Subgroups — #483459

Order 512.

455 conjugacy classes of subgroups.

The subgroup lattice has 299 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×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#1567	C₂	1	1	yes	maximal, characteristic
4	#12035	C₂	1	4	yes	maximal, complemented
5	#12033	C₂	1	2	yes	maximal, complemented
6	#12032	C₂	1	2	yes	maximal, complemented
7	#12025	C₂	1	4	yes	maximal, complemented
8	C₂×#432	—	2	1	no	—
9	C₂×#423	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2163	C₄	1	2	yes	—
12	#2163	C₄	1	2	yes	—
13	#2162	C₂²	1	2	yes	—
14	#2162	C₂²	1	2	yes	—
15	#2150	C₄	1	2	yes	—
16	#2112	C₄	1	2	yes	—
17	C₄×M₃₂	C₂²	1	2	yes	—
18	#1568	—	2	1	no	—
19	#1568	—	2	1	no	—
20	#1568	C₂²	1	2	yes	—
21	#1568	C₂²	1	2	yes	—
22	#1567	—	2	1	no	—
23	#1567	—	2	1	no	—
24	#1567	C₂²	1	2	yes	—
25	#1567	C₂²	1	2	yes	—
26	C₂×#335	—	2	1	no	—
27	C₂×#335	C₂²	1	1	yes	characteristic
28	C₂×#335	C₂²	1	1	yes	characteristic
29	C₂×#335	C₂²	1	2	yes	—
30	C₂×#334	—	2	1	no	—
31	#1372	C₂²	1	8	yes	complemented
32	#1371	C₂²	1	8	yes	complemented
33	C₂²×M₁₆	—	4	1	no	—
34	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₄²	D₄	1	1	yes	characteristic
37	C₂×M₃₂	C₂×C₄	1	4	yes	—
38	#432	—	2	2	no	—
39	#432	—	2	2	no	—
40	#432	—	4	1	no	—
41	#432	—	4	1	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	—	4	1	no	—
45	#423	—	4	1	no	—
46	C₂×C₄⋊C₈	—	2	1	no	—
47	C₂×C₄⋊C₈	—	2	1	no	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄⋊C₈	—	4	1	no	—
50	C₂×C₂²⋊C₈	—	4	1	no	—
51	C₂×#97	—	4	1	no	—
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	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	Q₈	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	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
65	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
66	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
67	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
68	#346	C₂×C₄	1	4	yes	—
69	C₄×C₁₆	C₂³	1	4	yes	—
70	#335	—	2	2	no	—
71	#335	—	2	2	no	—
72	#335	—	2	2	no	—
73	#335	—	2	1	no	—
74	#335	—	2	1	no	—
75	#335	C₂³	1	4	yes	—
76	#335	C₂³	1	2	yes	—
77	#335	C₂³	1	2	yes	—
78	#334	—	2	2	no	—
79	#334	—	2	2	no	—
80	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₄	D₈	1	1	yes	characteristic
82	C₂³×C₄	Q₁₆	1	1	yes	characteristic
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂×M₁₆	—	8	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	—	4	1	no	—
90	C₂²×C₈	—	4	1	no	—
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	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	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	D₈	1	1	yes	characteristic
107	C₂×C₄²	D₈	1	1	yes	characteristic
108	C₂×C₄²	D₈	1	1	yes	characteristic
109	C₂×C₄²	Q₁₆	1	1	yes	characteristic
110	C₂×C₄²	Q₁₆	1	1	yes	characteristic
111	C₂×C₄²	Q₁₆	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	M₃₂	—	4	4	no	—
116	C₂×C₁₆	C₂²×C₄	1	8	yes	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	4	2	no	—
121	C₄⋊C₈	—	4	1	no	—
122	C₄⋊C₈	—	4	1	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	#97	—	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	#107	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	#107	1	1	yes	characteristic
152	C₂²×C₄	#107	1	1	yes	characteristic
153	C₂²×C₄	#107	1	1	yes	characteristic
154	C₂²×C₄	#107	1	1	yes	characteristic
155	C₂²×C₄	#107	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	#107	1	1	yes	characteristic
159	C₂²×C₄	#107	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×C₄⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
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	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	8	4	no	—
187	C₂×C₈	—	8	2	no	—
188	C₂×C₈	—	8	2	no	—
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄⋊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	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
197	C₄²	#107	1	1	yes	characteristic
198	C₄²	#107	1	1	yes	characteristic
199	C₄²	#107	1	1	yes	characteristic
200	C₄²	#107	1	1	yes	characteristic
201	C₄²	#107	1	1	yes	characteristic
202	C₄²	#107	1	1	yes	characteristic
203	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄²	C₂×D₈	1	1	yes	characteristic
207	C₄²	C₂×D₈	1	1	yes	characteristic
208	C₄²	C₂×D₈	1	1	yes	characteristic
209	C₄²	C₂×Q₁₆	1	1	yes	characteristic
210	C₄²	C₂×Q₁₆	1	1	yes	characteristic
211	C₄²	C₂×Q₁₆	1	1	yes	characteristic
212	C₄²	C₂²×D₄	1	1	yes	characteristic
213	C₁₆	—	2	8	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₂×#107	1	1	yes	characteristic
218	C₂³	C₂×#107	1	1	yes	characteristic
219	C₂³	C₂×#107	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	C₂×#107	1	1	yes	characteristic
243	C₂×C₄	C₂×#107	1	1	yes	characteristic
244	C₂×C₄	C₂×#107	1	1	yes	characteristic
245	C₂×C₄	C₂×#107	1	1	yes	characteristic
246	C₂×C₄	C₂×#107	1	1	yes	characteristic
247	C₂×C₄	C₂×#107	1	1	yes	characteristic
248	C₂×C₄	C₂×#107	1	1	yes	characteristic
249	C₂×C₄	C₂×#107	1	1	yes	characteristic
250	C₂×C₄	C₂×#107	1	1	yes	characteristic
251	C₂×C₄	#509	1	2	yes	—
252	C₂×C₄	#510	1	2	yes	—
253	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	4	no	—
264	C₈	—	8	2	no	—
265	C₈	#430	1	2	yes	—
266	C₈	#568	1	2	yes	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	4	no	—
275	C₂²	#2159	1	2	yes	—
276	C₂²	#2161	1	2	yes	—
277	C₂²	C₂²×#107	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	#2159	1	2	yes	—
288	C₄	#2905	1	1	yes	characteristic
289	C₄	C₂×#430	1	1	yes	characteristic
290	C₄	C₂×#509	1	1	yes	characteristic
291	C₄	C₂×#510	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	#17814	1	1	yes	characteristic
296	C₂	C₂×#2159	1	1	yes	characteristic
297	C₂	C₂×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
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. #12035	C₂	semidirect
5. #12033	C₂	semidirect
6. #12032	C₂	semidirect
7. #12025	C₂	semidirect
31. #1372	C₂²	semidirect
32. #1371	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
C₂×#1568	2
C₂×#1567	2
#12035	2
#12033	2
#12032	2
#12025	2