Subgroups — #483421

Order 512.

480 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂²×#334	C₂	1	1	yes	maximal, characteristic
4	#12030	C₂	1	4	yes	maximal, complemented
5	#12028	C₂	1	4	yes	maximal, complemented
6	#12000	C₂	1	4	yes	maximal, complemented
7	#2920	—	2	1	no	—
8	C₂²×C₄⋊C₈	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2163	C₂²	1	2	yes	—
11	#2163	C₂²	1	2	yes	—
12	#2162	C₄	1	2	yes	—
13	#2162	C₄	1	2	yes	—
14	#2150	C₄	1	2	yes	—
15	#2112	C₄	1	2	yes	—
16	C₄×M₃₂	C₂²	1	2	yes	—
17	#1568	—	2	1	no	—
18	#1568	C₂²	1	2	yes	—
19	#1567	—	2	1	no	—
20	#1566	C₂²	1	2	yes	—
21	#1564	—	2	1	no	—
22	#1564	C₂²	1	2	yes	—
23	C₂×#334	—	2	1	no	—
24	C₂×#334	—	2	1	no	—
25	C₂×#334	—	2	1	no	—
26	C₂×#334	C₂²	1	2	yes	—
27	C₂×#334	C₂²	1	2	yes	—
28	C₂×#334	C₂²	1	2	yes	—
29	#1368	C₂²	1	16	yes	complemented
30	C₂³×C₈	—	4	1	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₄²	D₄	1	1	yes	characteristic
34	C₂×M₃₂	C₂×C₄	1	4	yes	—
35	#432	—	2	2	no	—
36	#432	—	4	1	no	—
37	#431	—	2	2	no	—
38	#431	—	4	1	no	—
39	#423	—	2	2	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₄⋊C₈	—	2	2	no	—
43	C₂×C₄⋊C₈	—	4	1	no	—
44	C₂×C₄⋊C₈	—	4	1	no	—
45	C₂×C₄⋊C₈	—	4	1	no	—
46	#407	—	4	1	no	—
47	C₄×M₁₆	—	4	1	no	—
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	D₄	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	Q₈	1	1	yes	characteristic
55	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
61	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
62	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
63	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
64	#346	C₂³	1	4	yes	—
65	C₄×C₁₆	C₂×C₄	1	4	yes	—
66	#335	—	2	2	no	—
67	#335	—	2	2	no	—
68	#334	—	2	2	no	—
69	#334	—	2	2	no	—
70	#334	—	2	2	no	—
71	#334	—	2	2	no	—
72	#334	C₂³	1	4	yes	—
73	#334	C₂³	1	4	yes	—
74	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₄	SD₁₆	1	1	yes	characteristic
76	C₂³×C₄	SD₁₆	1	1	yes	characteristic
77	C₂×M₁₆	—	4	2	no	—
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	—	4	2	no	—
81	C₂²×C₈	—	4	2	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	—	8	4	no	—
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	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊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	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₄²	C₄⋊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	SD₁₆	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	SD₁₆	1	1	yes	characteristic
104	C₂×C₄²	SD₁₆	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	M₃₂	—	2	8	no	—
109	C₂×C₁₆	C₂²×C₄	1	8	yes	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	2	no	—
117	C₂²⋊C₈	—	4	2	no	—
118	#97	—	4	2	no	—
119	C₄×C₈	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
129	C₄×C₈	C₂×D₄	1	1	yes	characteristic
130	C₄×C₈	C₂×D₄	1	1	yes	characteristic
131	C₄×C₈	C₂×D₄	1	1	yes	characteristic
132	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
133	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
134	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
135	C₄×C₈	C₂⁴	1	1	yes	characteristic
136	C₂⁴	#106	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	#106	1	1	yes	characteristic
142	C₂²×C₄	#106	1	1	yes	characteristic
143	C₂²×C₄	#106	1	1	yes	characteristic
144	C₂²×C₄	#106	1	1	yes	characteristic
145	C₂²×C₄	#106	1	1	yes	characteristic
146	C₂²×C₄	#106	1	1	yes	characteristic
147	C₂²×C₄	#106	1	1	yes	characteristic
148	C₂²×C₄	#106	1	1	yes	characteristic
149	C₂²×C₄	#106	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
159	M₁₆	—	8	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	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	—	2	2	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	4	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	8	4	no	—
174	C₂×C₈	—	8	4	no	—
175	C₂×C₈	—	8	4	no	—
176	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	#131	1	4	yes	—
183	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
185	C₄²	#106	1	1	yes	characteristic
186	C₄²	#106	1	1	yes	characteristic
187	C₄²	#106	1	1	yes	characteristic
188	C₄²	#106	1	1	yes	characteristic
189	C₄²	#106	1	1	yes	characteristic
190	C₄²	#106	1	1	yes	characteristic
191	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₄²	C₂×SD₁₆	1	1	yes	characteristic
195	C₄²	C₂×SD₁₆	1	1	yes	characteristic
196	C₄²	C₂×SD₁₆	1	1	yes	characteristic
197	C₄²	C₂×SD₁₆	1	1	yes	characteristic
198	C₄²	C₂×SD₁₆	1	1	yes	characteristic
199	C₄²	C₂×SD₁₆	1	1	yes	characteristic
200	C₄²	C₂²×D₄	1	1	yes	characteristic
201	C₁₆	—	2	8	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	C₂×#106	1	1	yes	characteristic
206	C₂³	C₂×#106	1	1	yes	characteristic
207	C₂³	C₂×#106	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
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	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
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	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	#365	1	4	yes	—
227	C₂×C₄	#424	1	2	yes	—
228	C₂×C₄	C₂×#106	1	1	yes	characteristic
229	C₂×C₄	C₂×#106	1	1	yes	characteristic
230	C₂×C₄	C₂×#106	1	1	yes	characteristic
231	C₂×C₄	C₂×#106	1	1	yes	characteristic
232	C₂×C₄	C₂×#106	1	1	yes	characteristic
233	C₂×C₄	C₂×#106	1	1	yes	characteristic
234	C₂×C₄	C₂×#106	1	1	yes	characteristic
235	C₂×C₄	C₂×#106	1	1	yes	characteristic
236	C₂×C₄	C₂×#106	1	1	yes	characteristic
237	C₂×C₄	#430	1	2	yes	—
238	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
240	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
241	C₈	—	2	1	no	—
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	—	8	4	no	—
248	C₈	—	8	4	no	—
249	C₈	#424	1	2	yes	—
250	C₈	C₂×#131	1	2	yes	—
251	C₂²	—	2	8	no	—
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	#1570	1	2	yes	—
259	C₂²	C₂×#365	1	2	yes	—
260	C₂²	#2155	1	2	yes	—
261	C₂²	C₂²×#106	1	1	yes	characteristic
262	C₄	—	2	1	no	—
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	#1570	1	2	yes	—
271	C₄	C₂×#365	1	2	yes	—
272	C₄	C₂×#365	1	2	yes	—
273	C₄	C₂×#424	1	1	yes	characteristic
274	C₄	C₂×#430	1	1	yes	characteristic
275	C₂	—	2	2	no	—
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	C₂×#1570	1	1	yes	characteristic
279	C₂	C₂²×#365	1	1	yes	characteristic
280	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. #12030	C₂	semidirect
5. #12028	C₂	semidirect
6. #12000	C₂	semidirect
29. #1368	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#17793	2
C₂²×#334	2
#12030	2
#12028	2
#12000	2