Subgroups — #483397

Order 512.

517 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	#12020	C₂	1	4	yes	maximal, complemented
5	#12019	C₂	1	4	yes	maximal, complemented
6	#12000	C₂	1	4	yes	maximal, complemented
7	C₂²×M₃₂	C₄	1	2	yes	—
8	#2920	—	2	1	no	—
9	C₂×#423	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×M₃₂	C₄	1	1	yes	characteristic
12	C₄×M₃₂	C₄	1	2	yes	—
13	C₄×M₃₂	C₄	1	1	yes	characteristic
14	C₄×M₃₂	C₂²	1	2	yes	—
15	C₄×M₃₂	C₂²	1	2	yes	—
16	C₂×#346	C₂²	1	2	yes	—
17	C₂×C₄×C₁₆	C₄	1	2	yes	—
18	#1568	—	2	1	no	—
19	#1568	C₂²	1	2	yes	—
20	#1567	—	2	1	no	—
21	#1566	—	2	1	no	—
22	#1566	—	2	1	no	—
23	#1566	C₂²	1	2	yes	—
24	#1566	C₂²	1	2	yes	—
25	#1566	C₂²	1	2	yes	—
26	#1564	—	2	1	no	—
27	#1564	C₂²	1	2	yes	—
28	C₂×#334	—	2	1	no	—
29	C₂×#334	C₂²	1	2	yes	—
30	#1368	C₂²	1	16	yes	complemented
31	C₂²×M₁₆	—	4	1	no	—
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₄²	D₄	1	1	yes	characteristic
35	C₂×M₃₂	—	2	8	no	—
36	C₂×M₃₂	C₂×C₄	1	2	yes	—
37	C₂×M₃₂	C₂×C₄	1	4	yes	—
38	C₂×M₃₂	C₂×C₄	1	2	yes	—
39	C₂²×C₁₆	C₂×C₄	1	4	yes	—
40	#432	—	2	2	no	—
41	#432	—	4	1	no	—
42	#431	—	2	2	no	—
43	#431	—	4	1	no	—
44	#423	—	2	2	no	—
45	#423	—	2	2	no	—
46	#423	—	2	2	no	—
47	#423	—	4	1	no	—
48	#423	—	4	1	no	—
49	C₂×C₄⋊C₈	—	2	2	no	—
50	C₂×C₄⋊C₈	—	4	1	no	—
51	#407	—	4	1	no	—
52	C₄×M₁₆	—	4	1	no	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	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	Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#346	C₂×C₄	1	2	yes	—
66	#346	C₂×C₄	1	2	yes	—
67	#346	C₂³	1	4	yes	—
68	C₄×C₁₆	C₂×C₄	1	2	yes	—
69	C₄×C₁₆	C₂×C₄	1	4	yes	—
70	C₄×C₁₆	C₂×C₄	1	2	yes	—
71	#335	—	2	2	no	—
72	#335	—	2	2	no	—
73	#334	—	2	2	no	—
74	#334	—	2	2	no	—
75	#334	—	2	2	no	—
76	#334	—	2	2	no	—
77	#334	C₂³	1	4	yes	—
78	#334	C₂³	1	4	yes	—
79	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₄	SD₁₆	1	1	yes	characteristic
81	C₂³×C₄	SD₁₆	1	1	yes	characteristic
82	C₂×M₁₆	—	4	2	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂×M₁₆	—	8	4	no	—
86	C₂²×C₈	—	2	4	no	—
87	C₂²×C₈	—	2	4	no	—
88	C₂²×C₈	—	4	2	no	—
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	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₄²	C₄⋊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	SD₁₆	1	1	yes	characteristic
105	C₂×C₄²	SD₁₆	1	1	yes	characteristic
106	C₂×C₄²	SD₁₆	1	1	yes	characteristic
107	C₂×C₄²	SD₁₆	1	1	yes	characteristic
108	C₂×C₄²	SD₁₆	1	1	yes	characteristic
109	C₂×C₄²	SD₁₆	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	8	no	—
115	M₃₂	M₁₆	1	8	yes	complemented
116	C₂×C₁₆	—	2	8	no	—
117	C₂×C₁₆	C₂²×C₄	1	4	yes	—
118	C₂×C₁₆	C₂²×C₄	1	4	yes	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₄⋊C₈	—	4	2	no	—
125	C₂²⋊C₈	—	4	2	no	—
126	C₂²⋊C₈	—	4	2	no	—
127	#97	—	4	2	no	—
128	C₄×C₈	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂×D₄	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×Q₈	1	1	yes	characteristic
142	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	C₄×C₈	C₂⁴	1	1	yes	characteristic
145	C₂⁴	#106	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	#106	1	1	yes	characteristic
151	C₂²×C₄	#106	1	1	yes	characteristic
152	C₂²×C₄	#106	1	1	yes	characteristic
