Subgroups — #490646

Order 512.

424 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₆×#41	C₂	1	1	yes	maximal, characteristic
3	#19126	C₂	1	2	yes	maximal
4	#12477	C₂	1	2	yes	maximal, complemented
5	#12469	C₂	1	2	yes	maximal, complemented
6	#12458	C₂	1	1	yes	maximal, characteristic, complemented
7	#12452	C₂	1	1	yes	maximal, characteristic, complemented
8	#12440	C₂	1	1	yes	maximal, characteristic, complemented
9	#12431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
11	#11949	C₂	1	2	yes	maximal, complemented
12	#3170	C₂²	1	2	yes	—
13	#3076	—	2	2	no	—
14	C₈×#41	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2247	C₂²	1	1	yes	characteristic
17	#2241	C₂²	1	2	yes	—
18	#2216	—	2	1	no	—
19	#2216	—	2	1	no	—
20	Q₈×C₁₆	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×Q₃₂	—	2	1	no	—
23	C₄×Q₃₂	—	2	1	no	—
24	D₄×C₁₆	—	2	1	no	—
25	D₄×C₁₆	—	2	1	no	—
26	D₄×C₁₆	C₂²	1	1	yes	characteristic
27	D₄×C₁₆	C₂²	1	1	yes	characteristic
28	#2162	—	2	1	no	—
29	#2162	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₁₆	—	2	1	no	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1684	—	2	2	no	—
33	#1683	—	2	2	no	—
34	#1676	C₂²	1	2	yes	—
35	#1675	C₂²	1	2	yes	—
36	#1670	C₂²	1	2	yes	—
37	#1669	C₂²	1	2	yes	—
38	C₂×#335	C₂²	1	2	yes	—
39	#1542	—	2	2	no	—
40	#1371	C₂²	1	1	yes	characteristic
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	2	yes	—
43	#1365	C₂²	1	2	yes	—
44	#1365	C₂²	1	2	yes	—
45	#1362	C₂²	1	2	yes	—
46	#1362	C₂²	1	2	yes	—
47	#533	—	2	2	no	—
48	C₄×#41	D₄	1	1	yes	characteristic
49	C₂×Q₃₂	—	2	1	no	—
50	C₂×Q₃₂	—	2	1	no	—
51	C₂²×C₁₆	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	D₄	1	1	yes	characteristic
54	C₂²×C₁₆	D₄	1	1	yes	characteristic
55	#500	C₂³	1	2	yes	—
56	#494	C₂³	1	2	yes	—
57	#487	—	2	2	no	—
58	#479	—	4	2	no	—
59	#478	—	4	2	no	—
60	#477	—	2	2	no	—
61	#476	—	2	2	no	—
62	#475	—	2	2	no	—
63	#474	—	2	2	no	—
64	#467	—	2	2	no	—
65	#467	—	2	2	no	—
66	C₈×Q₈	D₄	1	1	yes	characteristic
67	C₈×Q₈	C₂³	1	1	yes	characteristic
68	C₄×Q₁₆	—	2	2	no	—
69	C₄×Q₁₆	—	2	2	no	—
70	C₈×D₄	—	2	1	no	—
71	C₈×D₄	—	2	1	no	—
72	C₈×D₄	D₄	1	1	yes	characteristic
73	C₈×D₄	C₂³	1	1	yes	characteristic
74	#431	—	2	1	no	—
75	#431	D₄	1	1	yes	characteristic
76	#427	—	4	2	no	—
77	C₂×C₄⋊C₈	—	2	2	no	—
78	#416	—	4	2	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	2	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
87	#358	—	2	2	no	—
88	#358	—	2	2	no	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	C₄×C₁₆	—	2	1	no	—
94	C₄×C₁₆	—	2	1	no	—
95	C₄×C₁₆	D₄	1	1	yes	characteristic
96	C₄×C₁₆	D₄	1	1	yes	characteristic
97	C₄×C₁₆	C₂³	1	1	yes	characteristic
98	C₄×C₁₆	C₂³	1	1	yes	characteristic
99	#335	—	2	2	no	—
100	#335	C₂³	1	2	yes	—
101	#335	C₂³	1	2	yes	—
102	#333	—	2	2	no	—
103	#333	—	2	2	no	—
104	#332	—	2	2	no	—
105	#332	—	2	2	no	—
106	C₂×#41	D₈	1	1	yes	characteristic
107	C₂×Q₁₆	—	2	2	no	—
108	C₂×Q₁₆	—	2	2	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	4	2	no	—
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	2	no	—
116	#125	—	4	2	no	—
117	C₂²⋊Q₈	—	4	2	no	—
118	C₂²⋊Q₈	—	4	2	no	—
119	C₄×Q₈	—	4	2	no	—
120	C₄×Q₈	—	4	2	no	—
121	C₄×Q₈	D₈	1	1	yes	characteristic
122	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
123	C₄×D₄	—	2	1	no	—
124	C₄×D₄	—	2	1	no	—
125	C₄×D₄	D₈	1	1	yes	characteristic
126	C₄×D₄	C₂×D₄	1	1	yes	characteristic
127	#117	—	2	1	no	—
128	#117	—	4	2	no	—
129	#117	D₈	1	1	yes	characteristic
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
132	Q₃₂	—	4	2	no	—
