Subgroups — #213383

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#499	C₂	1	1	yes	maximal, characteristic
2	#21675	C₂	1	2	yes	maximal
3	#9875	C₂	1	2	yes	maximal
4	#9597	C₂	1	2	yes	maximal
5	#9594	C₂	1	1	yes	maximal, characteristic
6	#9594	C₂	1	1	yes	maximal, characteristic
7	#8807	C₂	1	1	yes	maximal, characteristic
8	#8290	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×#499	C₄	1	1	yes	characteristic
13	C₄×#125	C₂²	1	2	yes	—
14	#1991	C₄	1	2	yes	—
15	#1942	C₄	1	1	yes	characteristic
16	#1942	C₄	1	1	yes	characteristic
17	#1941	C₄	1	2	yes	—
18	#1939	C₂²	1	2	yes	—
19	#1939	C₂²	1	2	yes	—
20	#1917	—	2	2	no	—
21	#1917	—	2	2	no	—
22	#1846	C₄	1	1	yes	characteristic
23	#1831	C₂²	1	2	yes	—
24	C₄×#107	C₂²	1	1	yes	characteristic
25	C₄×#107	C₂²	1	1	yes	characteristic
26	C₄×#106	C₂²	1	2	yes	—
27	#1771	—	2	2	no	—
28	#1770	—	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×#334	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×#125	C₂×C₄	1	2	yes	—
46	#499	—	2	4	no	—
47	C₂×#107	C₂×C₄	1	1	yes	characteristic
48	C₂×#107	C₂×C₄	1	1	yes	characteristic
49	C₂×#107	C₂×C₄	1	1	yes	characteristic
50	C₂×#107	C₂×C₄	1	1	yes	characteristic
51	C₂×#106	C₂×C₄	1	2	yes	—
52	C₂×#106	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₈	—	2	2	no	—
54	C₂×C₄⋊C₈	—	2	2	no	—
55	C₂×C₄⋊C₈	C₂³	1	2	yes	—
56	C₂×C₄⋊C₈	C₂³	1	2	yes	—
57	C₂×#97	—	2	2	no	—
58	C₂×#97	D₄	1	1	yes	characteristic
59	C₂×#97	D₄	1	1	yes	characteristic
60	C₂×#97	D₄	1	1	yes	characteristic
61	C₂×#97	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	#389	C₂×C₄	1	2	yes	—
74	#389	C₂×C₄	1	2	yes	—
75	#387	—	2	2	no	—
76	#387	—	2	2	no	—
77	#382	C₂×C₄	1	2	yes	—
78	C₄×C₄⋊C₄	—	2	2	no	—
79	C₄×C₄⋊C₄	—	2	2	no	—
80	C₄×C₄⋊C₄	C₂³	1	2	yes	—
81	C₄×C₄⋊C₄	C₂³	1	2	yes	—
82	C₄³	C₂³	1	1	yes	characteristic
83	#340	C₂×C₄	1	2	yes	—
84	#340	C₂×C₄	1	2	yes	—
85	#340	C₂×C₄	1	2	yes	—
86	#340	C₂×C₄	1	2	yes	—
87	#336	—	2	2	no	—
88	#336	—	2	2	no	—
89	#336	—	2	2	no	—
90	#336	—	2	2	no	—
91	#336	—	2	2	no	—
92	#336	—	2	2	no	—
93	#336	—	2	1	no	—
94	#336	—	2	1	no	—
95	#335	—	2	2	no	—
96	#335	—	2	2	no	—
97	#334	—	2	4	no	—
98	#322	—	2	2	no	—
99	#322	Q₈	1	4	yes	—
100	C₈²	—	2	2	no	—
101	C₈²	Q₈	1	4	yes	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	#125	—	2	4	no	—
115	#125	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₄	1	2	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄²	#41	1	1	yes	characteristic
131	C₂×C₄²	#41	1	1	yes	characteristic
132	C₂×C₄²	C₂⁴	1	1	yes	characteristic
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	M₁₆	1	4	yes	—
137	#106	—	2	4	no	—
138	#106	—	4	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	4	4	no	—
144	C₄⋊C₈	—	4	4	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	#97	—	4	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	4	1	no	—
157	C₄×C₈	—	4	1	no	—
158	C₄×C₈	—	4	4	no	—
159	C₄×C₈	C₂×Q₈	1	2	yes	—
160	C₄×C₈	C₂×Q₈	1	2	yes	—
161	C₄×C₈	C₂×Q₈	1	2	yes	—
162	C₄×C₈	C₂×Q₈	1	2	yes	—
163	C₄×C₈	C₂×Q₈	1	2	yes	—
