Subgroups — #484115

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33735	C₂	1	1	yes	maximal, characteristic
2	#18069	C₂	1	2	yes	maximal, complemented
3	#12082	C₂	1	2	yes	maximal, complemented
4	#12072	C₂	1	2	yes	maximal, complemented
5	#12069	C₂	1	2	yes	maximal, complemented
6	C₈×SD₃₂	C₂	1	2	yes	maximal, complemented
7	#12008	C₂	1	1	yes	maximal, characteristic, complemented
8	#11647	C₂	1	2	yes	maximal, complemented
9	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#3168	C₂²	1	2	yes	—
11	#2969	—	2	2	no	—
12	#2253	C₄	1	1	yes	characteristic
13	#2249	C₄	1	1	yes	characteristic
14	#2242	C₄	1	2	yes	—
15	#2217	C₄	1	2	yes	—
16	#2216	C₄	1	1	yes	characteristic
17	#2215	C₄	1	1	yes	characteristic
18	C₄×Q₃₂	C₂²	1	1	yes	characteristic
19	C₄×SD₃₂	C₂²	1	2	yes	—
20	C₄×D₁₆	C₂²	1	1	yes	characteristic
21	#2162	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1591	C₂²	1	2	yes	—
24	#1589	C₂²	1	2	yes	—
25	#1583	—	2	2	no	—
26	#1582	—	2	2	no	—
27	C₈×Q₁₆	C₂²	1	2	yes	—
28	C₈×D₈	C₂²	1	2	yes	—
29	#1564	C₂²	1	2	yes	—
30	#1482	—	2	2	no	—
31	C₈×M₁₆	C₂²	1	1	yes	characteristic
32	#1372	C₂²	1	1	yes	characteristic
33	#1372	C₂²	1	1	yes	characteristic
34	#1371	C₂²	1	2	yes	—
35	#1332	C₂²	1	2	yes	—
36	#1332	C₂²	1	2	yes	—
37	#1331	C₂²	1	2	yes	—
38	#1331	C₂²	1	2	yes	—
39	#1312	C₂²	1	2	yes	—
40	C₈×C₁₆	C₂²	1	2	yes	—
41	#529	—	2	2	no	—
42	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
43	C₂×SD₃₂	C₂×C₄	1	2	yes	—
44	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
47	#499	C₂×C₄	1	2	yes	—
48	#495	C₂×C₄	1	2	yes	—
49	#488	—	2	2	no	—
50	#467	C₂×C₄	1	2	yes	—
51	#466	C₂×C₄	1	2	yes	—
52	#465	—	2	2	no	—
53	#446	—	2	2	no	—
54	C₈×Q₈	—	2	2	no	—
55	C₄×Q₁₆	C₂³	1	2	yes	—
56	C₄×SD₁₆	—	2	2	no	—
57	C₄×D₈	C₂³	1	2	yes	—
58	#436	—	2	2	no	—
59	#436	—	4	2	no	—
60	#435	—	4	2	no	—
61	C₈×D₄	—	2	2	no	—
62	#433	—	4	2	no	—
63	#431	—	2	2	no	—
64	#431	D₄	1	1	yes	characteristic
65	#405	—	4	2	no	—
66	C₄×M₁₆	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	#367	C₂×C₄	1	2	yes	—
69	#366	C₂×C₄	1	1	yes	characteristic
70	#366	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₁₆	—	2	1	no	—
72	#358	C₂×C₄	1	2	yes	—
73	#358	C₂×C₄	1	2	yes	—
74	#357	C₂×C₄	1	2	yes	—
75	#357	C₂×C₄	1	2	yes	—
76	C₂²⋊C₁₆	—	2	1	no	—
77	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
78	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
79	C₄×C₁₆	C₂³	1	2	yes	—
80	C₄×C₁₆	C₂³	1	1	yes	characteristic
81	C₄×C₁₆	C₂³	1	1	yes	characteristic
82	#335	C₂³	1	2	yes	—
83	#335	C₂³	1	2	yes	—
84	#334	—	2	2	no	—
85	Q₈⋊C₈	—	2	2	no	—
86	Q₈⋊C₈	—	2	2	no	—
87	D₄⋊C₈	—	2	2	no	—
88	D₄⋊C₈	—	2	2	no	—
89	#322	D₄	1	2	yes	—
90	C₈²	C₂³	1	2	yes	—
91	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
92	C₂×SD₁₆	—	2	2	no	—
93	C₂×D₈	C₂²×C₄	1	2	yes	—
94	C₂×M₁₆	—	4	2	no	—
95	C₂×M₁₆	D₈	1	2	yes	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	#126	—	4	2	no	—
100	#125	—	2	2	no	—
101	#124	—	2	2	no	—
102	#123	—	4	2	no	—
103	C₂²⋊Q₈	—	2	2	no	—
104	#121	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₄×D₄	—	2	2	no	—