133	Q₃₂	—	4	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	4	1	no	—
138	C₂×C₁₆	—	4	2	no	—
139	C₂×C₁₆	—	4	2	no	—
140	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
141	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
142	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
143	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
144	#107	—	2	2	no	—
145	#107	—	2	2	no	—
146	#107	—	4	2	no	—
147	#107	—	4	2	no	—
148	#106	—	4	2	no	—
149	#106	—	4	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
157	#103	—	4	2	no	—
158	#103	—	4	2	no	—
159	#103	—	4	2	no	—
160	#103	—	4	2	no	—
161	#102	—	4	2	no	—
162	#102	—	4	2	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂×D₄	1	1	yes	characteristic
170	C₄×C₈	C₂×D₄	1	1	yes	characteristic
171	C₄×C₈	#41	1	1	yes	characteristic
172	C₄×C₈	C₂⁴	1	1	yes	characteristic
173	#41	—	2	2	no	—
174	#41	Q₃₂	1	4	yes	complemented
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	C₂×D₈	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	#121	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
185	C₂²×C₄	#135	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	Q₁₆	—	4	2	no	—
188	Q₁₆	—	8	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	8	4	no	—
200	C₂×C₈	#121	1	1	yes	characteristic
201	C₂×C₈	#121	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₈	C₂×#41	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	8	2	no	—
211	C₄⋊C₄	—	8	2	no	—
212	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
213	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	8	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	8	2	no	—
222	C₄²	#121	1	1	yes	characteristic
223	C₄²	C₂×D₈	1	1	yes	characteristic
224	C₄²	#135	1	1	yes	characteristic
225	C₄²	C₂²×D₄	1	1	yes	characteristic
226	C₁₆	—	2	1	no	—
227	C₁₆	—	2	1	no	—
228	C₁₆	—	2	1	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	#135	1	2	yes	—
232	C₂³	—	2	1	no	—
233	C₂³	#466	1	1	yes	characteristic
234	Q₈	—	2	1	no	—
235	Q₈	—	8	2	no	—
236	Q₈	—	8	2	no	—
237	Q₈	—	16	2	no	—
238	Q₈	C₂×Q₃₂	1	2	yes	—
239	D₄	—	2	4	no	—
240	D₄	—	2	1	no	—
241	D₄	C₂×Q₃₂	1	2	yes	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#466	1	1	yes	characteristic
255	C₂×C₄	#466	1	1	yes	characteristic
256	C₂×C₄	#466	1	1	yes	characteristic
257	C₂×C₄	C₂×Q₃₂	1	2	yes	—
258	C₂×C₄	#508	1	2	yes	—
259	C₂×C₄	C₂×#121	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
261	C₂×C₄	C₂×#135	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	4	1	no	—
266	C₈	—	4	1	no	—
267	C₈	—	8	2	no	—
268	C₈	—	8	2	no	—
269	C₈	C₂×#135	1	1	yes	characteristic
270	C₈	#575	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#2216	1	2	yes	—
277	C₂²	C₂×#466	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	16	2	no	—
285	C₄	—	16	2	no	—
286	C₄	#2216	1	2	yes	—
287	C₄	#3035	1	1	yes	characteristic
288	C₄	#3067	1	1	yes	characteristic
289	C₄	C₂²×Q₃₂	1	1	yes	characteristic
290	C₄	C₂×#508	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	4	2	no	—
293	C₂	#19125	1	1	yes	characteristic
294	C₂	C₂×#2216	1	1	yes	characteristic
295	C₂	#33617	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41 ◁ 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. #33734	C₂	semidirect
4. #12477	C₂	semidirect
5. #12469	C₂	semidirect
6. #12458	C₂	semidirect
7. #12452	C₂	semidirect
8. #12440	C₂	semidirect
9. #12431	C₂	semidirect
10. C₂×#1371	C₂	semidirect
11. #11949	C₂	semidirect
174. #41	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
C₁₆×#41	2
#19126	2
#12477	2
#12469	2
#12458	2
#12452	2
#12440	2
#12431	2
C₂×#1371	2
#11949	2