Subgroups — #203904

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21177	C₂	1	2	yes	maximal
2	C₂×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#9624	C₂	1	8	yes	maximal, complemented
6	#8079	C₂	1	2	yes	maximal
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂×#430	—	2	2	no	—
9	C₂×#423	C₂²	1	2	yes	—
10	C₂×#423	C₂²	1	2	yes	—
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1950	—	2	4	no	—
14	#1950	C₂²	1	8	yes	—
15	#1738	—	2	2	no	—
16	#1567	—	2	1	no	—
17	#1567	—	2	1	no	—
18	#1567	C₄	1	2	yes	—
19	#1567	C₄	1	4	yes	—
20	#1567	C₄	1	2	yes	—
21	#1567	C₂²	1	4	yes	—
22	C₂×#335	—	2	2	no	—
23	C₂×#335	C₂²	1	4	yes	—
24	C₈×M₁₆	—	2	2	no	—
25	C₈×M₁₆	C₂²	1	4	yes	—
26	C₂×#322	—	2	1	no	—
27	C₂×C₈²	—	2	1	no	—
28	#1301	C₂²	1	8	yes	—
29	C₂²×M₁₆	—	2	2	no	—
30	C₂²×M₁₆	—	2	2	no	—
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂²×M₁₆	D₄	1	1	yes	characteristic
33	C₂³×C₈	Q₈	1	2	yes	—
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	Q₈	1	2	yes	—
38	#431	C₂³	1	2	yes	—
39	#430	—	4	8	no	—
40	C₂×#108	—	2	8	no	—
41	#423	—	2	2	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	—	2	2	no	—
45	#423	—	2	2	no	—
46	#423	C₂×C₄	1	4	yes	—
47	#423	C₂×C₄	1	4	yes	—
48	#423	C₂³	1	4	yes	—
49	C₂×C₄⋊C₈	—	2	2	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	—	2	2	no	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
55	C₄×M₁₆	—	2	1	no	—
56	C₄×M₁₆	—	2	1	no	—
57	C₄×M₁₆	D₄	1	2	yes	—
58	C₄×M₁₆	C₂³	1	2	yes	—
59	C₂×#97	—	2	1	no	—
60	C₂×C₄×C₈	—	2	1	no	—
61	C₂×C₄×C₈	—	2	1	no	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	2	yes	—
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	C₂×C₄×C₈	C₂³	1	2	yes	—
71	#341	—	2	8	no	—
72	#335	—	2	4	no	—
73	#335	—	2	4	no	—
74	#335	C₂×C₄	1	8	yes	—
75	#335	C₂×C₄	1	8	yes	—
76	#322	—	2	4	no	—
77	C₈²	—	2	4	no	—
78	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	4	no	—
86	C₂×M₁₆	—	2	4	no	—
87	C₂×M₁₆	—	2	4	no	—
88	C₂×M₁₆	—	4	4	no	—
89	C₂×M₁₆	—	4	4	no	—
90	C₂×M₁₆	—	4	2	no	—
91	C₂×M₁₆	—	4	4	no	—
92	C₂×M₁₆	—	4	4	no	—
93	C₂×M₁₆	D₈	1	4	yes	—
94	C₂×M₁₆	Q₁₆	1	4	yes	—
95	C₂×M₁₆	C₂×D₄	1	2	yes	—
96	C₂×M₁₆	C₂×D₄	1	2	yes	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	—	4	4	no	—
102	C₂²×C₈	—	4	2	no	—
103	C₂²×C₈	C₄⋊C₄	1	2	yes	—
104	C₂²×C₈	C₄⋊C₄	1	2	yes	—
105	C₂²×C₈	C₄⋊C₄	1	2	yes	—
106	C₂²×C₈	C₄⋊C₄	1	2	yes	—
107	C₂²×C₈	C₂×Q₈	1	2	yes	—
108	C₂²×C₈	C₂×Q₈	1	2	yes	—
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×Q₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄²	#41	1	1	yes	characteristic
117	C₂×C₄²	#41	1	1	yes	characteristic
118	C₂×C₄²	#41	1	1	yes	characteristic
119	C₂×C₄²	#41	1	1	yes	characteristic
120	C₂×C₄²	C₂⁴	1	1	yes	characteristic
121	#108	—	4	16	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	4	2	no	—
130	C₄⋊C₈	—	4	2	no	—
131	C₄⋊C₈	C₂²×C₄	1	8	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
134	#97	—	2	2	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄⋊C₄	1	2	yes	—
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈	C₄⋊C₄	1	2	yes	—
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	2	yes	—
147	C₄×C₈	C₂²×C₄	1	2	yes	—
148	C₄×C₈	C₂×D₄	1	2	yes	—
149	C₄×C₈	C₂×D₄	1	2	yes	—
150	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
156	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₄	#121	1	1	yes	characteristic
160	C₂²×C₄	#121	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂²×C₄	#125	1	1	yes	characteristic
