Subgroups — #109321

Order 512.

493 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20476	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂×#1568	C₂	1	1	yes	maximal, characteristic
4	#8125	C₂	1	2	yes	maximal
5	#7978	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#432	C₂²	1	2	yes	—
8	C₂×#432	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2283	C₂²	1	1	yes	characteristic
11	#1770	—	2	1	no	—
12	C₈×C₄⋊C₄	—	2	1	no	—
13	#1752	C₄	1	2	yes	—
14	C₈×C₂²⋊C₄	C₄	1	2	yes	—
15	#1746	—	2	1	no	—
16	#1726	—	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	C₄	1	2	yes	complemented
22	#1568	C₄	1	2	yes	complemented
23	#1567	—	2	1	no	—
24	#1566	—	2	1	no	—
25	#1564	C₄	1	2	yes	complemented
26	#1564	C₄	1	2	yes	complemented
27	C₂×#335	—	2	1	no	—
28	C₂×#334	—	2	1	no	—
29	#1308	—	2	4	no	—
30	#1288	—	2	4	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	#432	—	2	1	no	—
35	#432	—	2	1	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	C₂×C₄	1	4	yes	—
42	#432	D₄	1	2	yes	—
43	#432	Q₈	1	2	yes	—
44	#431	—	2	2	no	—
45	#431	C₂×C₄	1	4	yes	—
46	#423	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	1	no	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	#407	—	2	2	no	—
50	C₂×C₂²⋊C₈	—	2	2	no	—
51	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×M₁₆	—	2	2	no	—
53	C₂×#97	—	2	1	no	—
54	C₂×#97	—	2	2	no	—
55	C₂×#97	—	2	2	no	—
56	C₂×C₄×C₈	—	2	2	no	—
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	D₄	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	Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
69	#383	D₄	1	2	yes	—
70	#382	Q₈	1	2	yes	—
71	#381	—	2	1	no	—
72	#380	—	2	1	no	—
73	C₄×C₄⋊C₄	—	2	1	no	—
74	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
75	#376	—	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	#41	1	1	yes	characteristic
88	C₂³×C₄	#41	1	1	yes	characteristic
89	C₂×M₁₆	—	4	2	no	—
90	C₂²×C₈	—	2	2	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	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₂²⋊C₄	—	2	2	no	—
108	C₂×C₄²	—	2	2	no	—
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₄	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×Q₈	1	1	yes	characteristic
117	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄²	#41	1	1	yes	characteristic
120	C₂×C₄²	#41	1	1	yes	characteristic
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₂²⋊C₈	—	2	2	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	2	4	no	—
130	C₂²⋊C₈	—	4	2	no	—
131	C₂²⋊C₈	—	4	2	no	—
132	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
133	#97	—	2	4	no	—
134	#97	—	2	2	no	—
135	#97	—	2	2	no	—
136	#97	—	4	4	no	—
137	#97	—	4	2	no	—
138	#97	—	4	2	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	4	4	no	—
141	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₄⋊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	#95	—	2	2	no	—
158	#95	—	2	2	no	—
159	#95	—	2	2	no	—
160	#95	C₄⋊C₄	1	4	yes	—
161	C₂⁴	#117	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄⋊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	C₄×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
176	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₄	#121	1	1	yes	characteristic
178	C₂²×C₄	#121	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄	#124	1	1	yes	characteristic
182	C₂²×C₄	#124	1	1	yes	characteristic
183	C₂²×C₄	#125	1	1	yes	characteristic
184	C₂²×C₄	#125	1	1	yes	characteristic
185	M₁₆	—	8	4	no	—
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	#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	C₄⋊C₄	—	4	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₄²	—	4	2	no	—
215	C₄²	—	4	2	no	—
216	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₄²	C₄×D₄	1	1	yes	characteristic
221	C₄²	C₄×D₄	1	1	yes	characteristic
222	C₄²	C₄×Q₈	1	1	yes	characteristic
223	C₄²	C₄×Q₈	1	1	yes	characteristic
224	C₄²	#121	1	1	yes	characteristic
225	C₄²	#121	1	1	yes	characteristic
226	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
227	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
228	C₄²	#127	1	1	yes	characteristic
229	C₄²	C₄⋊Q₈	1	1	yes	characteristic
230	C₄²	C₄⋊Q₈	1	1	yes	characteristic
231	C₄²	C₄⋊Q₈	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂³	#380	1	1	yes	characteristic
237	C₂³	#382	1	1	yes	characteristic
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	1	no	—
243	C₂×C₄	—	2	1	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	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂×#95	1	1	yes	characteristic
263	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₄	#380	1	1	yes	characteristic
265	C₂×C₄	#384	1	1	yes	characteristic
266	C₂×C₄	#389	1	1	yes	characteristic
267	C₂×C₄	#389	1	1	yes	characteristic
268	C₂×C₄	#389	1	1	yes	characteristic
269	C₂×C₄	#389	1	1	yes	characteristic
270	C₂×C₄	#390	1	1	yes	characteristic
271	C₂×C₄	#390	1	1	yes	characteristic
272	C₂×C₄	#391	1	1	yes	characteristic
273	C₂×C₄	#391	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	4	4	no	—
281	C₈	—	8	4	no	—
282	C₈	—	8	4	no	—
283	C₈	#344	1	2	yes	—
284	C₈	#344	1	2	yes	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₄⋊#95	1	1	yes	characteristic
294	C₂²	#1814	1	2	yes	—
295	C₂²	#1833	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
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	—	8	4	no	—
308	C₄	C₂×#344	1	1	yes	characteristic
309	C₄	C₂×#344	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	—	4	1	no	—
313	C₂	#10856	1	1	yes	characteristic
314	C₂	#10868	1	1	yes	characteristic
315	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
21. #1568	C₄	semidirect
22. #1568	C₄	semidirect
25. #1564	C₄	semidirect
26. #1564	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20476	2
#17793	2
C₂×#1568	2
#8125	2
#7978	2