Subgroups — #483349

Order 512.

487 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#2163	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	#12014	C₂	1	4	yes	maximal, complemented
5	#12013	C₂	1	4	yes	maximal, complemented
6	#12005	C₂	1	2	yes	maximal, complemented
7	#11997	C₂	1	2	yes	maximal, complemented
8	C₂×#431	C₄	1	1	yes	characteristic
9	C₂×#431	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2163	—	2	1	no	—
12	#2163	—	2	1	no	—
13	#2163	—	2	1	no	—
14	#2163	C₂²	1	2	yes	—
15	#2163	C₂²	1	2	yes	—
16	#2163	C₂²	1	2	yes	—
17	#2162	—	2	1	no	—
18	#2162	C₂²	1	2	yes	—
19	#2150	C₂²	1	2	yes	—
20	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
21	#2112	—	2	1	no	—
22	C₂×C₂²⋊C₁₆	—	2	1	no	—
23	C₄×M₃₂	—	2	1	no	—
24	C₂×#346	—	2	1	no	—
25	#1450	C₄	1	2	yes	—
26	#1450	C₄	1	1	yes	characteristic
27	#1450	C₄	1	1	yes	characteristic
28	#1450	C₂²	1	1	yes	characteristic
29	#1450	C₂²	1	1	yes	characteristic
30	#1450	C₂²	1	2	yes	—
31	C₂×#322	C₄	1	2	yes	—
32	C₂×C₈²	C₂²	1	2	yes	—
33	#1370	C₂²	1	8	yes	complemented
34	#1367	C₂²	1	8	yes	complemented
35	C₂³×C₈	D₄	1	1	yes	characteristic
36	C₂³×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	2	no	—
39	C₂²×C₁₆	—	2	2	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	C₈	1	2	yes	—
43	#431	C₈	1	2	yes	—
44	#431	C₂×C₄	1	1	yes	characteristic
45	#431	C₂×C₄	1	1	yes	characteristic
46	#431	C₂×C₄	1	2	yes	—
47	#431	C₂×C₄	1	1	yes	characteristic
48	#431	C₂×C₄	1	1	yes	characteristic
49	#431	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	C₈	1	2	yes	—
52	C₂×C₂²⋊C₈	—	2	1	no	—
53	C₂×C₂²⋊C₈	C₈	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₄⋊C₁₆	—	2	2	no	—
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	C₂³	1	4	yes	—
71	C₄⋊C₁₆	C₂³	1	4	yes	—
72	C₂²⋊C₁₆	—	2	2	no	—
73	C₂²⋊C₁₆	—	2	2	no	—
74	C₂²⋊C₁₆	—	2	2	no	—
75	C₂²⋊C₁₆	—	2	2	no	—
76	#346	—	2	2	no	—
77	#346	—	2	2	no	—
78	#346	—	2	2	no	—
79	C₄×C₁₆	—	2	2	no	—
80	#322	C₂×C₄	1	2	yes	—
81	#322	C₂×C₄	1	2	yes	—
82	#322	C₂×C₄	1	2	yes	—
83	#322	C₂×C₄	1	2	yes	—
84	C₈²	C₂×C₄	1	2	yes	—
85	C₈²	C₂×C₄	1	2	yes	—
86	C₈²	C₂³	1	2	yes	—
87	C₈²	C₂³	1	2	yes	—
88	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
90	C₂³×C₄	M₁₆	1	1	yes	characteristic
91	C₂²×C₈	—	2	4	no	—
92	C₂²×C₈	—	2	4	no	—
93	C₂²×C₈	—	2	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₈	1	4	yes	—
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
113	C₂×C₄²	M₁₆	1	1	yes	characteristic
114	C₂×C₄²	M₁₆	1	1	yes	characteristic
115	C₂×C₄²	M₁₆	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	M₃₂	—	4	4	no	—
120	C₂×C₁₆	—	2	4	no	—
121	C₂×C₁₆	—	2	4	no	—
122	C₂×C₁₆	—	4	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	1	no	—
125	C₄⋊C₈	—	2	1	no	—
126	C₄⋊C₈	C₂×C₈	1	4	yes	—
127	C₄⋊C₈	C₂×C₈	1	2	yes	—
128	C₄⋊C₈	C₂×C₈	1	2	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	2	1	no	—
131	C₂²⋊C₈	—	2	1	no	—
132	C₂²⋊C₈	C₂×C₈	1	4	yes	—
133	C₂²⋊C₈	C₂×C₈	1	2	yes	—
134	C₂²⋊C₈	C₂×C₈	1	2	yes	—
135	C₄×C₈	—	2	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₂×C₈	1	4	yes	—
143	C₄×C₈	C₂²×C₄	1	2	yes	—
144	C₄×C₈	C₂²×C₄	1	2	yes	—
145	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₄	1	2	yes	—
149	C₄×C₈	C₂²×C₄	1	2	yes	—
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
154	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
155	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
156	C₄×C₈	C₂⁴	1	1	yes	characteristic
157	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
177	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
178	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	#108	1	4	yes	—
195	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₂²×C₈	1	4	yes	—
202	C₂×C₈	C₂²×C₈	1	4	yes	—
203	C₂×C₈	#135	1	4	yes	—
204	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
206	C₄²	C₄⋊C₈	1	1	yes	characteristic
207	C₄²	C₄⋊C₈	1	1	yes	characteristic
208	C₄²	C₄⋊C₈	1	1	yes	characteristic
209	C₄²	C₄⋊C₈	1	1	yes	characteristic
210	C₄²	C₄⋊C₈	1	1	yes	characteristic
211	C₄²	C₄⋊C₈	1	1	yes	characteristic
212	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₄²	C₂²×C₈	1	1	yes	characteristic
216	C₄²	C₂²×C₈	1	1	yes	characteristic
217	C₄²	C₂²×C₈	1	1	yes	characteristic
218	C₄²	C₂×M₁₆	1	1	yes	characteristic
219	C₄²	C₂×M₁₆	1	1	yes	characteristic
220	C₄²	C₂×M₁₆	1	1	yes	characteristic
221	C₄²	C₂³×C₄	1	1	yes	characteristic
222	C₁₆	—	4	4	no	—
223	C₁₆	—	4	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	1	no	—
227	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
228	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
229	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂×C₄⋊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	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
258	C₂×C₄	#427	1	2	yes	—
259	C₂×C₄	C₂×#108	1	2	yes	—
260	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
262	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
263	C₈	—	2	4	no	—
264	C₈	—	2	4	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	#427	1	2	yes	—
272	C₈	C₂×#108	1	2	yes	—
273	C₈	C₂×#108	1	2	yes	—
274	C₈	C₂×#135	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	4	no	—
283	C₂²	#1564	1	2	yes	—
284	C₂²	#2153	1	2	yes	—
285	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	#1564	1	2	yes	—
296	C₄	C₂×#427	1	1	yes	characteristic
297	C₄	C₂²×#108	1	1	yes	characteristic
298	C₄	#3407	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	4	1	no	—
302	C₂	C₂×#1564	1	1	yes	characteristic
303	C₂	#33397	1	1	yes	characteristic
304	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #12014	C₂	semidirect
5. #12013	C₂	semidirect
6. #12005	C₂	semidirect
7. #11997	C₂	semidirect
33. #1370	C₂²	semidirect
34. #1367	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×#2163	2
C₂×#1450	2
#12014	2
#12013	2
#12005	2
#11997	2