Subgroups — #213381

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#21713	C₂	1	2	yes	maximal
3	#9874	C₂	1	2	yes	maximal
4	#9598	C₂	1	2	yes	maximal
5	#9594	C₂	1	1	yes	maximal, characteristic
6	#9594	C₂	1	1	yes	maximal, characteristic
7	#8806	C₂	1	1	yes	maximal, characteristic
8	#8291	C₂	1	2	yes	maximal
9	#8282	C₂	1	1	yes	maximal, characteristic
10	C₈×#107	C₂	1	1	yes	maximal, characteristic
11	C₈×#97	C₂	1	1	yes	maximal, characteristic
12	C₂×#500	C₄	1	1	yes	characteristic
13	C₂×#446	—	2	2	no	—
14	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
15	#1990	C₄	1	2	yes	—
16	#1985	—	2	2	no	—
17	#1942	C₄	1	1	yes	characteristic
18	#1942	C₄	1	1	yes	characteristic
19	#1942	C₄	1	2	yes	—
20	#1940	—	2	2	no	—
21	#1939	C₂²	1	2	yes	—
22	#1939	C₂²	1	2	yes	—
23	#1845	C₄	1	1	yes	characteristic
24	#1842	C₂²	1	2	yes	—
25	C₄×#107	C₂²	1	1	yes	characteristic
26	C₄×#107	C₂²	1	1	yes	characteristic
27	C₄×#107	C₂²	1	2	yes	—
28	#1771	—	2	2	no	—
29	#1770	C₂²	1	2	yes	—
30	C₈×C₄⋊C₄	C₂²	1	2	yes	—
31	#1744	C₂²	1	1	yes	characteristic
32	C₄×#97	C₂²	1	1	yes	characteristic
33	C₄²×C₈	C₂²	1	1	yes	characteristic
34	C₂×#335	C₂²	1	1	yes	characteristic
35	C₂×#335	C₂²	1	1	yes	characteristic
36	C₂×#335	C₂²	1	2	yes	—
37	C₂×#322	C₂²	1	1	yes	characteristic
38	C₂×#322	C₂²	1	1	yes	characteristic
39	C₂×C₈²	C₂²	1	1	yes	characteristic
40	C₂×C₈²	C₂²	1	1	yes	characteristic
41	#1279	C₂²	1	2	yes	—
42	#1279	C₂²	1	2	yes	—
43	#1279	C₂²	1	2	yes	—
44	#1279	C₂²	1	2	yes	—
45	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×Q₈	—	2	2	no	—
47	#500	—	2	4	no	—
48	#446	—	4	8	no	—
49	C₂×#107	C₂×C₄	1	1	yes	characteristic
50	C₂×#107	C₂×C₄	1	1	yes	characteristic
51	C₂×#107	C₂×C₄	1	1	yes	characteristic
52	C₂×#107	C₂×C₄	1	2	yes	—
53	C₂×#107	C₂×C₄	1	2	yes	—
54	C₂×#107	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₈	—	2	2	no	—
56	C₂×C₄⋊C₈	—	2	2	no	—
57	C₂×C₄⋊C₈	—	2	2	no	—
58	C₂×C₄⋊C₈	—	2	2	no	—
59	C₂×C₄⋊C₈	C₂³	1	2	yes	—
60	C₂×C₄⋊C₈	C₂³	1	2	yes	—
61	C₂×#97	—	2	2	no	—
62	C₂×#97	D₄	1	1	yes	characteristic
63	C₂×#97	D₄	1	1	yes	characteristic
64	C₂×#97	D₄	1	1	yes	characteristic
65	C₂×#97	D₄	1	1	yes	characteristic
66	C₂×C₄×C₈	—	2	2	no	—
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	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	#391	—	2	2	no	—
78	#389	—	2	2	no	—
79	#389	C₂×C₄	1	2	yes	—
80	#389	C₂×C₄	1	2	yes	—
81	#384	C₂×C₄	1	2	yes	—
82	C₄×C₄⋊C₄	—	2	2	no	—
83	C₄×C₄⋊C₄	C₂³	1	2	yes	—
84	C₄×C₄⋊C₄	C₂³	1	2	yes	—
85	C₄³	C₂³	1	1	yes	characteristic
86	#340	C₂×C₄	1	2	yes	—
87	#340	C₂×C₄	1	2	yes	—
88	#340	C₂×C₄	1	2	yes	—
89	#340	C₂×C₄	1	2	yes	—
90	#336	—	2	2	no	—
91	#336	—	2	2	no	—
92	#336	—	2	2	no	—
93	#336	—	2	2	no	—
94	#336	—	2	1	no	—
95	#336	—	2	1	no	—
96	#335	—	2	2	no	—
97	#335	—	2	2	no	—
98	#335	—	2	4	no	—
99	#322	—	2	2	no	—
100	#322	Q₈	1	4	yes	complemented
101	C₈²	—	2	2	no	—
102	C₈²	Q₈	1	4	yes	complemented
103	C₂²×Q₈	—	2	2	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	2	no	—
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	4	no	—
119	C₄×Q₈	—	4	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	C₂×C₄²	#41	1	1	yes	characteristic
137	C₂×C₄²	C₂⁴	1	1	yes	characteristic
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	—	2	4	no	—
142	#107	—	4	2	no	—
143	#107	M₁₆	1	4	yes	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	4	4	no	—
149	C₄⋊C₈	—	4	4	no	—
150	C₄⋊C₈	—	4	4	no	—
