Subgroups — #126755

Order 512.

399 conjugacy classes of subgroups.

The subgroup lattice has 283 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	#33418	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#7166	C₂	1	4	yes	maximal, complemented
5	#2920	C₄	1	1	yes	characteristic
6	#2920	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2163	—	2	1	no	—
9	#2163	C₄	1	2	yes	—
10	#2163	C₄	1	2	yes	—
11	#2163	C₄	1	2	yes	—
12	#2162	—	2	1	no	—
13	#2162	—	2	1	no	—
14	#2162	—	2	1	no	—
15	#2162	C₂²	1	2	yes	—
16	#2150	—	2	1	no	—
17	#2150	C₄	1	2	yes	—
18	#2112	—	2	1	no	—
19	#2112	—	2	1	no	—
20	C₄×M₃₂	—	2	1	no	—
21	C₄×M₃₂	C₂²	1	2	yes	—
22	#1452	—	2	1	no	—
23	#1451	—	2	1	no	—
24	#1451	C₂²	1	2	yes	—
25	#1450	—	2	1	no	—
26	C₈×M₁₆	—	2	1	no	—
27	C₈×M₁₆	C₄	1	2	yes	—
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂³×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	#432	—	2	1	no	—
34	#432	—	2	1	no	—
35	#432	—	2	1	no	—
36	#432	C₈	1	2	yes	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	C₂×C₄	1	2	yes	—
41	#423	C₈	1	2	yes	—
42	#423	C₈	1	2	yes	—
43	#407	—	2	1	no	—
44	#407	C₈	1	2	yes	—
45	C₄×M₁₆	—	2	1	no	—
46	C₄×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
63	C₂²⋊C₁₆	—	2	2	no	—
64	C₂²⋊C₁₆	—	2	2	no	—
65	C₂²⋊C₁₆	—	2	2	no	—
66	C₂²⋊C₁₆	—	2	2	no	—
67	#346	—	2	2	no	—
68	#346	C₂×C₄	1	4	yes	—
69	C₄×C₁₆	—	2	2	no	—
70	C₄×C₁₆	—	2	2	no	—
71	#322	—	2	2	no	—
72	#322	—	2	2	no	—
73	#322	—	2	2	no	—
74	C₈²	—	2	2	no	—
75	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
77	C₂³×C₄	M₁₆	1	1	yes	characteristic
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	C₂×C₈	1	4	yes	—
80	C₂²×C₈	—	4	2	no	—
81	C₂²×C₈	—	4	2	no	—
82	C₂²×C₈	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄²	C₄²	1	1	yes	characteristic
95	C₂×C₄²	C₄²	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄²	M₁₆	1	1	yes	characteristic
103	C₂×C₄²	M₁₆	1	1	yes	characteristic
104	C₂×C₄²	M₁₆	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
106	M₃₂	—	4	4	no	—
107	M₃₂	—	4	4	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	—	2	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	C₂×C₈	1	4	yes	—
114	C₂²⋊C₈	—	2	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	2	2	no	—
117	C₂²⋊C₈	—	2	2	no	—
118	#97	—	2	2	no	—
119	#97	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊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	C₂²×C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₈	C₂×D₄	1	1	yes	characteristic
137	C₄×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
139	C₂²×C₄	—	4	1	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	#95	1	1	yes	characteristic
144	C₂²×C₄	#95	1	1	yes	characteristic
145	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
146	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
147	C₂²×C₄	#97	1	1	yes	characteristic
148	C₂²×C₄	#97	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
160	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
161	M₁₆	—	4	4	no	—
162	M₁₆	—	4	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
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	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	4	1	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	—	4	1	no	—
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	#95	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊C₄	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	C₄²	#95	1	1	yes	characteristic
186	C₄²	#95	1	1	yes	characteristic
187	C₄²	C₄×C₈	1	1	yes	characteristic
188	C₄²	C₄×C₈	1	1	yes	characteristic
189	C₄²	#97	1	1	yes	characteristic
190	C₄²	#97	1	1	yes	characteristic
191	C₄²	C₂²⋊C₈	1	1	yes	characteristic
192	C₄²	C₂²⋊C₈	1	1	yes	characteristic
193	C₄²	C₂²⋊C₈	1	1	yes	characteristic
194	C₄²	C₂²⋊C₈	1	1	yes	characteristic
195	C₄²	C₄⋊C₈	1	1	yes	characteristic
196	C₄²	C₄⋊C₈	1	1	yes	characteristic
197	C₄²	C₂×C₄²	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₂²×C₈	1	1	yes	characteristic
200	C₄²	C₂×M₁₆	1	1	yes	characteristic
201	C₁₆	—	4	4	no	—
202	C₁₆	—	4	4	no	—
203	C₂³	—	4	1	no	—
204	C₂³	—	4	1	no	—
205	C₂³	—	4	1	no	—
206	C₂³	#336	1	1	yes	characteristic
207	C₂³	#336	1	1	yes	characteristic
208	C₂³	C₂×C₄⋊C₈	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	1	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	—	2	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	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	#336	1	1	yes	characteristic
231	C₂×C₄	#336	1	1	yes	characteristic
232	C₂×C₄	#336	1	1	yes	characteristic
233	C₂×C₄	#336	1	1	yes	characteristic
234	C₂×C₄	#336	1	1	yes	characteristic
235	C₂×C₄	#336	1	1	yes	characteristic
236	C₂×C₄	C₂×#95	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
238	C₂×C₄	C₂×#97	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	4	no	—
249	C₈	—	4	4	no	—
250	C₈	#421	1	2	yes	—
251	C₈	#430	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	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	—	4	4	no	—
262	C₂²	C₂×#336	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
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	#1733	1	1	yes	characteristic
273	C₄	#1738	1	1	yes	characteristic
274	C₄	#2115	1	2	yes	—
275	C₄	#2117	1	2	yes	—
276	C₂	—	4	1	no	—
277	C₂	—	4	1	no	—
278	C₂	—	4	1	no	—
279	C₂	#7882	1	1	yes	characteristic
280	C₂	#12551	1	1	yes	characteristic
281	C₂	#12554	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ 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. #7166	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#17323	2
#7166	2