Subgroups — #479951

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33479	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#41	C₂	1	1	yes	maximal, characteristic
3	#17328	C₂	1	1	yes	maximal, characteristic
4	#11412	C₂	1	2	yes	maximal, complemented
5	#11410	C₂	1	1	yes	maximal, characteristic, complemented
6	#11409	C₂	1	2	yes	maximal, complemented
7	#11408	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×M₁₆	C₂	1	2	yes	maximal, complemented
10	#11401	C₂	1	2	yes	maximal, complemented
11	C₂×#1312	C₂	1	1	yes	maximal, characteristic, complemented
12	#2965	C₄	1	1	yes	characteristic
13	C₈×#41	C₂²	1	1	yes	characteristic
14	#2922	C₄	1	1	yes	characteristic
15	#2183	C₄	1	2	yes	—
16	Q₈×C₁₆	C₄	1	2	yes	—
17	#2169	C₄	1	2	yes	—
18	#2168	C₄	1	4	yes	—
19	D₄×C₁₆	C₄	1	2	yes	—
20	D₄×C₁₆	C₄	1	4	yes	—
21	#2162	C₂²	1	1	yes	characteristic
22	#2162	C₂²	1	2	yes	—
23	#2162	C₂²	1	1	yes	characteristic
24	#2162	C₂²	1	2	yes	—
25	C₄×M₃₂	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₁₆	C₂²	1	2	yes	—
29	#1454	C₄	1	2	yes	—
30	#1453	C₄	1	2	yes	—
31	#1452	C₄	1	1	yes	characteristic
32	#1451	C₄	1	1	yes	characteristic
33	#1450	C₂²	1	2	yes	—
34	C₈×M₁₆	C₂²	1	1	yes	characteristic
35	C₈×M₁₆	C₂²	1	2	yes	—
36	C₂×C₈²	C₂²	1	1	yes	characteristic
37	#1312	C₂²	1	1	yes	characteristic
38	#1312	C₂²	1	1	yes	characteristic
39	#1312	C₂²	1	1	yes	characteristic
40	#1312	C₂²	1	1	yes	characteristic
41	#1312	C₂²	1	2	yes	—
42	#1312	C₂²	1	2	yes	—
43	#1312	C₂²	1	2	yes	—
44	#1311	C₂²	1	2	yes	—
45	C₈×C₁₆	C₂²	1	2	yes	—
46	C₈×C₁₆	C₂²	1	2	yes	—
47	C₄×#41	C₂×C₄	1	1	yes	characteristic
48	C₂×M₃₂	C₂×C₄	1	4	yes	—
49	C₂²×C₁₆	C₂×C₄	1	2	yes	—
50	C₂²×C₁₆	C₂×C₄	1	4	yes	—
51	C₂²×C₁₆	C₂×C₄	1	2	yes	—
52	#446	C₈	1	2	yes	—
53	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
54	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
55	#436	C₈	1	2	yes	—
56	#435	C₈	1	4	yes	—
57	C₈×D₄	C₂×C₄	1	1	yes	characteristic
58	C₈×D₄	C₂×C₄	1	2	yes	—
59	C₈×D₄	C₂×C₄	1	2	yes	—
60	C₈×D₄	C₂×C₄	1	1	yes	characteristic
61	#433	C₈	1	4	yes	—
62	#431	C₂×C₄	1	1	yes	characteristic
63	#431	C₂×C₄	1	2	yes	—
64	#431	C₂×C₄	1	2	yes	—
65	#431	C₂×C₄	1	2	yes	—
66	#431	C₂×C₄	1	1	yes	characteristic
67	#431	C₂×C₄	1	2	yes	—
68	#424	C₈	1	2	yes	—
69	#408	C₈	1	2	yes	—
70	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
71	C₄×M₁₆	C₂×C₄	1	2	yes	—
72	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂³	1	2	yes	—
76	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
77	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
78	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
79	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
80	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