151	C₄⋊C₈	—	4	4	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	#97	—	4	4	no	—
155	#97	D₈	1	4	yes	—
156	#97	Q₁₆	1	4	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	4	1	no	—
165	C₄×C₈	—	4	4	no	—
166	C₄×C₈	C₂×Q₈	1	2	yes	—
167	C₄×C₈	C₂×Q₈	1	2	yes	—
168	C₄×C₈	C₂×Q₈	1	2	yes	—
169	C₄×C₈	C₂×Q₈	1	2	yes	—
170	C₄×C₈	C₂×Q₈	1	2	yes	—
171	C₄×C₈	C₂×Q₈	1	2	yes	—
172	C₄×C₈	#41	1	2	yes	—
173	C₄×C₈	#41	1	2	yes	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	C₂×Q₈	—	4	4	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	8	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×#41	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	C₄×Q₈	1	2	yes	—
217	C₂×C₈	C₄×Q₈	1	2	yes	—
218	C₂×C₈	C₄×Q₈	1	2	yes	—
219	C₂×C₈	C₄×Q₈	1	2	yes	—
220	C₂×C₈	C₄⋊Q₈	1	2	yes	—
221	C₂×C₈	C₄⋊Q₈	1	2	yes	—
222	C₂×C₈	C₂×M₁₆	1	2	yes	—
223	C₂×C₈	#131	1	2	yes	—
224	C₂×C₈	C₂×D₈	1	2	yes	—
225	C₂×C₈	C₂×D₈	1	2	yes	—
226	C₂×C₈	C₂×Q₁₆	1	2	yes	—
227	C₂×C₈	C₂×Q₁₆	1	2	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₄⋊C₄	C₂×M₁₆	1	4	yes	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₂×D₈	1	2	yes	—
252	C₄²	C₂×Q₁₆	1	2	yes	—
253	C₄²	#135	1	2	yes	—
254	C₄²	#135	1	2	yes	—
255	C₄²	C₂²×Q₈	1	1	yes	characteristic
256	C₄²	C₂²×Q₈	1	1	yes	characteristic
257	C₄²	C₂×#41	1	1	yes	characteristic
258	C₄²	C₂×#41	1	1	yes	characteristic
259	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
260	Q₈	—	8	8	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₄×D₈	1	2	yes	—
284	C₂×C₄	C₄×D₈	1	2	yes	—
285	C₂×C₄	C₄×Q₁₆	1	2	yes	—
286	C₂×C₄	C₄×Q₁₆	1	2	yes	—
287	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
290	C₂×C₄	#532	1	1	yes	characteristic
291	C₂×C₄	#533	1	1	yes	characteristic
292	C₂×C₄	#533	1	1	yes	characteristic
293	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
294	C₂×C₄	C₂×#131	1	1	yes	characteristic
295	C₂×C₄	#568	1	1	yes	characteristic
296	C₂×C₄	#568	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
298	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
299	C₂×C₄	C₂×#135	1	1	yes	characteristic
300	C₂×C₄	C₂×#135	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	8	2	no	—
310	C₈	—	8	2	no	—
311	C₈	#446	1	4	yes	complemented
312	C₈	#500	1	4	yes	complemented
313	C₂²	C₄×C₄⋊Q₈	1	1	yes	characteristic
314	C₂²	D₄×M₁₆	1	1	yes	characteristic
315	C₂²	#2935	1	1	yes	characteristic
316	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
317	C₂²	C₂×C₄×Q₁₆	1	1	yes	characteristic
318	C₂²	#2957	1	1	yes	characteristic
319	C₂²	#2957	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₂×#446	1	1	yes	characteristic
331	C₄	#2961	1	1	yes	characteristic
332	C₄	#2962	1	1	yes	characteristic
333	C₄	Q₈×M₁₆	1	1	yes	characteristic
334	C₄	C₂×#500	1	1	yes	characteristic
335	C₄	#3160	1	1	yes	characteristic
336	C₄	#3169	1	1	yes	characteristic
337	C₄	#3170	1	1	yes	characteristic
338	C₂	M₁₆×D₈	1	1	yes	characteristic
339	C₂	M₁₆×Q₁₆	1	1	yes	characteristic
340	C₂	#18066	1	1	yes	characteristic
341	C₂	#18068	1	1	yes	characteristic
342	C₂	#21713	1	1	yes	characteristic
343	C₂	C₄×#500	1	1	yes	characteristic
344	C₂	#21733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
100. #322	Q₈	semidirect
102. C₈²	Q₈	semidirect
311. C₈	#446	semidirect
312. C₈	#500	semidirect

Maximal subgroups

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

subgroup	index
C₄×#500	2
#21713	2
#9874	2
#9598	2
#9594	2
#8806	2
#8291	2
#8282	2
C₈×#107	2
C₈×#97	2