Subgroups — #491177

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₆×#41	C₂	1	1	yes	maximal, characteristic
3	#19273	C₂	1	2	yes	maximal
4	#12459	C₂	1	2	yes	maximal, complemented
5	#12458	C₂	1	1	yes	maximal, characteristic, complemented
6	#12457	C₂	1	1	yes	maximal, characteristic, complemented
7	#12453	C₂	1	2	yes	maximal, complemented
8	#12452	C₂	1	1	yes	maximal, characteristic, complemented
9	#12451	C₂	1	1	yes	maximal, characteristic, complemented
10	#12024	C₂	1	1	yes	maximal, characteristic, complemented
11	#11973	C₂	1	2	yes	maximal, complemented
12	#3160	C₂²	1	2	yes	—
13	#3116	—	2	2	no	—
14	C₈×#41	C₂²	1	1	yes	characteristic
15	#2254	C₂²	1	2	yes	—
16	#2253	—	2	1	no	—
17	#2253	—	2	1	no	—
18	#2252	C₂²	1	1	yes	characteristic
19	#2252	C₂²	1	1	yes	characteristic
20	Q₈×C₁₆	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	D₄×C₁₆	—	2	1	no	—
23	D₄×C₁₆	—	2	1	no	—
24	D₄×C₁₆	C₂²	1	1	yes	characteristic
25	D₄×C₁₆	C₂²	1	1	yes	characteristic
26	#2162	—	2	1	no	—
27	#2162	C₂²	1	1	yes	characteristic
28	#2158	—	2	1	no	—
29	#2158	—	2	1	no	—
30	C₂×C₄×C₁₆	—	2	1	no	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1676	C₂²	1	2	yes	—
33	#1675	C₂²	1	2	yes	—
34	#1674	C₂²	1	2	yes	—
35	#1673	C₂²	1	2	yes	—
36	#1672	—	2	2	no	—
37	#1671	—	2	2	no	—
38	#1568	C₂²	1	2	yes	—
39	#1554	—	2	2	no	—
40	#1372	C₂²	1	2	yes	—
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	1	yes	characteristic
43	#1365	C₂²	1	2	yes	—
44	#1365	C₂²	1	2	yes	—
45	#1364	C₂²	1	2	yes	—
46	#1364	C₂²	1	2	yes	—
47	#533	—	2	2	no	—
48	C₄×#41	D₄	1	1	yes	characteristic
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	—	2	1	no	—
51	C₂²×C₁₆	D₄	1	1	yes	characteristic
52	C₂²×C₁₆	D₄	1	1	yes	characteristic
53	#500	C₂³	1	2	yes	—
54	#500	C₂³	1	2	yes	—
55	#499	—	2	2	no	—
56	#499	—	2	2	no	—
57	#498	—	2	2	no	—
58	#485	—	4	2	no	—
59	#477	—	2	2	no	—
60	#476	—	2	2	no	—
61	#462	—	2	2	no	—
62	#461	—	2	2	no	—
63	#452	—	4	2	no	—
64	C₈×Q₈	D₄	1	1	yes	characteristic
65	C₈×Q₈	C₂³	1	1	yes	characteristic
66	#441	—	4	2	no	—
67	#440	—	4	2	no	—
68	C₈×D₄	—	2	1	no	—
69	C₈×D₄	—	2	1	no	—
70	C₈×D₄	D₄	1	1	yes	characteristic
71	C₈×D₄	C₂³	1	1	yes	characteristic
72	#432	—	2	2	no	—
73	#431	—	2	1	no	—
74	#431	D₄	1	1	yes	characteristic
75	#427	—	2	2	no	—
76	#427	—	2	2	no	—
77	C₂×C₄×C₈	—	2	1	no	—
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	#367	—	2	2	no	—
80	#367	—	2	2	no	—
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	#366	—	2	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
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	C₂³	1	2	yes	—
100	#335	C₂³	1	2	yes	—
101	#334	—	2	2	no	—
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	—	4	2	no	—
108	C₂×SD₁₆	—	4	2	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	—	2	2	no	—
115	#125	—	4	2	no	—
116	C₂²⋊Q₈	—	4	2	no	—
117	C₂²⋊Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	C₄×Q₈	—	4	2	no	—
120	C₄×Q₈	D₈	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
122	C₄×D₄	—	2	1	no	—
123	C₄×D₄	—	2	1	no	—
124	C₄×D₄	D₈	1	1	yes	characteristic
125	C₄×D₄	C₂×D₄	1	1	yes	characteristic
126	#117	—	2	1	no	—
127	#117	—	4	2	no	—
128	#117	D₈	1	1	yes	characteristic