81	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
82	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
83	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
84	#346	C₂×C₄	1	4	yes	—
85	C₄×C₁₆	C₂×C₄	1	2	yes	—
86	C₄×C₁₆	C₂×C₄	1	4	yes	—
87	C₄×C₁₆	C₂×C₄	1	2	yes	—
88	C₄×C₁₆	C₂³	1	1	yes	characteristic
89	C₄×C₁₆	C₂³	1	2	yes	—
90	C₄×C₁₆	C₂³	1	2	yes	—
91	C₄×C₁₆	C₂³	1	1	yes	characteristic
92	C₄×C₁₆	C₂³	1	1	yes	characteristic
93	C₄×C₁₆	C₂³	1	1	yes	characteristic
94	#322	C₂×C₄	1	2	yes	—
95	#322	C₂×C₄	1	2	yes	—
96	#322	C₂×C₄	1	2	yes	—
97	#322	C₂×C₄	1	2	yes	—
98	#322	C₂×C₄	1	2	yes	—
99	C₈²	C₂×C₄	1	1	yes	characteristic
100	C₈²	C₂×C₄	1	1	yes	characteristic
101	C₈²	C₂³	1	1	yes	characteristic
102	C₈²	C₂³	1	2	yes	—
103	C₈²	C₂³	1	1	yes	characteristic
104	C₂×#41	C₂×C₈	1	1	yes	characteristic
105	C₂×M₁₆	C₂×C₈	1	2	yes	—
106	C₂×M₁₆	C₂×C₈	1	4	yes	—
107	C₂²×C₈	C₂×C₈	1	2	yes	—
108	C₂²×C₈	C₂×C₈	1	2	yes	—
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₄	1	2	yes	—
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×C₄	1	2	yes	—
113	C₄×Q₈	C₄²	1	1	yes	characteristic
114	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
115	C₄×D₄	C₄²	1	2	yes	—
116	C₄×D₄	C₄²	1	1	yes	characteristic
117	C₄×D₄	C₂×C₈	1	2	yes	—
118	C₄×D₄	C₂×C₈	1	1	yes	characteristic
119	#117	C₂×C₈	1	1	yes	characteristic
120	#117	C₂×C₈	1	2	yes	—
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	2	yes	—
123	M₃₂	—	4	4	no	—
124	C₂×C₁₆	—	4	2	no	—
125	C₂×C₁₆	—	4	4	no	—
126	C₂×C₁₆	—	4	2	no	—
127	C₂×C₁₆	C₂²×C₄	1	2	yes	—
128	C₂×C₁₆	C₂²×C₄	1	2	yes	—
129	C₂×C₁₆	C₂²×C₄	1	4	yes	—
130	C₂×C₁₆	C₂²×C₄	1	2	yes	—
131	C₂×C₁₆	C₂²×C₄	1	2	yes	—
132	C₂×C₁₆	C₂²×C₄	1	4	yes	—
133	C₄⋊C₈	C₄²	1	1	yes	characteristic
134	C₄⋊C₈	C₄²	1	1	yes	characteristic
135	C₄⋊C₈	C₄²	1	2	yes	—
136	C₄⋊C₈	C₄²	1	2	yes	—
137	C₄⋊C₈	C₂×C₈	1	4	yes	—
138	C₄⋊C₈	C₂×C₈	1	4	yes	—
139	C₄⋊C₈	C₂×C₈	1	4	yes	—
140	C₄⋊C₈	C₂×C₈	1	2	yes	—
141	C₂²⋊C₈	C₄²	1	2	yes	—
142	C₂²⋊C₈	C₄²	1	2	yes	—
143	C₂²⋊C₈	C₄²	1	2	yes	—
144	C₂²⋊C₈	C₂×C₈	1	4	yes	—
145	C₂²⋊C₈	C₂×C₈	1	4	yes	—
146	C₂²⋊C₈	C₂×C₈	1	2	yes	—
147	C₂²⋊C₈	C₂×C₈	1	4	yes	—
148	#97	C₂×C₈	1	2	yes	—
149	#97	C₂×C₈	1	4	yes	—
150	C₄×C₈	C₂×C₈	1	2	yes	—
151	C₄×C₈	C₂×C₈	1	2	yes	—
152	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₄	1	2	yes	—
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	2	yes	—
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²×C₄	1	2	yes	—
158	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
162	C₄×C₈	C₂²×C₄	1	2	yes	—
163	C₄×C₈	C₂²×C₄	1	2	yes	—
164	C₄×C₈	C₂⁴	1	1	yes	characteristic
165	#41	—	2	4	no	—
166	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
167	C₂×D₄	C₄×C₈	1	2	yes	—
