Subgroups — #497826

Order 512.

646 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	C₂×#2171	C₂	1	2	yes	maximal
2	#33398	C₂	1	2	yes	maximal, complemented
3	C₄×#443	C₂	1	1	yes	maximal, characteristic
4	#13162	C₂	1	2	yes	maximal, complemented
5	#12917	C₂	1	4	yes	maximal, complemented
6	#12900	C₂	1	1	yes	maximal, characteristic, complemented
7	#12787	C₂	1	2	yes	maximal, complemented
8	#12653	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504	C₂²	1	4	yes	—
10	C₂×#443	C₄	1	1	yes	characteristic
11	C₂×#443	C₂²	1	1	yes	characteristic
12	C₄×#135	C₄	1	1	yes	characteristic
13	C₄×#131	C₂²	1	2	yes	—
14	#2171	—	2	8	no	—
15	C₂×#364	C₂²	1	2	yes	—
16	C₂×#364	C₂²	1	2	yes	—
17	#2151	—	2	4	no	—
18	#2151	C₂²	1	4	yes	—
19	#2150	—	2	2	no	—
20	#2150	C₂²	1	2	yes	—
21	C₂×C₄⋊C₁₆	—	2	2	no	—
22	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂×#350	C₂²	1	4	yes	—
25	C₂×#346	C₂²	1	2	yes	—
26	C₄×#108	C₄	1	1	yes	characteristic
27	#1772	C₄	1	1	yes	characteristic
28	#1772	C₂²	1	1	yes	characteristic
29	#1764	C₄	1	2	yes	—
30	#1764	C₂²	1	2	yes	—
31	C₄×#104	C₄	1	2	yes	—
32	C₄²×C₈	C₂²	1	1	yes	characteristic
33	#1379	C₂²	1	4	yes	—
34	#1374	C₂²	1	2	yes	—
35	C₂×#135	C₂×C₄	1	1	yes	characteristic
36	C₂×#131	C₂×C₄	1	2	yes	—
37	C₂×#131	C₂³	1	2	yes	—
38	C₄×#41	C₂×C₄	1	2	yes	—
39	#504	—	2	8	no	—
40	#504	—	4	4	no	—
41	C₂×M₃₂	—	2	4	no	—
42	C₂×M₃₂	C₂³	1	4	yes	—
43	C₂²×C₁₆	—	2	4	no	—
44	C₂²×C₁₆	C₂³	1	2	yes	—
45	C₂²×C₁₆	C₂³	1	2	yes	—
46	#443	—	2	4	no	—
47	#443	D₄	1	4	yes	—
48	#443	Q₈	1	4	yes	—
49	C₄×Q₁₆	C₈	1	1	yes	characteristic
50	C₄×Q₁₆	C₈	1	1	yes	characteristic
51	C₄×SD₁₆	C₈	1	2	yes	—
52	C₄×SD₁₆	C₈	1	2	yes	—
53	C₄×D₈	C₈	1	1	yes	characteristic
54	C₄×D₈	C₈	1	1	yes	characteristic
55	C₂×#108	C₂×C₄	1	1	yes	characteristic
56	C₂×#108	C₂×C₄	1	1	yes	characteristic
57	#427	C₂×C₄	1	1	yes	characteristic
58	C₂×#104	C₂×C₄	1	2	yes	—
59	C₂×#104	C₂×C₄	1	2	yes	—
60	#416	C₂×C₄	1	2	yes	—
61	#405	—	2	2	no	—
62	#405	—	2	2	no	—
63	#405	C₂×C₄	1	2	yes	—
64	#405	C₂³	1	2	yes	—
65	C₄×M₁₆	—	2	2	no	—
66	C₄×M₁₆	C₂×C₄	1	2	yes	—
67	C₂×C₄×C₈	—	2	2	no	—
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	C₄³	C₂×C₄	1	1	yes	characteristic
75	#364	—	2	4	no	—
76	#364	—	4	2	no	—
77	C₄⋊C₁₆	—	2	4	no	—
78	C₄⋊C₁₆	—	2	4	no	—
79	C₄⋊C₁₆	—	2	2	no	—
80	C₄⋊C₁₆	—	2	2	no	—
81	C₄⋊C₁₆	—	2	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	D₄	1	2	yes	—
84	C₄⋊C₁₆	D₄	1	2	yes	—
85	#350	—	2	8	no	—
86	#346	—	2	4	no	—
87	#341	C₂×C₄	1	2	yes	—
88	#339	C₂×C₄	1	2	yes	—
89	#339	C₂×C₄	1	2	yes	—
90	C₂×#41	C₂²×C₄	1	2	yes	—
91	#135	C₄⋊C₄	1	4	yes	—
92	#135	M₁₆	1	4	yes	—
93	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
94	C₂×SD₁₆	C₂×C₈	1	2	yes	—
95	C₂×D₈	C₂×C₈	1	1	yes	characteristic
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	#131	—	2	4	no	—
99	#131	—	4	2	no	—
100	#131	C₂×D₄	1	4	yes	—
101	#131	C₂×Q₈	1	4	yes	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	C₂²×C₄	1	2	yes	—
105	C₂×M₁₆	C₂²×C₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
112	C₄×Q₈	C₂×C₈	1	2	yes	—
113	C₄×Q₈	C₂×C₈	1	2	yes	—
114	C₄×D₄	—	2	2	no	—
115	C₄×D₄	—	2	2	no	—
116	C₄×D₄	C₂×C₈	1	2	yes	—
117	C₄×D₄	C₂×C₈	1	2	yes	—
118	#117	—	2	2	no	—
119	#117	C₂²×C₄	1	2	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	4	8	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	4	8	no	—
136	C₂×C₁₆	C₂×D₄	1	4	yes	—
137	#108	—	2	2	no	—
138	#108	C₄⋊C₄	1	4	yes	—
139	#107	C₂×C₈	1	1	yes	characteristic
140	#107	C₂×C₈	1	1	yes	characteristic
141	#106	C₂×C₈	1	2	yes	—
142	#104	—	2	4	no	—
143	#104	C₄⋊C₄	1	8	yes	—
144	#103	C₂×C₈	1	2	yes	—
145	#103	C₂×C₈	1	2	yes	—
146	#102	C₂×C₈	1	2	yes	—
147	#102	C₂×C₈	1	2	yes	—
148	#97	—	2	2	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	4	1	no	—
