Subgroups — #490644

Order 512.

478 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33733	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₆×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	#19125	C₂	1	2	yes	maximal, complemented
4	#12475	C₂	1	2	yes	maximal, complemented
5	#12467	C₂	1	2	yes	maximal, complemented
6	#12457	C₂	1	1	yes	maximal, characteristic, complemented
7	#12451	C₂	1	1	yes	maximal, characteristic, complemented
8	#12439	C₂	1	1	yes	maximal, characteristic, complemented
9	#12430	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
11	#11947	C₂	1	2	yes	maximal, complemented
12	#3169	C₂²	1	2	yes	—
13	#3035	—	2	2	no	—
14	C₈×#41	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2246	C₂²	1	1	yes	characteristic
17	#2240	C₂²	1	2	yes	—
18	#2215	—	2	1	no	—
19	#2215	—	2	1	no	—
20	Q₈×C₁₆	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×D₁₆	—	2	1	no	—
23	C₄×D₁₆	—	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	#1682	—	2	2	no	—
33	#1681	—	2	2	no	—
34	#1674	C₂²	1	2	yes	—
35	#1673	C₂²	1	2	yes	—
36	#1668	C₂²	1	2	yes	—
37	#1667	C₂²	1	2	yes	—
38	C₂×#335	C₂²	1	2	yes	—
39	#1540	—	2	2	no	—
40	#1371	C₂²	1	1	yes	characteristic
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	2	yes	—
43	#1364	C₂²	1	2	yes	—
44	#1364	C₂²	1	2	yes	—
45	#1361	C₂²	1	2	yes	—
46	#1361	C₂²	1	2	yes	—
47	C₂×#135	—	4	2	no	—
48	#532	—	2	2	no	—
49	C₄×#41	D₄	1	1	yes	characteristic
50	C₂×D₁₆	—	2	1	no	—
51	C₂×D₁₆	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	—	2	1	no	—
54	C₂²×C₁₆	D₄	1	1	yes	characteristic
55	C₂²×C₁₆	D₄	1	1	yes	characteristic
56	#500	C₂³	1	2	yes	—
57	#493	C₂³	1	2	yes	—
58	#486	—	2	2	no	—
59	#466	—	2	2	no	—
60	#466	—	2	2	no	—
61	#464	—	4	2	no	—
62	#463	—	4	2	no	—
63	#462	—	2	2	no	—
64	#461	—	2	2	no	—
65	#460	—	2	2	no	—
66	#459	—	2	2	no	—
67	C₈×Q₈	D₄	1	1	yes	characteristic
68	C₈×Q₈	C₂³	1	1	yes	characteristic
69	C₄×D₈	—	2	2	no	—
70	C₄×D₈	—	2	2	no	—
71	C₈×D₄	—	2	1	no	—
72	C₈×D₄	—	2	1	no	—
73	C₈×D₄	D₄	1	1	yes	characteristic
74	C₈×D₄	C₂³	1	1	yes	characteristic
75	#431	—	2	1	no	—
76	#431	D₄	1	1	yes	characteristic
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	#357	—	2	2	no	—
88	#357	—	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	#332	—	2	2	no	—
103	#332	—	2	2	no	—
104	#331	—	2	2	no	—
105	#331	—	2	2	no	—
106	C₂×#41	—	4	2	no	—
107	C₂×#41	D₈	1	1	yes	characteristic
108	#135	—	8	8	no	—
109	C₂×Q₁₆	—	4	2	no	—
110	C₂×SD₁₆	—	4	2	no	—
111	C₂×SD₁₆	—	4	2	no	—
112	C₂×D₈	—	2	2	no	—
113	C₂×D₈	—	2	2	no	—
114	C₂×D₈	—	4	2	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	4	2	no	—
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#127	—	2	2	no	—
122	#124	—	4	2	no	—
123	#121	—	4	2	no	—
124	#121	—	4	2	no	—
125	C₄×Q₈	D₈	1	1	yes	characteristic
126	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
127	C₄×D₄	—	2	1	no	—
128	C₄×D₄	—	2	1	no	—
129	C₄×D₄	—	4	2	no	—
130	C₄×D₄	—	4	2	no	—
131	C₄×D₄	D₈	1	1	yes	characteristic
132	C₄×D₄	C₂×D₄	1	1	yes	characteristic
133	#117	—	2	1	no	—
134	#117	D₈	1	1	yes	characteristic
135	C₂×C₄²	—	2	1	no	—
136	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
137	D₁₆	—	4	2	no	—
138	D₁₆	—	4	2	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	4	1	no	—
143	C₂×C₁₆	—	4	2	no	—
144	C₂×C₁₆	—	4	2	no	—
145	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
146	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
147	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
148	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	4	2	no	—
