Subgroups — #498077

Order 512.

410 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20518	C₂	1	1	yes	maximal, characteristic
2	#13268	C₂	1	2	yes	maximal
3	#13107	C₂	1	2	yes	maximal
4	#13107	C₂	1	2	yes	maximal
5	#12901	C₂	1	1	yes	maximal, characteristic
6	#12882	C₂	1	1	yes	maximal, characteristic
7	#12881	C₂	1	1	yes	maximal, characteristic
8	#12666	C₂	1	2	yes	maximal
9	#12660	C₂	1	1	yes	maximal, characteristic
10	#12659	C₂	1	1	yes	maximal, characteristic
11	#12656	C₂	1	1	yes	maximal, characteristic
12	C₂×#446	C₄	1	1	yes	characteristic
13	C₂×#446	C₂²	1	1	yes	characteristic
14	#2295	C₄	1	1	yes	characteristic
15	C₂×#364	C₂²	1	1	yes	characteristic
16	C₂×#364	C₂²	1	1	yes	characteristic
17	C₂×#364	C₂²	1	1	yes	characteristic
18	C₂×#364	C₂²	1	2	yes	—
19	C₂×#364	C₂²	1	1	yes	characteristic
20	#2150	C₂²	1	2	yes	—
21	C₂×#347	C₂²	1	2	yes	—
22	C₂×#347	C₂²	1	2	yes	—
23	C₄×M₃₂	C₂²	1	2	yes	—
24	#1771	C₄	1	1	yes	characteristic
25	#1771	C₄	1	1	yes	characteristic
26	#1771	C₂²	1	1	yes	characteristic
27	#1771	C₂²	1	1	yes	characteristic
28	#1770	C₄	1	1	yes	characteristic
29	#1770	C₂²	1	1	yes	characteristic
30	#1768	C₄	1	1	yes	characteristic
31	#1768	C₄	1	1	yes	characteristic
32	#1767	C₄	1	1	yes	characteristic
33	#1726	C₂²	1	1	yes	characteristic
34	#1726	C₂²	1	1	yes	characteristic
35	C₄×#97	C₂²	1	1	yes	characteristic
36	#1377	C₂²	1	2	yes	—
37	#1377	C₂²	1	2	yes	—
38	#1377	C₂²	1	2	yes	—
39	#1377	C₂²	1	2	yes	—
40	#1376	C₂²	1	2	yes	—
41	#1375	C₂²	1	2	yes	—
42	#1375	C₂²	1	2	yes	—
43	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂×M₃₂	C₂³	1	2	yes	—
45	C₂×M₃₂	C₂³	1	2	yes	—
46	C₂×M₃₂	C₂³	1	2	yes	—
47	C₂×M₃₂	C₂³	1	2	yes	—
48	#446	—	2	4	no	—
49	#446	—	2	2	no	—
50	#446	—	2	2	no	—
51	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×#97	C₂×C₄	1	1	yes	characteristic
58	C₂×#97	C₂×C₄	1	1	yes	characteristic
59	C₂×#97	C₂×C₄	1	1	yes	characteristic
60	C₂×#97	C₂×C₄	1	1	yes	characteristic
61	C₂×#97	C₂×C₄	1	1	yes	characteristic
62	C₂×#97	C₂×C₄	1	1	yes	characteristic
63	C₂×#97	C₂³	1	1	yes	characteristic
64	C₂×#97	C₂³	1	1	yes	characteristic
65	C₂×#97	C₂³	1	1	yes	characteristic
66	C₂×#97	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	#391	—	2	1	no	—
74	#389	—	2	1	no	—
75	#387	—	2	1	no	—
76	#387	—	2	1	no	—
77	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	#376	C₂×C₄	1	1	yes	characteristic
81	#376	C₂×C₄	1	1	yes	characteristic
82	C₄³	C₂×C₄	1	1	yes	characteristic
83	#364	—	2	2	no	—
84	#364	—	2	2	no	—
85	#364	—	2	2	no	—
86	#364	—	4	2	no	—
87	#364	Q₈	1	4	yes	complemented
88	C₄⋊C₁₆	—	2	4	no	—
89	#347	—	4	2	no	—
90	#347	—	4	2	no	—
91	#346	—	2	2	no	—
92	C₄×C₁₆	—	2	2	no	—
93	#336	C₂×C₄	1	1	yes	characteristic
94	#336	C₂×C₄	1	1	yes	characteristic
95	#336	C₂×C₄	1	1	yes	characteristic
96	#336	C₂×C₄	1	1	yes	characteristic
97	#336	C₂×C₄	1	1	yes	characteristic
98	#336	C₂×C₄	1	1	yes	characteristic
99	C₂²×Q₈	M₁₆	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	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂⁴	1	1	yes	characteristic
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
114	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
