Subgroups — #127265

Order 512.

444 conjugacy classes of subgroups.

The subgroup lattice has 287 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	C₂×#1835	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#7221	C₂	1	4	yes	maximal, complemented
5	#2920	—	2	1	no	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×#382	—	2	1	no	—
8	#2163	C₄	1	2	yes	complemented
9	#2163	C₂²	1	2	yes	—
10	#2162	—	2	1	no	—
11	#2162	—	2	1	no	—
12	#2150	—	2	1	no	—
13	#2112	—	2	1	no	—
14	C₄×M₃₂	C₄	1	2	yes	complemented
15	#1835	—	2	1	no	—
16	#1835	—	2	1	no	—
17	#1835	C₄	1	2	yes	—
18	#1835	C₂²	1	2	yes	—
19	C₂×#340	—	2	1	no	—
20	C₂×#340	C₄	1	2	yes	—
21	#1568	—	2	1	no	—
22	#1568	C₄	1	2	yes	complemented
23	#1567	—	2	1	no	—
24	#1566	C₂²	1	2	yes	—
25	#1564	—	2	1	no	—
26	#1564	C₄	1	2	yes	complemented
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄⋊C₄	—	2	2	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	#432	—	2	2	no	—
33	#432	—	4	1	no	—
34	#431	—	2	2	no	—
35	#431	—	4	1	no	—
36	#423	—	2	2	no	—
37	#407	—	4	1	no	—
38	C₄×M₁₆	—	4	1	no	—
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	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	#382	—	2	2	no	—
52	#382	—	2	2	no	—
53	#382	—	4	1	no	—
54	#382	—	4	1	no	—
55	C₂×#95	—	4	1	no	—
56	C₂×#95	—	4	1	no	—
57	C₄⋊C₁₆	—	2	2	no	—
58	C₄⋊C₁₆	—	2	2	no	—
59	C₂²⋊C₁₆	—	2	2	no	—
60	C₂²⋊C₁₆	—	2	2	no	—
61	#346	C₂×C₄	1	4	yes	complemented
62	C₄×C₁₆	—	2	2	no	—
63	#340	—	2	2	no	—
64	#340	—	2	2	no	—
65	#340	—	2	2	no	—
66	#340	C₂×C₄	1	4	yes	—
67	#335	—	2	2	no	—
68	#335	—	2	2	no	—
69	#334	—	2	2	no	—
70	#334	C₂×C₄	1	4	yes	complemented
71	C₂³×C₄	—	4	2	no	—
72	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₄	D₈	1	1	yes	characteristic
74	C₂³×C₄	SD₁₆	1	1	yes	characteristic
75	C₂×M₁₆	—	4	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	—	4	2	no	—
78	C₂²×C₈	C₄²	1	1	yes	characteristic
79	C₂²×C₈	C₄²	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	4	4	no	—
92	C₂×C₄⋊C₄	—	4	4	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	D₈	1	1	yes	characteristic
100	C₂×C₄²	SD₁₆	1	1	yes	characteristic
101	C₂×C₄²	SD₁₆	1	1	yes	characteristic
102	C₂×C₄²	SD₁₆	1	1	yes	characteristic
103	C₂×C₄²	Q₁₆	1	1	yes	characteristic
104	C₂×C₄²	Q₁₆	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	M₃₂	—	4	4	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	4	2	no	—
110	C₂²⋊C₈	—	4	2	no	—
111	C₂²⋊C₈	—	4	2	no	—
112	#97	—	4	2	no	—
113	C₄×C₈	—	4	2	no	—
114	C₄×C₈	C₄²	1	1	yes	characteristic
115	C₄×C₈	C₄²	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₄⋊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	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
130	#95	—	4	2	no	—
131	#95	—	4	2	no	—
132	#95	—	4	2	no	—
133	#95	—	4	2	no	—
134	C₂⁴	#102	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	—	8	4	no	—
141	C₂²×C₄	—	8	4	no	—
142	C₂²×C₄	—	8	4	no	—
143	C₂²×C₄	#95	1	1	yes	characteristic
144	C₂²×C₄	#95	1	1	yes	characteristic
145	C₂²×C₄	#102	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
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	#103	1	1	yes	characteristic
151	C₂²×C₄	#103	1	1	yes	characteristic
152	C₂²×C₄	#103	1	1	yes	characteristic
153	C₂²×C₄	#103	1	1	yes	characteristic
154	C₂²×C₄	#106	1	1	yes	characteristic
155	C₂²×C₄	#106	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
160	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	M₁₆	—	8	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	#95	1	1	yes	characteristic
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	C₂×C₄²	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	8	4	no	—
186	C₄²	#95	1	1	yes	characteristic
187	C₄²	#95	1	1	yes	characteristic
188	C₄²	#102	1	1	yes	characteristic
189	C₄²	#102	1	1	yes	characteristic
190	C₄²	#103	1	1	yes	characteristic
191	C₄²	#103	1	1	yes	characteristic
192	C₄²	#103	1	1	yes	characteristic
193	C₄²	#103	1	1	yes	characteristic
194	C₄²	#106	1	1	yes	characteristic
195	C₄²	#106	1	1	yes	characteristic
196	C₄²	#107	1	1	yes	characteristic
197	C₄²	#107	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₄²	C₂×SD₁₆	1	1	yes	characteristic
201	C₄²	C₂×Q₁₆	1	1	yes	characteristic
202	C₁₆	—	4	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	#340	1	1	yes	characteristic
207	C₂³	#340	1	1	yes	characteristic
208	C₂³	C₂×#102	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	16	8	no	—
232	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
233	C₂×C₄	#340	1	1	yes	characteristic
234	C₂×C₄	#340	1	1	yes	characteristic
235	C₂×C₄	#340	1	1	yes	characteristic
236	C₂×C₄	#340	1	1	yes	characteristic
237	C₂×C₄	#340	1	1	yes	characteristic
238	C₂×C₄	#340	1	1	yes	characteristic
239	C₂×C₄	#344	1	2	yes	—
240	C₂×C₄	C₂×#95	1	1	yes	characteristic
241	C₂×C₄	C₂×#102	1	1	yes	characteristic
242	C₂×C₄	C₂×#103	1	1	yes	characteristic
243	C₂×C₄	C₂×#103	1	1	yes	characteristic
244	C₂×C₄	C₂×#106	1	1	yes	characteristic
245	C₂×C₄	C₂×#107	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	8	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	#1286	1	2	yes	—
263	C₂²	#1389	1	2	yes	—
264	C₂²	C₂×#340	1	1	yes	characteristic
265	C₂²	#2145	1	2	yes	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	16	4	no	—
277	C₄	#1286	1	2	yes	—
278	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
279	C₄	C₂×#344	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	2	2	no	—
283	C₂	C₂×#1286	1	1	yes	characteristic
284	C₂	#12567	1	1	yes	characteristic
285	C₂	C₂×#1389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7221	C₂	semidirect
8. #2163	C₄	semidirect
14. C₄×M₃₂	C₄	semidirect
22. #1568	C₄	semidirect
26. #1564	C₄	semidirect
61. #346	C₂×C₄	semidirect
70. #334	C₂×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
C₂×#1835	2
#17793	2
#7221	2