164	C₄×C₈	C₂×Q₈	1	2	yes	—
165	C₄×C₈	#41	1	2	yes	—
166	C₄×C₈	#41	1	2	yes	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×#41	1	1	yes	characteristic
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
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	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₄×Q₈	1	2	yes	—
209	C₂×C₈	C₄×Q₈	1	2	yes	—
210	C₂×C₈	C₄×Q₈	1	2	yes	—
211	C₂×C₈	C₄×Q₈	1	2	yes	—
212	C₂×C₈	C₄⋊Q₈	1	2	yes	—
213	C₂×C₈	C₄⋊Q₈	1	2	yes	—
214	C₂×C₈	C₂×M₁₆	1	2	yes	—
215	C₂×C₈	#131	1	2	yes	—
216	C₂×C₈	#135	1	2	yes	—
217	C₂×C₈	#135	1	2	yes	—
218	C₂×C₈	#135	1	2	yes	—
219	C₂×C₈	#135	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄⋊C₄	C₂×M₁₆	1	4	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₂²×Q₈	1	1	yes	characteristic
247	C₄²	C₂²×Q₈	1	1	yes	characteristic
248	C₄²	C₂×#41	1	1	yes	characteristic
249	C₄²	C₂×#41	1	1	yes	characteristic
250	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
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	4	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	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	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	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	C₂×C₄×Q₈	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	#532	1	1	yes	characteristic
281	C₂×C₄	#533	1	1	yes	characteristic
282	C₂×C₄	#533	1	1	yes	characteristic
283	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
284	C₂×C₄	C₂×#131	1	1	yes	characteristic
285	C₂×C₄	#568	1	1	yes	characteristic
286	C₂×C₄	#568	1	1	yes	characteristic
287	C₂×C₄	C₂×#135	1	1	yes	characteristic
288	C₂×C₄	C₂×#135	1	1	yes	characteristic
289	C₂×C₄	C₂×#135	1	1	yes	characteristic
290	C₂×C₄	C₂×#135	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	8	2	no	—
300	C₈	—	8	2	no	—
301	C₈	#446	1	4	yes	—
302	C₈	#499	1	4	yes	complemented
303	C₂²	C₄×C₄⋊Q₈	1	1	yes	characteristic
304	C₂²	D₄×M₁₆	1	1	yes	characteristic
305	C₂²	#2935	1	1	yes	characteristic
306	C₂²	C₄×#135	1	1	yes	characteristic
307	C₂²	C₄×#135	1	1	yes	characteristic
308	C₂²	#2957	1	1	yes	characteristic
309	C₂²	#2957	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	4	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₂×#446	1	1	yes	characteristic
321	C₄	#2961	1	1	yes	characteristic
322	C₄	#2962	1	1	yes	characteristic
323	C₄	Q₈×M₁₆	1	1	yes	characteristic
324	C₄	C₂×#499	1	1	yes	characteristic
325	C₄	#3160	1	1	yes	characteristic
326	C₄	#3168	1	1	yes	characteristic
327	C₄	#3168	1	1	yes	characteristic
328	C₂	#18035	1	1	yes	characteristic
329	C₂	#18039	1	1	yes	characteristic
330	C₂	#18069	1	1	yes	characteristic
331	C₂	#18069	1	1	yes	characteristic
332	C₂	#21713	1	1	yes	characteristic
333	C₂	C₄×#499	1	1	yes	characteristic
334	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
302. C₈	#499	semidirect

Maximal subgroups

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

subgroup	index
C₄×#499	2
#21675	2
#9875	2
#9597	2
#9594	2
#8807	2
#8290	2
#8282	2
C₈×#107	2
C₈×#97	2