Subgroups — #109281

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20473	C₂	1	1	yes	maximal, characteristic
2	C₂×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#1568	C₂	1	1	yes	maximal, characteristic
4	#8089	C₂	1	2	yes	maximal
5	#7930	C₂	1	2	yes	maximal
6	C₂×#432	C₂²	1	2	yes	—
7	C₂×#432	C₂²	1	2	yes	—
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄×#117	C₂²	1	1	yes	characteristic
11	#1770	—	2	1	no	—
12	#1770	—	2	1	no	—
13	#1752	C₄	1	2	yes	—
14	#1752	C₄	1	2	yes	—
15	#1744	—	2	1	no	—
16	C₄×#97	—	2	1	no	—
17	#1568	—	2	1	no	—
18	#1568	—	2	1	no	—
19	#1568	—	2	1	no	—
20	#1568	—	2	1	no	—
21	#1568	—	2	1	no	—
22	#1568	—	2	1	no	—
23	#1568	C₄	1	2	yes	complemented
24	#1568	C₄	1	2	yes	complemented
25	C₂×#335	—	2	1	no	—
26	C₂×#335	C₄	1	2	yes	complemented
27	C₂×#334	—	2	1	no	—
28	C₂×#334	C₄	1	2	yes	complemented
29	#1302	—	2	4	no	—
30	#1282	—	2	4	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×#117	Q₈	1	2	yes	—
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#432	—	2	2	no	—
36	#432	—	2	2	no	—
37	#432	—	2	2	no	—
38	#432	—	2	2	no	—
39	#432	—	2	2	no	—
40	#432	—	2	2	no	—
41	#432	—	2	2	no	—
42	#432	C₂×C₄	1	4	yes	—
43	#431	—	2	1	no	—
44	#431	—	2	1	no	—
45	#431	D₄	1	2	yes	—
46	#431	Q₈	1	2	yes	—
47	C₂×C₄⋊C₈	—	2	1	no	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊C₈	—	2	2	no	—
51	C₂×C₂²⋊C₈	—	2	2	no	—
52	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
53	C₂×#97	—	2	2	no	—
54	C₂×#97	—	2	2	no	—
55	C₂×#97	—	2	2	no	—
56	C₂×#97	—	2	2	no	—
57	C₂×C₄×C₈	—	2	1	no	—
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	D₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
70	C₄×C₄⋊C₄	—	2	1	no	—
71	C₄×C₄⋊C₄	—	2	1	no	—
72	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
73	C₄×C₂²⋊C₄	D₄	1	2	yes	—
74	#376	—	2	1	no	—
75	C₄³	—	2	1	no	—
76	#336	—	2	2	no	—
77	#336	—	2	2	no	—
78	#335	—	2	2	no	—
79	#335	—	2	2	no	—
80	#335	—	2	2	no	—
81	#335	—	2	2	no	—
82	#334	—	2	2	no	—
83	#334	—	2	2	no	—
84	#334	—	2	2	no	—
85	#334	—	2	2	no	—
86	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	—	4	2	no	—
92	C₂²×C₈	—	4	2	no	—
93	C₂²×C₈	—	4	2	no	—
94	C₂²×C₈	C₄²	1	1	yes	characteristic
95	C₂²×C₈	C₄²	1	1	yes	characteristic
96	C₂²×C₈	C₄²	1	1	yes	characteristic
97	C₂²×C₈	C₄²	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊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	#117	—	4	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₄	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	C₂×C₄²	#41	1	1	yes	characteristic
123	C₂×C₄²	#41	1	1	yes	characteristic
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	4	2	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₂²⋊C₈	—	2	4	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	—	4	2	no	—
134	C₂²⋊C₈	—	4	2	no	—
135	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
136	#97	—	2	4	no	—
137	#97	—	4	4	no	—
138	#97	—	4	4	no	—
139	#97	—	4	2	no	—
140	#97	—	4	2	no	—
141	#97	—	4	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
160	#95	—	2	2	no	—
161	#95	—	2	2	no	—
162	C₂⁴	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	#117	1	1	yes	characteristic
173	C₂²×C₄	#117	1	1	yes	characteristic
174	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
178	C₂²×C₄	#121	1	1	yes	characteristic
179	C₂²×C₄	#121	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂²×C₄	#127	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	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	4	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	8	4	no	—
204	C₂×C₈	—	8	4	no	—
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	#95	1	1	yes	characteristic
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	2	8	no	—
215	C₄²	—	4	2	no	—
216	C₄²	—	4	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₄²	C₄×D₄	1	1	yes	characteristic
224	C₄²	C₄×D₄	1	1	yes	characteristic
225	C₄²	C₄×Q₈	1	1	yes	characteristic
226	C₄²	C₄×Q₈	1	1	yes	characteristic
227	C₄²	#121	1	1	yes	characteristic
228	C₄²	#121	1	1	yes	characteristic
229	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
230	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
231	C₄²	#124	1	1	yes	characteristic
232	C₄²	#124	1	1	yes	characteristic
233	C₄²	#125	1	1	yes	characteristic
234	C₄²	#125	1	1	yes	characteristic
235	C₂³	—	4	1	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
238	C₂³	#380	1	1	yes	characteristic
239	C₂³	#384	1	1	yes	characteristic
240	C₂³	#430	1	4	yes	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×#95	1	1	yes	characteristic
266	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	#380	1	1	yes	characteristic
268	C₂×C₄	#382	1	1	yes	characteristic
269	C₂×C₄	#389	1	1	yes	characteristic
270	C₂×C₄	#389	1	1	yes	characteristic
271	C₂×C₄	#389	1	1	yes	characteristic
272	C₂×C₄	#389	1	1	yes	characteristic
273	C₂×C₄	#390	1	1	yes	characteristic
274	C₂×C₄	#390	1	1	yes	characteristic
275	C₂×C₄	#391	1	1	yes	characteristic
276	C₂×C₄	#391	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	4	no	—
284	C₈	—	8	4	no	—
285	C₈	—	8	4	no	—
286	C₈	#344	1	2	yes	—
287	C₈	#344	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₄⋊#95	1	1	yes	characteristic
296	C₂²	#1778	1	2	yes	—
297	C₂²	#1811	1	2	yes	—
298	C₂²	#1849	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	4	no	—
310	C₄	C₂×#344	1	1	yes	characteristic
311	C₄	C₂×#344	1	1	yes	characteristic
312	C₄	#1976	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#10856	1	1	yes	characteristic
317	C₂	#10865	1	1	yes	characteristic
318	C₂	#10871	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
23. #1568	C₄	semidirect
24. #1568	C₄	semidirect
26. C₂×#335	C₄	semidirect
28. C₂×#334	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20473	2
C₂×#1568	2
#8089	2
#7930	2