	no	—
17	#1564	—	2	1	no	—
18	#1564	—	2	1	no	—
19	#1564	C₄	1	2	yes	—
20	#1564	C₄	1	2	yes	—
21	C₂×#335	—	2	1	no	—
22	C₂×#334	—	2	1	no	—
23	C₈×M₁₆	—	2	1	no	—
24	C₈×M₁₆	—	2	1	no	—
25	C₈×M₁₆	C₄	1	2	yes	—
26	C₈×M₁₆	C₄	1	2	yes	—
27	C₂×#322	—	2	1	no	—
28	C₂×C₈²	—	2	1	no	—
29	#1302	—	2	4	no	—
30	#1278	—	2	4	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂³×C₈	Q₈	1	1	yes	characteristic
33	C₂³×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#432	—	2	2	no	—
36	#432	—	2	2	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	2	no	—
40	#431	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	—	2	2	no	—
43	#431	—	2	2	no	—
44	#431	C₂×C₄	1	4	yes	—
45	#431	C₂×C₄	1	2	yes	—
46	#431	D₄	1	2	yes	—
47	#423	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄⋊C₈	Q₈	1	2	yes	—
50	#407	—	2	2	no	—
51	C₂×C₂²⋊C₈	—	2	1	no	—
52	C₂×C₂²⋊C₈	—	2	2	no	—
53	C₄×M₁₆	—	2	1	no	—
54	C₄×M₁₆	—	2	1	no	—
55	C₄×M₁₆	—	2	2	no	—
56	C₄×M₁₆	C₂×C₄	1	2	yes	—
57	C₄×M₁₆	Q₈	1	2	yes	—
58	C₂×#97	D₄	1	2	yes	—
59	C₂×C₄×C₈	—	2	1	no	—
60	C₂×C₄×C₈	—	2	1	no	—
61	C₂×C₄×C₈	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	#335	—	2	2	no	—
75	#335	—	2	2	no	—
76	#335	—	2	2	no	—
77	#335	—	2	2	no	—
78	#334	—	2	2	no	—
79	#334	—	2	2	no	—
80	#334	—	2	2	no	—
81	#334	—	2	2	no	—
82	#322	—	2	2	no	—
83	#322	—	2	2	no	—
84	C₈²	—	2	2	no	—
85	C₈²	—	2	2	no	—
86	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₄	#41	1	1	yes	characteristic
88	C₂³×C₄	#41	1	1	yes	characteristic
89	C₂×M₁₆	—	2	8	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	2	4	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	4	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	—	4	2	no	—
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₄²	#41	1	1	yes	characteristic
121	C₂×C₄²	#41	1	1	yes	characteristic
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
128	C₂²⋊C₈	—	2	2	no	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	2	4	no	—
131	C₂²⋊C₈	—	4	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	—	4	2	no	—
134	#97	—	2	2	no	—
135	#97	—	4	2	no	—
136	#97	C₄⋊C₄	1	4	yes	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	4	2	no	—
144	C₄×C₈	C₄²	1	1	yes	characteristic
145	C₄×C₈	C₄²	1	1	yes	characteristic
146	C₄×C₈	C₄²	1	1	yes	characteristic
147	C₄×C₈	C₄²	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁴	#125	1	1	yes	characteristic
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	#117	1	1	yes	characteristic
169	C₂²×C₄	#117	1	1	yes	characteristic
170	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
171	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
173	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
174	C₂²×C₄	#121	1	1	yes	characteristic
175	C₂²×C₄	#121	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
178	C₂²×C₄	#124	1	1	yes	characteristic
179	C₂²×C₄	#124	1	1	yes	characteristic
180	C₂²×C₄	#125	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
183	M₁₆	—	2	8	no	—
184	M₁₆	—	2	8	no	—
185	M₁₆	—	4	8	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	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	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	8	4	no	—
204	C₂×C₈	#95	1	1	yes	characteristic
205	C₂×C₈	#95	1	1	yes	characteristic
206	C₂×C₈	#95	1	1	yes	characteristic
207	C₂×C₈	#95	1	1	yes	characteristic
208	C₂×C₈	#95	1	1	yes	characteristic
209	C₂×C₈	#95	1	1	yes	characteristic
210	C₂×C₈	#95	1	1	yes	characteristic
211	C₂×C₈	#95	1	1	yes	characteristic
212	C₂×C₈	#108	1	4	yes	—
213	C₄²	C₂×C₄²	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₄²	C₄×D₄	1	1	yes	characteristic
220	C₄²	C₄×D₄	1	1	yes	characteristic
221	C₄²	C₄×Q₈	1	1	yes	characteristic
222	C₄²	C₄×Q₈	1	1	yes	characteristic
223	C₄²	#121	1	1	yes	characteristic
224	C₄²	#121	1	1	yes	characteristic
225	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
226	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
227	C₄²	#127	1	1	yes	characteristic
228	C₄²	C₄⋊Q₈	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	4	1	no	—
232	C₂³	#382	1	1	yes	characteristic
233	C₂³	#389	1	1	yes	characteristic
234	C₂³	#389	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂×#95	1	1	yes	characteristic
254	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#380	1	1	yes	characteristic
257	C₂×C₄	#380	1	1	yes	characteristic
258	C₂×C₄	#384	1	1	yes	characteristic
259	C₂×C₄	#389	1	1	yes	characteristic
260	C₂×C₄	#389	1	1	yes	characteristic
261	C₂×C₄	#390	1	1	yes	characteristic
262	C₂×C₄	#390	1	1	yes	characteristic
263	C₂×C₄	#391	1	1	yes	characteristic
264	C₂×C₄	#391	1	1	yes	characteristic
265	C₂×C₄	C₂×#108	1	2	yes	—
266	C₂×C₄	#430	1	2	yes	—
267	C₂×C₄	#443	1	4	yes	—
268	C₈	—	2	8	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	4	no	—
276	C₈	—	8	4	no	—
277	C₈	—	8	4	no	—
278	C₈	#341	1	2	yes	—
279	C₈	#341	1	2	yes	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₄⋊#95	1	1	yes	characteristic
289	C₂²	#1945	1	2	yes	—
290	C₂²	#1950	1	2	yes	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	C₂×#341	1	1	yes	characteristic
299	C₄	#1738	1	1	yes	characteristic
300	C₄	#1772	1	2	yes	—
301	C₄	#1814	1	2	yes	—
302	C₄	#1950	1	2	yes	—
303	C₄	#1974	1	2	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	4	1	no	—
307	C₂	#10853	1	1	yes	characteristic
308	C₂	#10859	1	1	yes	characteristic
309	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#17793	2
C₂×#1564	2
C₂×C₈×M₁₆	2
#8084	2
#7884	2