153	C₂²×C₄	#106	1	1	yes	characteristic
154	C₂²×C₄	#106	1	1	yes	characteristic
155	C₂²×C₄	#106	1	1	yes	characteristic
156	C₂²×C₄	#106	1	1	yes	characteristic
157	C₂²×C₄	#106	1	1	yes	characteristic
158	C₂²×C₄	#106	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₄⋊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×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
168	M₁₆	—	8	4	no	—
169	M₁₆	—	8	4	no	—
170	M₁₆	—	8	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	8	4	no	—
184	C₂×C₈	#108	1	4	yes	—
185	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×M₁₆	1	4	yes	—
192	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
194	C₄²	#106	1	1	yes	characteristic
195	C₄²	#106	1	1	yes	characteristic
196	C₄²	#106	1	1	yes	characteristic
197	C₄²	#106	1	1	yes	characteristic
198	C₄²	#106	1	1	yes	characteristic
199	C₄²	#106	1	1	yes	characteristic
200	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₄²	C₂×SD₁₆	1	1	yes	characteristic
204	C₄²	C₂×SD₁₆	1	1	yes	characteristic
205	C₄²	C₂×SD₁₆	1	1	yes	characteristic
206	C₄²	C₂×SD₁₆	1	1	yes	characteristic
207	C₄²	C₂×SD₁₆	1	1	yes	characteristic
208	C₄²	C₂×SD₁₆	1	1	yes	characteristic
209	C₄²	C₂²×D₄	1	1	yes	characteristic
210	C₁₆	—	2	4	no	—
211	C₁₆	C₂×M₁₆	1	8	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×#106	1	1	yes	characteristic
216	C₂³	C₂×#106	1	1	yes	characteristic
217	C₂³	C₂×#106	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	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	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	#365	1	4	yes	—
237	C₂×C₄	#423	1	2	yes	—
238	C₂×C₄	C₂×#106	1	1	yes	characteristic
239	C₂×C₄	C₂×#106	1	1	yes	characteristic
240	C₂×C₄	C₂×#106	1	1	yes	characteristic
241	C₂×C₄	C₂×#106	1	1	yes	characteristic
242	C₂×C₄	C₂×#106	1	1	yes	characteristic
243	C₂×C₄	C₂×#106	1	1	yes	characteristic
244	C₂×C₄	C₂×#106	1	1	yes	characteristic
245	C₂×C₄	C₂×#106	1	1	yes	characteristic
246	C₂×C₄	C₂×#106	1	1	yes	characteristic
247	C₂×C₄	C₂×#108	1	2	yes	—
248	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
250	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	8	4	no	—
256	C₈	—	8	4	no	—
257	C₈	#423	1	2	yes	—
258	C₈	C₂×#108	1	2	yes	—
259	C₈	C₂×#108	1	2	yes	—
260	C₈	C₂²×M₁₆	1	2	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
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	—	4	4	no	—
269	C₂²	#1566	1	2	yes	—
270	C₂²	C₂×#365	1	2	yes	—
271	C₂²	C₂²×#106	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
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	#1566	1	2	yes	—
280	C₄	C₂×#365	1	2	yes	—
281	C₄	C₂×#365	1	2	yes	—
282	C₄	#2155	1	2	yes	—
283	C₄	C₂×#423	1	1	yes	characteristic
284	C₄	C₂²×#108	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	C₂×#1566	1	1	yes	characteristic
289	C₂	C₂²×#365	1	1	yes	characteristic
290	C₂	C₂×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ 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
4. #12020	C₂	semidirect
5. #12019	C₂	semidirect
6. #12000	C₂	semidirect
30. #1368	C₂²	semidirect
115. M₃₂	M₁₆	semidirect
211. C₁₆	C₂×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
#17793	2
C₂×#1566	2
#12020	2
#12019	2
#12000	2