Subgroups — #483632

Order 512.

450 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33480	C₂	1	1	yes	maximal, characteristic, complemented
2	#33434	C₂	1	1	yes	maximal, characteristic, complemented
3	#17964	C₂	1	1	yes	maximal, characteristic
4	#12063	C₂	1	2	yes	maximal, complemented
5	#12056	C₂	1	2	yes	maximal, complemented
6	#12049	C₂	1	2	yes	maximal, complemented
7	#12038	C₂	1	2	yes	maximal, complemented
8	#12017	C₂	1	1	yes	maximal, characteristic, complemented
9	#11632	C₂	1	1	yes	maximal, characteristic, complemented
10	#11624	C₂	1	1	yes	maximal, characteristic, complemented
11	#11408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#504	—	2	1	no	—
13	#3171	C₄	1	1	yes	characteristic
14	#2966	C₂²	1	1	yes	characteristic
15	#2931	C₂²	1	1	yes	characteristic
16	#2183	C₂²	1	1	yes	characteristic
17	Q₈×C₁₆	C₂²	1	1	yes	characteristic
18	#2169	—	2	1	no	—
19	#2169	C₂²	1	1	yes	characteristic
20	#2169	C₂²	1	2	yes	—
21	#2168	—	2	1	no	—
22	D₄×C₁₆	—	2	1	no	—
23	D₄×C₁₆	—	2	1	no	—
24	D₄×C₁₆	C₂²	1	2	yes	—
25	D₄×C₁₆	C₂²	1	1	yes	characteristic
26	#2164	—	2	1	no	—
27	#2163	C₂²	1	1	yes	characteristic
28	#2162	C₂²	1	1	yes	characteristic
29	#2150	C₂²	1	1	yes	characteristic
30	#2113	—	2	1	no	—
31	#2108	—	2	1	no	—
32	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#1594	C₄	1	2	yes	—
34	#1592	C₄	1	2	yes	—
35	C₈×SD₁₆	C₂²	1	2	yes	—
36	C₈×D₈	C₂²	1	2	yes	—
37	#1569	C₄	1	1	yes	characteristic
38	#1496	C₄	1	1	yes	characteristic
39	#1495	C₄	1	1	yes	characteristic
40	C₈×M₁₆	C₂²	1	1	yes	characteristic
41	#1370	C₂²	1	2	yes	—
42	#1367	C₂²	1	2	yes	—
43	Q₈⋊C₁₆	C₂²	1	2	yes	—
44	D₄⋊C₁₆	C₂²	1	2	yes	—
45	D₄⋊C₁₆	C₂²	1	2	yes	—
46	D₄⋊C₁₆	C₂²	1	2	yes	—
47	#1312	C₂²	1	2	yes	—
48	#1311	C₂²	1	2	yes	—
49	C₂×#131	—	2	1	no	—
50	#532	C₂×C₄	1	1	yes	characteristic
51	#529	C₂×C₄	1	1	yes	characteristic
52	#504	—	4	4	no	—
53	C₂×M₃₂	—	2	1	no	—
54	C₂×M₃₂	—	2	1	no	—
55	C₂×M₃₂	—	2	2	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	—	2	2	no	—
58	C₂²×C₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₁₆	D₄	1	1	yes	characteristic
60	#501	C₈	1	1	yes	characteristic
61	#496	C₈	1	1	yes	characteristic
62	#490	C₈	1	1	yes	characteristic
63	#489	C₈	1	1	yes	characteristic
64	#469	C₈	1	2	yes	—
65	#468	C₈	1	2	yes	—
66	#446	C₂×C₄	1	1	yes	characteristic
67	C₈×Q₈	C₂³	1	1	yes	characteristic
68	C₄×SD₁₆	C₂×C₄	1	2	yes	—
69	C₄×D₈	C₂×C₄	1	2	yes	—
70	#436	—	2	1	no	—
71	#436	C₂×C₄	1	2	yes	—
72	#436	C₂×C₄	1	1	yes	characteristic
73	#435	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	—	2	1	no	—
76	C₈×D₄	C₂³	1	1	yes	characteristic
77	C₈×D₄	C₂³	1	2	yes	—
78	#433	—	2	1	no	—
79	#432	C₂×C₄	1	1	yes	characteristic
80	#431	C₂×C₄	1	1	yes	characteristic
81	#423	C₂×C₄	1	1	yes	characteristic
82	#408	—	2	1	no	—
83	#405	—	2	1	no	—
84	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	—	2	1	no	—
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
92	C₄⋊C₁₆	C₂³	1	2	yes	—
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₂²⋊C₁₆	—	2	1	no	—
95	C₂²⋊C₁₆	—	2	2	no	—
96	C₂²⋊C₁₆	—	2	2	no	—
97	C₂²⋊C₁₆	—	2	1	no	—
98	C₂²⋊C₁₆	—	2	1	no	—
99	#346	—	2	1	no	—
100	#346	—	2	1	no	—
101	C₄×C₁₆	—	2	1	no	—
102	C₄×C₁₆	D₄	1	1	yes	characteristic
103	C₄×C₁₆	D₄	1	1	yes	characteristic
104	C₄×C₁₆	C₂³	1	2	yes	—
105	C₄×C₁₆	C₂³	1	1	yes	characteristic
106	C₄×C₁₆	C₂³	1	1	yes	characteristic
107	#335	C₂×C₄	1	2	yes	—
108	#334	C₂×C₄	1	2	yes	—
109	Q₈⋊C₈	C₂×C₄	1	2	yes	—
110	D₄⋊C₈	C₂×C₄	1	2	yes	—
111	D₄⋊C₈	C₂×C₄	1	2	yes	—
112	D₄⋊C₈	C₂×C₄	1	2	yes	—
113	#322	C₂×C₄	1	2	yes	—
114	C₈²	C₂³	1	2	yes	—
115	C₂×#41	—	2	1	no	—
116	C₂×SD₁₆	C₂×C₈	1	2	yes	—
117	C₂×D₈	C₂×C₈	1	2	yes	—
118	#131	—	4	2	no	—
119	#131	—	4	2	no	—
120	C₂×M₁₆	—	2	1	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	C₂×C₈	1	2	yes	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
129	#127	C₂×C₈	1	1	yes	characteristic
130	#126	—	2	1	no	—