107	C₄×D₄	—	4	2	no	—
108	#117	—	4	2	no	—
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	Q₃₂	—	4	1	no	—
111	SD₃₂	—	2	4	no	—
112	D₁₆	—	4	1	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	4	1	no	—
116	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
120	#107	C₂²×C₄	1	2	yes	—
121	#107	C₂²×C₄	1	2	yes	—
122	#106	—	2	2	no	—
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	#103	—	2	2	no	—
129	#103	—	2	2	no	—
130	#102	—	2	2	no	—
131	#102	—	2	2	no	—
132	C₂²⋊C₈	—	2	1	no	—
133	C₂²⋊C₈	—	4	2	no	—
134	C₂²⋊C₈	—	4	2	no	—
135	C₂²⋊C₈	—	4	2	no	—
136	#97	—	4	2	no	—
137	#97	D₈	1	1	yes	characteristic
138	#97	D₈	1	1	yes	characteristic
139	C₄×C₈	—	4	2	no	—
140	C₄×C₈	—	4	2	no	—
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂×D₄	1	2	yes	—
143	C₄×C₈	C₂×D₄	1	1	yes	characteristic
144	C₄×C₈	C₂×D₄	1	2	yes	—
145	C₄×C₈	#41	1	1	yes	characteristic
146	C₄×C₈	C₂⁴	1	1	yes	characteristic
147	C₂×Q₈	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂²×C₄	#135	1	1	yes	characteristic
154	Q₁₆	—	2	2	no	—
155	Q₁₆	—	2	2	no	—
156	SD₁₆	—	8	2	no	—
157	D₈	—	2	2	no	—
158	D₈	—	2	2	no	—
159	M₁₆	—	2	4	no	—
160	M₁₆	—	8	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	8	4	no	—
168	C₂×C₈	C₄×D₄	1	1	yes	characteristic
169	C₂×C₈	C₄×D₄	1	1	yes	characteristic
170	C₂×C₈	C₂×D₈	1	2	yes	—
171	C₂×C₈	C₂×D₈	1	2	yes	—
172	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₈	C₂×#41	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₄×D₄	1	1	yes	characteristic
187	C₄²	C₂×D₈	1	1	yes	characteristic
188	C₄²	#135	1	1	yes	characteristic
189	C₄²	C₂²×D₄	1	1	yes	characteristic
190	C₁₆	—	2	1	no	—
191	C₁₆	—	2	1	no	—
192	C₁₆	—	2	1	no	—
193	C₁₆	—	4	2	no	—
194	C₁₆	#131	1	2	yes	—
195	C₂³	—	4	2	no	—
196	C₂³	C₄×D₈	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	16	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₄×D₈	1	1	yes	characteristic
212	C₂×C₄	C₄×D₈	1	1	yes	characteristic
213	C₂×C₄	C₄×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
216	C₂×C₄	C₂×#135	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	8	2	no	—
223	C₈	—	8	2	no	—
224	C₈	#508	1	4	yes	—
225	C₈	C₂×#131	1	1	yes	characteristic
226	C₈	#568	1	1	yes	characteristic
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	8	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	8	2	no	—
241	C₄	#2935	1	1	yes	characteristic
242	C₄	#2957	1	1	yes	characteristic
243	C₄	C₂×#508	1	2	yes	—
244	C₂	—	4	1	no	—
245	C₂	—	16	2	no	—
246	C₂	#18066	1	1	yes	characteristic
247	C₂	C₄×#508	1	1	yes	characteristic
248	C₂	#33465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#2215	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆ ◁ 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
2. #18069	C₂	semidirect
3. #12082	C₂	semidirect
4. #12072	C₂	semidirect
5. #12069	C₂	semidirect
6. C₈×SD₃₂	C₂	semidirect
7. #12008	C₂	semidirect
8. #11647	C₂	semidirect
9. C₁₆×M₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
#18069	2
#12082	2
#12072	2
#12069	2
C₈×SD₃₂	2
#12008	2
#11647	2
C₁₆×M₁₆	2