115	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
116	C₂×C₄²	—	2	1	no	—
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	C₂×Q₈	1	4	yes	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	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	#97	—	2	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	2	no	—
146	#97	—	2	1	no	—
147	#97	—	2	1	no	—
148	#97	—	2	1	no	—
149	#97	—	2	1	no	—
150	#97	—	2	1	no	—
151	#97	—	2	1	no	—
152	#97	—	2	1	no	—
153	#97	—	2	1	no	—
154	#97	—	4	1	no	—
155	#97	—	4	1	no	—
156	#97	M₁₆	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	—	4	1	no	—
162	C₄×C₈	C₂×Q₈	1	2	yes	—
163	C₄×C₈	#41	1	2	yes	—
164	C₄×C₈	#41	1	2	yes	—
165	C₄×C₈	#41	1	2	yes	—
166	#95	—	2	1	no	—
167	#95	—	2	1	no	—
168	#95	—	2	1	no	—
169	#95	—	2	1	no	—
170	#95	—	2	1	no	—
171	#95	—	2	1	no	—
172	#95	—	2	1	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	8	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
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	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	#117	1	2	yes	—
203	C₂×C₈	C₄×Q₈	1	2	yes	—
204	C₂×C₈	C₂×M₁₆	1	2	yes	—
205	C₂×C₈	C₂×M₁₆	1	2	yes	—
206	C₂×C₈	#131	1	2	yes	—
207	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
208	C₂×C₈	C₂×#41	1	1	yes	characteristic
209	C₂×C₈	C₂×#41	1	1	yes	characteristic
210	C₂×C₈	C₂×#41	1	1	yes	characteristic
211	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	8	1	no	—
217	C₄⋊C₄	—	8	1	no	—
218	C₄⋊C₄	—	8	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	8	1	no	—
231	C₄²	—	8	1	no	—
232	C₄²	#117	1	2	yes	—
233	C₄²	C₄×Q₈	1	2	yes	—
234	C₄²	C₂×M₁₆	1	2	yes	—
235	C₁₆	—	2	4	no	—
236	C₁₆	—	4	2	no	—
237	C₁₆	—	4	2	no	—
238	C₁₆	—	4	2	no	—
239	C₂³	C₂²×M₁₆	1	1	yes	characteristic
240	Q₈	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	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	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	#432	1	2	yes	—
264	C₂×C₄	#432	1	2	yes	—
265	C₂×C₄	C₂×#117	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
267	C₂×C₄	#520	1	1	yes	characteristic
268	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×#131	1	1	yes	characteristic
270	C₂×C₄	#568	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	#446	1	4	yes	—
278	C₈	#554	1	1	yes	characteristic
279	C₈	#556	1	1	yes	characteristic
280	C₈	#557	1	1	yes	characteristic
281	C₈	#559	1	1	yes	characteristic
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×#432	1	1	yes	characteristic
285	C₂²	Q₈×M₁₆	1	1	yes	characteristic
286	C₂²	#2991	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	8	1	no	—
295	C₄	—	8	1	no	—
296	C₄	—	8	1	no	—
297	C₄	#2355	1	1	yes	characteristic
298	C₄	#2925	1	1	yes	characteristic
299	C₄	C₂×#446	1	1	yes	characteristic
300	C₄	#2989	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	#22825	1	1	yes	characteristic
304	C₂	#33426	1	1	yes	characteristic
305	C₂	#33427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
87. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#20518	2
#13268	2
#13107	2
#12901	2
#12882	2
#12881	2
#12666	2
#12660	2
#12659	2
#12656	2