Subgroups — #127166

Order 512.

471 conjugacy classes of subgroups.

The subgroup lattice has 294 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×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#7212	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	C₄	1	2	yes	—
6	C₂×#431	C₄	1	1	yes	characteristic
7	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	#2163	—	2	1	no	—
10	#2163	C₄	1	2	yes	—
11	#2162	—	2	1	no	—
12	#2162	C₂²	1	2	yes	—
13	#2150	—	2	1	no	—
14	#2150	—	2	1	no	—
15	#2150	—	2	1	no	—
16	#2150	C₄	1	2	yes	—
17	#2150	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	—	2	1	no	—
19	C₂×C₄⋊C₁₆	—	2	1	no	—
20	#2112	C₄	1	2	yes	—
21	C₄×M₃₂	—	2	1	no	—
22	C₈×M₁₆	—	2	1	no	—
23	C₈×M₁₆	—	2	1	no	—
24	C₈×M₁₆	C₄	1	2	yes	—
25	C₈×M₁₆	C₂²	1	2	yes	—
26	C₂×#322	—	2	1	no	—
27	C₂×C₈²	—	2	1	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	4	4	no	—
35	C₂×M₃₂	C₂×C₄	1	4	yes	—
36	C₂²×C₁₆	—	2	2	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	C₈	1	2	yes	—
40	#431	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	C₈	1	2	yes	—
42	C₂×C₂²⋊C₈	—	2	1	no	—
43	C₄×M₁₆	—	2	1	no	—
44	C₄×M₁₆	—	2	1	no	—
45	C₄×M₁₆	C₈	1	2	yes	—
46	C₄×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×#97	C₈	1	2	yes	—
48	C₂×C₄×C₈	—	2	1	no	—
49	C₂×C₄×C₈	—	2	1	no	—
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₄⋊C₁₆	—	2	2	no	—
65	C₄⋊C₁₆	—	2	2	no	—
66	C₄⋊C₁₆	—	2	2	no	—
67	C₄⋊C₁₆	—	2	2	no	—
68	C₂²⋊C₁₆	—	2	2	no	—
69	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
70	#346	—	2	2	no	—
71	C₄×C₁₆	—	2	2	no	—
72	#322	—	2	2	no	—
73	#322	—	2	2	no	—
74	C₈²	—	2	2	no	—
75	C₈²	—	2	2	no	—
76	C₂³×C₄	C₄²	1	1	yes	characteristic
77	C₂³×C₄	M₁₆	1	1	yes	characteristic
78	C₂³×C₄	M₁₆	1	1	yes	characteristic
79	C₂×M₁₆	—	2	8	no	—
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	2	2	no	—
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₄²	1	1	yes	characteristic
88	C₂²×C₈	C₄²	1	1	yes	characteristic
89	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₄²	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄²	M₁₆	1	1	yes	characteristic
109	C₂×C₄²	M₁₆	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
111	M₃₂	—	4	4	no	—
112	M₃₂	—	4	4	no	—
113	M₃₂	—	4	4	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	4	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	C₂×C₈	1	4	yes	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	#97	C₂×C₈	1	4	yes	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊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	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₂×D₄	1	1	yes	characteristic
140	C₄×C₈	C₂×D₄	1	1	yes	characteristic
141	C₄×C₈	C₂×D₄	1	1	yes	characteristic
142	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
143	C₂⁴	#97	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	#95	1	1	yes	characteristic
149	C₂²×C₄	#95	1	1	yes	characteristic
150	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
151	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
152	C₂²×C₄	#97	1	1	yes	characteristic
153	C₂²×C₄	#97	1	1	yes	characteristic
154	C₂²×C₄	#97	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
164	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
166	M₁₆	—	2	8	no	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	#95	1	1	yes	characteristic
186	C₂×C₈	#95	1	1	yes	characteristic
187	C₂×C₈	#95	1	1	yes	characteristic
188	C₂×C₈	#95	1	1	yes	characteristic
189	C₂×C₈	#100	1	2	yes	—
190	C₂×C₈	#101	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₄²	#95	1	1	yes	characteristic
196	C₄²	#95	1	1	yes	characteristic
197	C₄²	C₄×C₈	1	1	yes	characteristic
198	C₄²	C₄×C₈	1	1	yes	characteristic
199	C₄²	C₂²⋊C₈	1	1	yes	characteristic
200	C₄²	C₂²⋊C₈	1	1	yes	characteristic
201	C₄²	C₂²⋊C₈	1	1	yes	characteristic
202	C₄²	C₂²⋊C₈	1	1	yes	characteristic
203	C₄²	C₄⋊C₈	1	1	yes	characteristic
204	C₄²	C₄⋊C₈	1	1	yes	characteristic
205	C₄²	C₄⋊C₈	1	1	yes	characteristic
206	C₄²	C₄⋊C₈	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄²	C₂²×C₈	1	1	yes	characteristic
210	C₄²	C₂²×C₈	1	1	yes	characteristic
211	C₁₆	—	4	4	no	—
212	C₁₆	—	4	4	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	#336	1	1	yes	characteristic
217	C₂³	#336	1	1	yes	characteristic
218	C₂³	C₂×#97	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
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	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	#336	1	1	yes	characteristic
238	C₂×C₄	#336	1	1	yes	characteristic
239	C₂×C₄	#336	1	1	yes	characteristic
240	C₂×C₄	#336	1	1	yes	characteristic
241	C₂×C₄	#336	1	1	yes	characteristic
242	C₂×C₄	#336	1	1	yes	characteristic
243	C₂×C₄	#343	1	2	yes	—
244	C₂×C₄	#344	1	2	yes	—
245	C₂×C₄	#364	1	4	yes	—
246	C₂×C₄	C₂×#95	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	4	no	—
260	C₈	#343	1	2	yes	—
261	C₈	C₂×#100	1	1	yes	characteristic
262	C₈	C₂×#101	1	1	yes	characteristic
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	#1283	1	2	yes	—
272	C₂²	#1376	1	2	yes	—
273	C₂²	C₂×#336	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	#1283	1	2	yes	—
282	C₄	#1376	1	2	yes	—
283	C₄	C₂×#343	1	1	yes	characteristic
284	C₄	C₂×#344	1	1	yes	characteristic
285	C₄	#2117	1	2	yes	—
286	C₄	C₂×#364	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	—	4	1	no	—
290	C₂	C₂×#1283	1	1	yes	characteristic
291	C₂	C₂×#1376	1	1	yes	characteristic
292	C₂	#12554	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #7212	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
C₂×#2150	2
C₂×C₈×M₁₆	2
#7212	2