156	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
157	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
158	#103	—	4	2	no	—
159	#103	—	4	2	no	—
160	#102	—	4	2	no	—
161	#102	—	4	2	no	—
162	#102	—	4	2	no	—
163	#102	—	4	2	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×D₄	1	1	yes	characteristic
171	C₄×C₈	C₂×D₄	1	1	yes	characteristic
172	C₄×C₈	#41	1	1	yes	characteristic
173	C₄×C₈	C₂⁴	1	1	yes	characteristic
174	#41	—	2	2	no	—
175	#41	—	8	4	no	—
176	#41	—	8	4	no	—
177	#41	D₁₆	1	4	yes	complemented
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	8	2	no	—
185	C₂×D₄	C₂×D₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	8	2	no	—
190	C₂²×C₄	#121	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
192	C₂²×C₄	#135	1	1	yes	characteristic
193	Q₁₆	—	8	4	no	—
194	SD₁₆	—	8	4	no	—
195	SD₁₆	—	8	4	no	—
196	D₈	—	4	2	no	—
197	D₈	—	4	2	no	—
198	D₈	—	8	2	no	—
199	D₈	—	8	4	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	8	4	no	—
211	C₂×C₈	#121	1	1	yes	characteristic
212	C₂×C₈	#121	1	1	yes	characteristic
213	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₈	C₂×#41	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
220	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	8	2	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	#121	1	1	yes	characteristic
230	C₄²	C₂×D₈	1	1	yes	characteristic
231	C₄²	#135	1	1	yes	characteristic
232	C₄²	C₂²×D₄	1	1	yes	characteristic
233	C₁₆	—	2	1	no	—
234	C₁₆	—	2	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	4	1	no	—
238	C₁₆	#135	1	2	yes	complemented
239	C₂³	—	2	1	no	—
240	C₂³	—	8	2	no	—
241	C₂³	#466	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	8	2	no	—
244	Q₈	—	8	2	no	—
245	Q₈	C₂×D₁₆	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	8	2	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	16	2	no	—
253	D₄	—	16	4	no	—
254	D₄	C₂×D₁₆	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	16	4	no	—
267	C₂×C₄	#466	1	1	yes	characteristic
268	C₂×C₄	#466	1	1	yes	characteristic
269	C₂×C₄	#466	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₆	1	2	yes	—
271	C₂×C₄	#508	1	2	yes	—
272	C₂×C₄	C₂×#121	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
274	C₂×C₄	C₂×#135	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	8	2	no	—
281	C₈	—	8	2	no	—
282	C₈	C₂×#135	1	1	yes	characteristic
283	C₈	#575	1	1	yes	characteristic
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	16	2	no	—
290	C₂²	—	16	2	no	—
291	C₂²	—	16	2	no	—
292	C₂²	#2215	1	2	yes	—
293	C₂²	C₂×#466	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	16	2	no	—
301	C₄	#2215	1	2	yes	—
302	C₄	#3035	1	1	yes	characteristic
303	C₄	#3067	1	1	yes	characteristic
304	C₄	C₂²×D₁₆	1	1	yes	characteristic
305	C₄	C₂×#508	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	—	32	2	no	—
309	C₂	#19125	1	1	yes	characteristic
310	C₂	C₂×#2215	1	1	yes	characteristic
311	C₂	#33617	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#466
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. #33733	C₂	semidirect
2. C₁₆×#41	C₂	semidirect
3. #19125	C₂	semidirect
4. #12475	C₂	semidirect
5. #12467	C₂	semidirect
6. #12457	C₂	semidirect
7. #12451	C₂	semidirect
8. #12439	C₂	semidirect
9. #12430	C₂	semidirect
10. C₂×#1371	C₂	semidirect
11. #11947	C₂	semidirect
177. #41	D₁₆	semidirect
238. C₁₆	#135	semidirect

Maximal subgroups

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

subgroup	index
#33733	2
C₁₆×#41	2
#19125	2
#12475	2
#12467	2
#12457	2
#12451	2
#12439	2
#12430	2
C₂×#1371	2
#11947	2