164	C₂²×C₄	#125	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₈	1	2	yes	—
169	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
170	C₂²×C₄	#135	1	2	yes	—
171	C₂²×C₄	#135	1	2	yes	—
172	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	4	no	—
178	M₁₆	—	2	4	no	—
179	M₁₆	—	4	8	no	—
180	M₁₆	—	4	8	no	—
181	M₁₆	—	4	4	no	—
182	M₁₆	—	4	4	no	—
183	M₁₆	—	8	4	no	—
184	M₁₆	C₂×D₈	1	4	yes	—
185	M₁₆	C₂×Q₁₆	1	4	yes	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	8	4	no	—
203	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
204	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
205	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
206	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
207	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄²	C₄×D₄	1	1	yes	characteristic
210	C₄²	C₄×D₄	1	1	yes	characteristic
211	C₄²	C₄×Q₈	1	1	yes	characteristic
212	C₄²	C₄×Q₈	1	1	yes	characteristic
213	C₄²	#121	1	1	yes	characteristic
214	C₄²	#121	1	1	yes	characteristic
215	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
216	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
217	C₄²	#125	1	1	yes	characteristic
218	C₄²	#125	1	1	yes	characteristic
219	C₄²	C₂³×C₄	1	1	yes	characteristic
220	C₄²	C₂²×D₄	1	1	yes	characteristic
221	C₄²	C₂×#41	1	1	yes	characteristic
222	C₄²	C₂×#41	1	1	yes	characteristic
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	#389	1	1	yes	characteristic
226	C₂³	#389	1	1	yes	characteristic
227	C₂³	#499	1	2	yes	—
228	C₂³	#500	1	2	yes	—
229	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	#389	1	1	yes	characteristic
243	C₂×C₄	#389	1	1	yes	characteristic
244	C₂×C₄	#389	1	1	yes	characteristic
245	C₂×C₄	#389	1	1	yes	characteristic
246	C₂×C₄	#389	1	1	yes	characteristic
247	C₂×C₄	#389	1	1	yes	characteristic
248	C₂×C₄	C₄×D₈	1	2	yes	—
249	C₂×C₄	C₄×D₈	1	2	yes	—
250	C₂×C₄	C₄×Q₁₆	1	2	yes	—
251	C₂×C₄	C₄×Q₁₆	1	2	yes	—
252	C₂×C₄	#466	1	2	yes	—
253	C₂×C₄	#466	1	2	yes	—
254	C₂×C₄	#467	1	2	yes	—
255	C₂×C₄	#467	1	2	yes	—
256	C₂×C₄	#499	1	2	yes	—
257	C₂×C₄	#500	1	2	yes	—
258	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
261	C₂×C₄	C₂×#121	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂×C₄	C₂×#125	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
266	C₂×C₄	C₂×#135	1	1	yes	characteristic
267	C₂×C₄	C₂×#135	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	4	no	—
272	C₈	—	2	4	no	—
273	C₈	—	4	4	no	—
274	C₈	—	4	8	no	—
275	C₈	—	8	4	no	—
276	C₈	#430	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#1942	1	2	yes	—
286	C₂²	#1942	1	2	yes	—
287	C₂²	C₂×#389	1	1	yes	characteristic
288	C₂²	C₂×#499	1	1	yes	characteristic
289	C₂²	C₂×#500	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	#1942	1	2	yes	—
297	C₄	#1942	1	2	yes	—
298	C₄	#1953	1	2	yes	—
299	C₄	C₂×#430	1	2	yes	—
300	C₄	C₂×C₄×D₈	1	1	yes	characteristic
301	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
302	C₄	C₂×#466	1	1	yes	characteristic
303	C₄	C₂×#467	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	2	no	—
308	C₂	C₂×#1942	1	1	yes	characteristic
309	C₂	#21177	1	1	yes	characteristic
310	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #9624	C₂	semidirect
276. C₈	#430	semidirect

Maximal subgroups

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

subgroup	index
#21177	2
C₂×#1567	2
C₂×C₈×M₁₆	2
#9624	2
#8079	2