Subgroups — #483629

Order 512.

467 conjugacy classes of subgroups.

The subgroup lattice has 330 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	C₂×#2169	C₂	1	1	yes	maximal, characteristic, complemented
3	#17910	C₂	1	1	yes	maximal, characteristic
4	#12063	C₂	1	2	yes	maximal, complemented
5	#12056	C₂	1	2	yes	maximal, complemented
6	#12051	C₂	1	2	yes	maximal, complemented
7	#12039	C₂	1	2	yes	maximal, complemented
8	#12016	C₂	1	1	yes	maximal, characteristic, complemented
9	#11631	C₂	1	1	yes	maximal, characteristic, complemented
10	#11622	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₃₂	—	2	1	no	—
13	#3105	C₄	1	1	yes	characteristic
14	#2966	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₄	C₂²	1	1	yes	characteristic
16	#2183	C₂²	1	1	yes	characteristic, complemented
17	Q₈×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	#2169	—	2	1	no	—
19	#2169	—	2	1	no	—
20	#2169	C₂²	1	2	yes	—
21	#2169	C₂²	1	2	yes	—
22	#2169	C₂²	1	1	yes	characteristic, complemented
23	#2168	—	2	1	no	—
24	D₄×C₁₆	—	2	1	no	—
25	D₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
26	#2164	—	2	1	no	—
27	#2163	C₂²	1	1	yes	characteristic
28	#2162	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₁₆	—	2	1	no	—
31	#2108	—	2	1	no	—
32	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#1586	C₄	1	2	yes	—
34	#1585	C₄	1	2	yes	—
35	C₈×SD₁₆	C₂²	1	2	yes	—
36	C₈×D₈	C₂²	1	2	yes	—
37	#1573	C₄	1	1	yes	characteristic
38	#1490	C₄	1	1	yes	characteristic
39	#1483	C₄	1	1	yes	characteristic
40	C₈×M₁₆	C₂²	1	1	yes	characteristic
41	#1370	C₂²	1	2	yes	complemented
42	#1367	C₂²	1	2	yes	complemented
43	Q₈⋊C₁₆	C₂²	1	2	yes	complemented
44	D₄⋊C₁₆	C₂²	1	2	yes	—
45	D₄⋊C₁₆	C₂²	1	2	yes	—
46	D₄⋊C₁₆	C₂²	1	2	yes	complemented
47	#1312	C₂²	1	2	yes	complemented
48	C₈×C₁₆	C₂²	1	2	yes	complemented
49	C₂³×C₈	—	2	1	no	—
50	#529	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₄	C₂×C₄	1	1	yes	characteristic
52	C₂×M₃₂	—	2	1	no	—
53	C₂×M₃₂	—	2	2	no	—
54	C₂×M₃₂	—	2	2	no	—
55	C₂×M₃₂	—	4	4	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	—	2	1	no	—
58	C₂²×C₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₁₆	D₄	1	1	yes	characteristic
60	#484	C₈	1	1	yes	characteristic
61	#481	C₈	1	1	yes	characteristic
62	#478	C₈	1	2	yes	—
63	#463	C₈	1	2	yes	—
64	#450	C₈	1	1	yes	characteristic
65	#447	C₈	1	1	yes	characteristic
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	C₂×C₄	1	1	yes	characteristic
71	#435	—	2	1	no	—
72	C₈×D₄	—	2	1	no	—
73	C₈×D₄	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	C₂×C₄	1	2	yes	—
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	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₈	—	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	C₂²×D₄	C₂×C₈	1	1	yes	characteristic
116	C₂³×C₄	—	2	1	no	—
117	C₂×SD₁₆	C₂×C₈	1	2	yes	—
118	C₂×D₈	C₂×C₈	1	2	yes	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	2	1	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	4	2	no	—
126	C₂²×C₈	—	4	2	no	—
127	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	#126	—	2	1	no	—
130	#125	C₂×C₈	1	1	yes	characteristic
131	#124	C₂×C₈	1	1	yes	characteristic
132	#123	—	2	1	no	—
133	C₂²⋊Q₈	C₂×C₈	1	1	yes	characteristic
134	#121	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	2	yes	—
140	C₄×D₄	C₂²×C₄	1	2	yes	—
141	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
142	#117	—	2	1	no	—
143	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	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	4	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	2	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	2	yes	—
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	2	no	—
176	C₂²⋊C₈	—	2	1	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	1	no	—
179	C₂²⋊C₈	C₂×C₈	1	1	yes	characteristic
180	C₂²⋊C₈	C₂×C₈	1	1	yes	characteristic
181	#97	—	2	1	no	—
182	#97	—	2	1	no	—
183	#97	—	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	C₂⁴	—	2	1	no	—
192	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	C₂²×C₈	1	2	yes	—
197	C₂×D₄	C₂²×C₈	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
206	C₂²×C₄	#131	1	1	yes	characteristic
207	SD₁₆	—	4	2	no	—
208	D₈	—	4	2	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	2	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	—	4	1	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₄×D₄	1	1	yes	characteristic
225	C₂×C₈	C₄×D₄	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₈	1	2	yes	—
227	C₂×C₈	C₂²×C₈	1	2	yes	—
228	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂×#41	1	1	yes	characteristic
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	—	2	1	no	—
235	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
236	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
237	C₄⋊C₄	C₂²×C₈	1	2	yes	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₄×D₄	1	1	yes	characteristic
247	C₄²	C₂²×C₈	1	1	yes	characteristic
248	C₄²	#131	1	1	yes	characteristic
249	C₄²	C₂³×C₄	1	1	yes	characteristic
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	2	1	no	—
253	C₁₆	—	4	2	no	—
254	C₁₆	—	4	2	no	—
255	C₁₆	—	4	1	no	—
256	C₁₆	—	4	1	no	—
257	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	C₈×D₄	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	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	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	C₈×D₄	1	1	yes	characteristic
285	C₂×C₄	C₈×D₄	1	1	yes	characteristic
286	C₂×C₄	C₈×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
289	C₂×C₄	C₂×#131	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	4	1	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	1	no	—
299	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₈	#575	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	8	4	no	—
310	C₂²	#2170	1	2	yes	—
311	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	2	no	—
320	C₄	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	#2955	1	1	yes	characteristic
322	C₄	#3407	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	—	8	1	no	—
325	C₂	—	8	2	no	—
326	C₂	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂	#33433	1	1	yes	characteristic
328	C₂	C₂×#2170	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	#447	C₈
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. C₂×#2169	C₂	semidirect
4. #12063	C₂	semidirect
5. #12056	C₂	semidirect
6. #12051	C₂	semidirect
7. #12039	C₂	semidirect
8. #12016	C₂	semidirect
9. #11631	C₂	semidirect
10. #11622	C₂	semidirect
11. C₁₆×M₁₆	C₂	semidirect
16. #2183	C₂²	semidirect
17. Q₈×C₁₆	C₂²	semidirect
22. #2169	C₂²	semidirect
25. D₄×C₁₆	C₂²	semidirect
41. #1370	C₂²	semidirect
42. #1367	C₂²	semidirect
43. Q₈⋊C₁₆	C₂²	semidirect
46. D₄⋊C₁₆	C₂²	semidirect
47. #1312	C₂²	semidirect
48. C₈×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33480	2
C₂×#2169	2
#17910	2
#12063	2
#12056	2
#12051	2
#12039	2
#12016	2
#11631	2
#11622	2
C₁₆×M₁₆	2