158	C₄×C₈	C₂×C₈	1	1	yes	characteristic
159	C₄×C₈	C₂×C₈	1	1	yes	characteristic
160	C₄×C₈	C₂×C₈	1	1	yes	characteristic
161	C₄×C₈	C₂×C₈	1	1	yes	characteristic
162	C₄×C₈	C₂×D₄	1	2	yes	—
163	C₄×C₈	C₂×D₄	1	1	yes	characteristic
164	C₄×C₈	C₂×D₄	1	1	yes	characteristic
165	C₄×C₈	C₂×Q₈	1	2	yes	—
166	C₄×C₈	#41	1	1	yes	characteristic
167	C₄×C₈	#41	1	1	yes	characteristic
168	#41	—	2	2	no	—
169	#41	—	2	2	no	—
170	#41	C₂×C₄⋊C₄	1	4	yes	—
171	#41	C₂×M₁₆	1	4	yes	—
172	C₂×Q₈	C₂²×C₈	1	2	yes	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	C₂²×C₈	1	2	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	#131	1	1	yes	characteristic
181	C₂²×C₄	#131	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
183	Q₁₆	C₄⋊C₈	1	4	yes	—
184	SD₁₆	C₄⋊C₈	1	8	yes	—
185	D₈	C₄⋊C₈	1	4	yes	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	2	4	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	4	4	no	—
192	M₁₆	C₂×C₄⋊C₄	1	4	yes	—
193	C₂×C₈	—	2	2	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	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
203	C₂×C₈	C₄×D₄	1	1	yes	characteristic
204	C₂×C₈	C₄×D₄	1	1	yes	characteristic
205	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
209	C₂×C₈	C₂×M₁₆	1	2	yes	—
210	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
213	C₂×C₈	C₂×#41	1	1	yes	characteristic
214	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
215	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	C₂²×C₈	1	2	yes	—
219	C₄⋊C₄	C₂²×C₈	1	2	yes	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₄⋊C₄	1	2	yes	—
232	C₄²	C₄×D₄	1	1	yes	characteristic
233	C₄²	C₄×D₄	1	1	yes	characteristic
234	C₄²	C₂²×C₈	1	1	yes	characteristic
235	C₄²	C₂²×C₈	1	1	yes	characteristic
236	C₁₆	—	2	4	no	—
237	C₁₆	—	4	4	no	—
238	C₁₆	—	4	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₂×#131	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	C₂×C₄⋊C₈	1	4	yes	—
243	D₄	—	2	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	2	no	—
246	D₄	C₂×C₄⋊C₈	1	4	yes	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	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	4	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	C₈×D₄	1	1	yes	characteristic
263	C₂×C₄	C₈×D₄	1	1	yes	characteristic
264	C₂×C₄	C₈×D₄	1	1	yes	characteristic
265	C₂×C₄	C₈×D₄	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂×C₄	#520	1	1	yes	characteristic
269	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
270	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
271	C₂×C₄	#568	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	2	no	—
278	C₈	C₂×C₄⋊C₈	1	2	yes	—
279	C₈	C₂×C₄⋊C₈	1	2	yes	—
280	C₈	D₄²	1	1	yes	characteristic
281	C₈	#547	1	1	yes	characteristic
282	C₈	D₄×Q₈	1	1	yes	characteristic
283	C₈	#554	1	1	yes	characteristic
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#2906	1	1	yes	characteristic
290	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
291	C₂²	#2990	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	#2171	1	2	yes	complemented
301	C₄	D₄×C₄⋊C₄	1	1	yes	characteristic
302	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
303	C₄	D₄×M₁₆	1	1	yes	characteristic
304	C₄	#2988	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	4	4	no	—
308	C₂	D₄×C₄⋊C₈	1	1	yes	characteristic
309	C₂	C₂×#2171	1	1	yes	characteristic
310	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #33398	C₂	semidirect
4. #13162	C₂	semidirect
5. #12917	C₂	semidirect
6. #12900	C₂	semidirect
7. #12787	C₂	semidirect
8. #12653	C₂	semidirect
300. C₄	#2171	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2171	2
#33398	2
C₄×#443	2
#13162	2
#12917	2
#12900	2
#12787	2
#12653	2