Subgroups — #127553

Order 512.

543 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	#20816	C₂	1	1	yes	maximal, characteristic
3	C₄×#427	C₂	1	1	yes	maximal, characteristic
4	#7262	C₂	1	4	yes	maximal
5	C₂×#427	—	2	1	no	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#2285	—	2	1	no	—
8	C₄×#117	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2163	C₄	1	2	yes	complemented
11	#2162	—	2	1	no	—
12	#2162	C₄	1	2	yes	complemented
13	#2150	—	2	1	no	—
14	#2112	C₂²	1	2	yes	—
15	C₄×M₃₂	—	2	1	no	—
16	#1844	C₄	1	2	yes	complemented
17	#1834	—	2	1	no	—
18	#1807	—	2	1	no	—
19	#1807	C₂²	1	2	yes	—
20	C₄×#107	—	2	1	no	—
21	C₄×#107	C₄	1	2	yes	complemented
22	C₄×#106	—	2	1	no	—
23	C₄×#106	—	2	1	no	—
24	C₄×#106	C₄	1	2	yes	complemented
25	C₄×#106	C₄	1	2	yes	complemented
26	#1735	C₂²	1	2	yes	—
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	2	2	no	—
30	C₂×#117	—	4	1	no	—
31	C₂²×C₄²	D₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	2	no	—
33	#427	—	2	8	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#106	—	2	2	no	—
36	C₂×#106	—	2	2	no	—
37	C₂×#106	C₂×C₄	1	4	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
49	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
50	#384	—	2	2	no	—
51	#382	—	4	1	no	—
52	#380	—	2	2	no	—
53	#380	—	4	1	no	—
54	C₄×C₄⋊C₄	—	2	2	no	—
55	C₄×C₄⋊C₄	—	2	2	no	—
56	C₄×C₄⋊C₄	—	2	2	no	—
57	C₄×C₄⋊C₄	—	4	1	no	—
58	C₄×C₂²⋊C₄	—	4	1	no	—
59	C₄×C₂²⋊C₄	—	4	1	no	—
60	#376	—	4	1	no	—
61	C₄³	—	4	1	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₂²⋊C₁₆	—	2	2	no	—
65	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
66	#346	—	2	2	no	—
67	C₄×C₁₆	—	2	2	no	—
68	#340	—	2	2	no	—
69	#340	—	2	2	no	—
70	#340	—	2	2	no	—
71	#340	C₂×C₄	1	4	yes	—
72	C₂³×C₄	D₈	1	1	yes	characteristic
73	C₂³×C₄	D₈	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	—	2	4	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
89	#117	—	2	8	no	—
90	#117	—	4	4	no	—
91	#117	—	8	4	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	4	2	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₂²⋊C₄	—	4	2	no	—
97	C₂×C₂²⋊C₄	—	4	2	no	—
98	C₂×C₄²	—	4	2	no	—
99	C₂×C₄²	—	4	2	no	—
100	C₂×C₄²	—	4	2	no	—
101	C₂×C₄²	—	4	2	no	—
102	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	SD₁₆	1	1	yes	characteristic
109	C₂×C₄²	SD₁₆	1	1	yes	characteristic
110	C₂×C₄²	SD₁₆	1	1	yes	characteristic
111	C₂×C₄²	SD₁₆	1	1	yes	characteristic
112	C₂×C₄²	Q₁₆	1	1	yes	characteristic
113	C₂×C₄²	Q₁₆	1	1	yes	characteristic
114	M₃₂	—	4	4	no	—
115	C₂×C₁₆	—	2	4	no	—
116	#107	—	2	8	no	—
117	#106	—	2	8	no	—
118	#106	—	4	4	no	—
119	#106	—	4	4	no	—
120	C₄×C₈	C₄²	1	1	yes	characteristic
121	C₄×C₈	C₄²	1	1	yes	characteristic
122	C₄×C₈	C₄²	1	1	yes	characteristic
123	C₄×C₈	C₄²	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	#95	—	4	2	no	—
137	#95	—	4	2	no	—
138	#95	—	4	2	no	—
139	#95	—	4	2	no	—
140	C₂⁴	C₂×D₈	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	#102	1	1	yes	characteristic
148	C₂²×C₄	#102	1	1	yes	characteristic