168	C₂×D₄	C₄×C₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄²	1	2	yes	—
170	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₈	1	2	yes	—
172	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₈	1	2	yes	—
174	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
175	M₁₆	—	2	4	no	—
176	M₁₆	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
184	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
185	C₂×C₈	C₂×C₄²	1	2	yes	—
186	C₂×C₈	C₂×C₄²	1	2	yes	—
187	C₂×C₈	C₂²×C₈	1	2	yes	—
188	C₂×C₈	C₂²×C₈	1	2	yes	—
189	C₂×C₈	C₂²×C₈	1	2	yes	—
190	C₂×C₈	C₂²×C₈	1	4	yes	—
191	C₂×C₈	C₂²×C₈	1	4	yes	—
192	C₂×C₈	C₂²×C₈	1	2	yes	—
193	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
195	C₄⋊C₄	C₄×C₈	1	2	yes	—
196	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
197	C₄⋊C₄	C₄×C₈	1	2	yes	—
198	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
199	C₂²⋊C₄	C₄×C₈	1	2	yes	—
200	C₂²⋊C₄	C₄×C₈	1	2	yes	—
201	C₂²⋊C₄	C₄×C₈	1	2	yes	—
202	C₄²	C₂×C₄²	1	2	yes	—
203	C₄²	C₂×C₄²	1	1	yes	characteristic
204	C₄²	C₂²×C₈	1	1	yes	characteristic
205	C₄²	C₂²×C₈	1	1	yes	characteristic
206	C₄²	C₂²×C₈	1	2	yes	—
207	C₄²	C₂²×C₈	1	2	yes	—
208	C₄²	C₂³×C₄	1	1	yes	characteristic
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	2	2	no	—
212	C₁₆	—	2	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	4	no	—
215	C₂³	C₂×C₄×C₈	1	2	yes	—
216	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
229	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
230	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
232	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
233	C₂×C₄	#504	1	4	yes	—
234	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
236	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	4	no	—
245	C₈	—	2	4	no	—
246	C₈	#504	1	4	yes	—
247	C₈	#568	1	2	yes	—
248	C₈	#568	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	#2108	1	2	yes	—
255	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	#2108	1	2	yes	—
262	C₄	#2875	1	1	yes	characteristic
263	C₄	#2875	1	1	yes	characteristic
264	C₄	C₂×#504	1	2	yes	—
265	C₄	C₂×#504	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	#17316	1	1	yes	characteristic
269	C₂	C₂×#2108	1	1	yes	characteristic
270	C₂	C₄×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ 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. #11412	C₂	semidirect
5. #11410	C₂	semidirect
6. #11409	C₂	semidirect
7. #11408	C₂	semidirect
8. C₁₆×M₁₆	C₂	semidirect
9. C₁₆×M₁₆	C₂	semidirect
10. #11401	C₂	semidirect
11. C₂×#1312	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33479	2
C₁₆×#41	2
#17328	2
#11412	2
#11410	2
#11409	2
#11408	2
C₁₆×M₁₆	2
#11401	2
C₂×#1312	2