131	#125	C₂×C₈	1	1	yes	characteristic
132	#124	—	2	1	no	—
133	#124	C₂×C₈	1	1	yes	characteristic
134	#123	—	2	1	no	—
135	C₂²⋊Q₈	C₂×C₈	1	1	yes	characteristic
136	#121	—	2	1	no	—
137	#121	C₂×C₈	1	1	yes	characteristic
138	#121	C₂×C₈	1	2	yes	—
139	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₄	1	2	yes	—
144	#117	—	2	1	no	—
145	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
146	M₃₂	—	4	4	no	—
147	M₃₂	—	4	2	no	—
148	M₃₂	—	4	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	4	1	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₂×C₁₆	—	4	4	no	—
157	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
158	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
159	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
160	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
161	#107	C₂×C₈	1	2	yes	—
162	#106	C₂×C₈	1	2	yes	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
168	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₄	1	2	yes	—
170	#103	C₂×C₈	1	2	yes	—
171	#102	C₂×C₈	1	2	yes	—
172	#102	C₂×C₈	1	2	yes	—
173	#102	C₂×C₈	1	2	yes	—
174	C₂²⋊C₈	—	2	1	no	—
175	C₂²⋊C₈	—	2	1	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	1	no	—
180	#97	—	2	1	no	—
181	#97	—	2	1	no	—
182	#97	C₂×C₈	1	1	yes	characteristic
183	#97	C₂×C₈	1	1	yes	characteristic
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	C₂²×C₄	1	2	yes	—
186	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
187	C₄×C₈	C₂²×C₄	1	2	yes	—
188	C₄×C₈	C₂×D₄	1	1	yes	characteristic
189	C₄×C₈	#41	1	1	yes	characteristic
190	C₄×C₈	C₂⁴	1	1	yes	characteristic
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂²×C₈	1	1	yes	characteristic
200	C₂×D₄	C₂²×C₈	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
205	C₂²×C₄	#131	1	1	yes	characteristic
206	SD₁₆	—	4	2	no	—
207	D₈	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₄×D₄	1	1	yes	characteristic
224	C₂×C₈	C₄×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₈	1	2	yes	—
226	C₂×C₈	C₂²×C₈	1	2	yes	—
227	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₈	C₂×#41	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
235	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
236	C₄⋊C₄	C₂²×C₈	1	2	yes	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	C₄×D₄	1	1	yes	characteristic
246	C₄²	C₂²×C₈	1	1	yes	characteristic
247	C₄²	#131	1	1	yes	characteristic
248	C₄²	C₂³×C₄	1	1	yes	characteristic
249	C₁₆	—	2	1	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	4	2	no	—
253	C₁₆	—	4	2	no	—
254	C₁₆	—	4	1	no	—
255	C₁₆	—	4	1	no	—
256	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₈×D₄	1	1	yes	characteristic
260	Q₈	—	2	2	no	—
261	Q₈	—	4	2	no	—
262	D₄	—	2	2	no	—
263	D₄	—	2	2	no	—
264	D₄	—	2	2	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₈×D₄	1	1	yes	characteristic
282	C₂×C₄	C₈×D₄	1	1	yes	characteristic
283	C₂×C₄	C₈×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
285	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×#131	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	1	no	—
296	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₈	#575	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
317	C₄	#2955	1	1	yes	characteristic
318	C₄	#3407	1	2	yes	—
319	C₂	—	4	1	no	—
320	C₂	—	8	2	no	—
321	C₂	—	8	1	no	—
322	C₂	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic
323	C₂	#33435	1	1	yes	characteristic
324	C₂	#33441	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3171	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33480	C₂	semidirect
2. #33434	C₂	semidirect
4. #12063	C₂	semidirect
5. #12056	C₂	semidirect
6. #12049	C₂	semidirect
7. #12038	C₂	semidirect
8. #12017	C₂	semidirect
9. #11632	C₂	semidirect
10. #11624	C₂	semidirect
11. #11408	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33480	2
#33434	2
#17964	2
#12063	2
#12056	2
#12049	2
#12038	2
#12017	2
#11632	2
#11624	2
#11408	2