Subgroups — #483622

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#33434	C₂	1	1	yes	maximal, characteristic, complemented
3	#17912	C₂	1	1	yes	maximal, characteristic
4	#12063	C₂	1	2	yes	maximal, complemented
5	#12056	C₂	1	2	yes	maximal, complemented
6	C₁₆×SD₁₆	C₂	1	2	yes	maximal, complemented
7	C₁₆×D₈	C₂	1	2	yes	maximal, complemented
8	#12018	C₂	1	1	yes	maximal, characteristic, complemented
9	#11630	C₂	1	1	yes	maximal, characteristic, complemented
10	#11623	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#504	—	2	1	no	—
13	#3106	C₄	1	1	yes	characteristic
14	C₈×#41	C₂²	1	1	yes	characteristic
15	#2931	C₂²	1	1	yes	characteristic
16	Q₈×C₁₆	C₂²	1	1	yes	characteristic
17	Q₈×C₁₆	C₂²	1	1	yes	characteristic
18	#2169	—	2	1	no	—
19	#2169	C₂²	1	2	yes	—
20	D₄×C₁₆	—	2	1	no	—
21	D₄×C₁₆	—	2	1	no	—
22	D₄×C₁₆	—	2	1	no	—
23	D₄×C₁₆	C₂²	1	1	yes	characteristic
24	D₄×C₁₆	C₂²	1	2	yes	—
25	D₄×C₁₆	C₂²	1	1	yes	characteristic
26	#2162	—	2	1	no	—
27	#2162	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	C₂×C₄×C₁₆	—	2	1	no	—
32	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#1588	C₄	1	2	yes	—
34	#1586	C₄	1	2	yes	—
35	C₈×SD₁₆	C₂²	1	2	yes	—
36	C₈×D₈	C₂²	1	2	yes	—
37	#1574	C₄	1	1	yes	characteristic
38	#1489	C₄	1	1	yes	characteristic
39	#1484	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	C₈×C₁₆	C₂²	1	2	yes	—
49	C₂×#131	—	2	1	no	—
50	#532	C₂×C₄	1	1	yes	characteristic
51	C₄×#41	C₂×C₄	1	1	yes	characteristic
52	#504	—	4	4	no	—
53	C₂×M₃₂	—	2	1	no	—
54	C₂×M₃₂	—	2	2	no	—
55	C₂²×C₁₆	—	2	1	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	#482	C₈	1	2	yes	—
61	#477	C₈	1	1	yes	characteristic
62	#476	C₈	1	1	yes	characteristic
63	#460	C₈	1	1	yes	characteristic
64	#459	C₈	1	1	yes	characteristic
65	#449	C₈	1	2	yes	—
66	C₈×Q₈	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	C₈×D₄	—	2	1	no	—
73	C₈×D₄	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	C₂×C₄	1	1	yes	characteristic
76	C₈×D₄	C₂³	1	1	yes	characteristic
77	C₈×D₄	C₂³	1	2	yes	—
78	#431	—	2	1	no	—
79	#431	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	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×C₈	—	2	1	no	—
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	C₄×C₁₆	—	2	1	no	—
100	C₄×C₁₆	—	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×#41	C₂×C₈	1	1	yes	characteristic
117	C₂×SD₁₆	C₂×C₈	1	2	yes	—
118	C₂×D₈	C₂×C₈	1	2	yes	—
119	#131	—	4	2	no	—
120	#131	—	4	2	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
130	#127	C₂×C₈	1	1	yes	characteristic
131	#124	—	2	1	no	—
132	#121	—	2	1	no	—
133	#121	C₂×C₈	1	2	yes	—
134	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
135	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	C₂×C₈	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₄	1	2	yes	—
142	#117	—	2	1	no	—
143	#117	C₂×C₈	1	1	yes	characteristic
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
146	M₃₂	—	4	4	no	—
147	M₃₂	—	4	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	4	1	no	—
154	C₂×C₁₆	—	4	2	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	C₂×C₈	1	1	yes	characteristic
167	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
168	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
170	C₄⋊C₈	C₂²×C₄	1	2	yes	—
171	#103	C₂×C₈	1	2	yes	—
172	#102	C₂×C₈	1	2	yes	—
173	#102	C₂×C₈	1	2	yes	—
174	#102	C₂×C₈	1	2	yes	—
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	C₂²⋊C₈	C₂×C₈	1	2	yes	—
181	#97	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
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	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	—	4	2	no	—
194	#41	—	4	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	C₂²×C₈	1	2	yes	—
202	C₂×D₄	C₂²×C₈	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
209	C₂²×C₄	#131	1	1	yes	characteristic
210	SD₁₆	—	2	4	no	—
211	D₈	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	4	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₄×D₄	1	1	yes	characteristic
228	C₂×C₈	C₄×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₈	1	2	yes	—
230	C₂×C₈	C₂²×C₈	1	2	yes	—
231	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₈	C₂×#41	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
237	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
238	C₄⋊C₄	C₂²×C₈	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	C₄×D₄	1	1	yes	characteristic
248	C₄²	C₂²×C₈	1	1	yes	characteristic
249	C₄²	#131	1	1	yes	characteristic
250	C₄²	C₂³×C₄	1	1	yes	characteristic
251	C₁₆	—	2	1	no	—
252	C₁₆	—	2	1	no	—
253	C₁₆	—	2	1	no	—
254	C₁₆	—	4	2	no	—
255	C₁₆	—	4	1	no	—
256	C₁₆	—	4	2	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₈×D₄	1	1	yes	characteristic
262	Q₈	—	2	1	no	—
263	Q₈	—	4	2	no	—
264	Q₈	#504	1	2	yes	complemented
265	D₄	—	2	4	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	4	no	—
271	D₄	—	4	2	no	—
272	D₄	#504	1	2	yes	complemented
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₈×D₄	1	1	yes	characteristic
289	C₂×C₄	C₈×D₄	1	1	yes	characteristic
290	C₂×C₄	C₈×D₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×#131	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	1	no	—
303	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₈	#575	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	2	no	—
322	C₄	#2170	1	2	yes	—
323	C₄	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₄	#2955	1	1	yes	characteristic
325	C₄	C₂×#504	1	1	yes	characteristic
326	C₄	C₂×#504	1	1	yes	characteristic
327	C₂	—	4	1	no	—
328	C₂	—	4	2	no	—
329	C₂	—	8	2	no	—
330	C₂	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₂	#33432	1	1	yes	characteristic
332	C₂	C₂×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3106	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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. C₁₆×#41	C₂	semidirect
2. #33434	C₂	semidirect
4. #12063	C₂	semidirect
5. #12056	C₂	semidirect
6. C₁₆×SD₁₆	C₂	semidirect
7. C₁₆×D₈	C₂	semidirect
8. #12018	C₂	semidirect
9. #11630	C₂	semidirect
10. #11623	C₂	semidirect
11. C₁₆×M₁₆	C₂	semidirect
264. Q₈	#504	semidirect
272. D₄	#504	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×#41	2
#33434	2
#17912	2
#12063	2
#12056	2
C₁₆×SD₁₆	2
C₁₆×D₈	2
#12018	2
#11630	2
#11623	2
C₁₆×M₁₆	2