149	C₂²×C₄	#102	1	1	yes	characteristic
150	C₂²×C₄	#102	1	1	yes	characteristic
151	C₂²×C₄	#102	1	1	yes	characteristic
152	C₂²×C₄	#102	1	1	yes	characteristic
153	C₂²×C₄	#102	1	1	yes	characteristic
154	C₂²×C₄	#102	1	1	yes	characteristic
155	C₂²×C₄	#107	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	#107	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	#95	1	1	yes	characteristic
177	C₂×C₈	#95	1	1	yes	characteristic
178	C₂×C₈	#95	1	1	yes	characteristic
179	C₂×C₈	#95	1	1	yes	characteristic
180	C₂×C₈	#104	1	4	yes	—
181	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	8	4	no	—
190	C₄⋊C₄	—	8	4	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₂²⋊C₄	—	8	4	no	—
193	C₄²	—	4	8	no	—
194	C₄²	—	8	4	no	—
195	C₄²	—	8	4	no	—
196	C₄²	—	8	4	no	—
197	C₄²	#95	1	1	yes	characteristic
198	C₄²	#95	1	1	yes	characteristic
199	C₄²	#95	1	1	yes	characteristic
200	C₄²	#95	1	1	yes	characteristic
201	C₄²	#103	1	1	yes	characteristic
202	C₄²	#103	1	1	yes	characteristic
203	C₄²	#103	1	1	yes	characteristic
204	C₄²	#103	1	1	yes	characteristic
205	C₄²	#103	1	1	yes	characteristic
206	C₄²	#103	1	1	yes	characteristic
207	C₄²	#103	1	1	yes	characteristic
208	C₄²	#103	1	1	yes	characteristic
209	C₄²	#106	1	1	yes	characteristic
210	C₄²	#106	1	1	yes	characteristic
211	C₄²	#106	1	1	yes	characteristic
212	C₄²	#106	1	1	yes	characteristic
213	C₁₆	—	4	4	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₂×#102	1	1	yes	characteristic
218	C₂³	C₂×#102	1	1	yes	characteristic
219	C₂³	C₂×#107	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	#340	1	1	yes	characteristic
245	C₂×C₄	#340	1	1	yes	characteristic
246	C₂×C₄	#340	1	1	yes	characteristic
247	C₂×C₄	#340	1	1	yes	characteristic
248	C₂×C₄	#340	1	1	yes	characteristic
249	C₂×C₄	#340	1	1	yes	characteristic
250	C₂×C₄	#340	1	1	yes	characteristic
251	C₂×C₄	#340	1	1	yes	characteristic
252	C₂×C₄	#360	1	4	yes	—
253	C₂×C₄	C₂×#95	1	1	yes	characteristic
254	C₂×C₄	#410	1	2	yes	—
255	C₂×C₄	C₂×#103	1	1	yes	characteristic
256	C₂×C₄	C₂×#103	1	1	yes	characteristic
257	C₂×C₄	C₂×#104	1	2	yes	—
258	C₂×C₄	C₂×#106	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	#339	1	2	yes	—
266	C₈	#339	1	2	yes	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	4	no	—
275	C₂²	#1502	1	2	yes	—
276	C₂²	C₂×#340	1	1	yes	characteristic
277	C₂²	C₂×#360	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	8	no	—
286	C₄	—	16	4	no	—
287	C₄	#1384	1	2	yes	—
288	C₄	#1386	1	2	yes	—
289	C₄	#1386	1	2	yes	—
290	C₄	#1503	1	2	yes	—
291	C₄	C₂×#339	1	1	yes	characteristic
292	C₄	#1740	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	—	4	1	no	—
296	C₂	#8014	1	1	yes	characteristic
297	C₂	C₂×#1384	1	1	yes	characteristic
298	C₂	C₂×#1386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. #2163	C₄	semidirect
12. #2162	C₄	semidirect
16. #1844	C₄	semidirect
21. C₄×#107	C₄	semidirect
24. C₄×#106	C₄	semidirect
25. C₄×#106	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#20816	2
C₄×#427	2
#7262	2