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	4	2	no	—
136	C₂×C₁₆	—	4	2	no	—
137	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
138	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
139	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
140	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	—	4	2	no	—
146	#106	—	4	2	no	—
147	#106	—	4	2	no	—
148	#106	—	4	2	no	—
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
155	#103	—	4	2	no	—
156	#103	—	4	2	no	—
157	#103	—	4	2	no	—
158	#102	—	4	2	no	—
159	C₂²⋊C₈	—	2	1	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₂²⋊C₈	—	4	2	no	—
163	#97	—	4	2	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	C₂×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂×D₄	1	1	yes	characteristic
169	C₄×C₈	#41	1	1	yes	characteristic
170	C₄×C₈	C₂⁴	1	1	yes	characteristic
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂×D₈	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	#121	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
183	C₂²×C₄	#135	1	1	yes	characteristic
184	Q₁₆	—	8	4	no	—
185	SD₁₆	—	8	4	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	#121	1	1	yes	characteristic
197	C₂×C₈	#121	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₈	C₂×#41	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	8	2	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
209	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	8	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	8	2	no	—
218	C₄²	#121	1	1	yes	characteristic
219	C₄²	C₂×D₈	1	1	yes	characteristic
220	C₄²	#135	1	1	yes	characteristic
221	C₄²	C₂²×D₄	1	1	yes	characteristic
222	C₁₆	—	2	1	no	—
223	C₁₆	—	2	1	no	—
224	C₁₆	—	4	1	no	—
225	C₁₆	—	4	1	no	—
226	C₁₆	#137	1	2	yes	—
227	C₁₆	#137	1	2	yes	—
228	C₂³	—	2	1	no	—
229	C₂³	#466	1	1	yes	characteristic
230	Q₈	—	2	1	no	—
231	Q₈	—	8	2	no	—
232	Q₈	—	8	2	no	—
233	Q₈	—	16	2	no	—
234	Q₈	#508	1	2	yes	—
235	D₄	—	2	4	no	—
236	D₄	—	2	1	no	—
237	D₄	#508	1	2	yes	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	#466	1	1	yes	characteristic
253	C₂×C₄	#466	1	1	yes	characteristic
254	C₂×C₄	#466	1	1	yes	characteristic
255	C₂×C₄	C₂×#121	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
257	C₂×C₄	C₂×#135	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	#473	1	2	yes	—
265	C₈	C₂×#137	1	1	yes	characteristic
266	C₈	C₂×#137	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂×#466	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	16	2	no	—
282	C₄	—	16	2	no	—
283	C₄	#3039	1	1	yes	characteristic
284	C₄	C₂×#473	1	1	yes	characteristic
285	C₄	C₂×#508	1	1	yes	characteristic
286	C₄	C₂×#508	1	1	yes	characteristic
287	C₂	—	4	1	no	—
288	C₂	—	4	2	no	—
289	C₂	#19271	1	1	yes	characteristic
290	C₂	#33600	1	1	yes	characteristic
291	C₂	#33617	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #33726	C₂	semidirect
4. #12459	C₂	semidirect
5. #12458	C₂	semidirect
6. #12457	C₂	semidirect
7. #12453	C₂	semidirect
8. #12452	C₂	semidirect
9. #12451	C₂	semidirect
10. #12024	C₂	semidirect
11. #11973	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33726	2
C₁₆×#41	2
#19273	2
#12459	2
#12458	2
#12457	2
#12453	2
#12452	2
#12451	2
#12024	2
#11973	2