Subgroups — #134709

Order 512.

364 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#12012	C₂	1	2	yes	maximal, complemented
3	#11459	C₂	1	2	yes	maximal
4	#7145	C₂	1	2	yes	maximal, complemented
5	C₂×#431	C₂²	1	1	yes	characteristic
6	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
8	#2162	C₂²	1	4	yes	—
9	C₈×M₁₆	—	2	2	no	—
10	C₈×M₁₆	—	2	2	no	—
11	C₂×#322	C₄	1	1	yes	characteristic
12	C₂×#322	C₂²	1	1	yes	characteristic
13	C₂×C₈²	C₄	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1370	—	2	2	no	—
16	#1367	—	2	2	no	—
17	#1322	—	2	2	no	—
18	#1314	—	2	2	no	—
19	C₂²×M₁₆	C₈	1	1	yes	characteristic
20	C₂²×M₁₆	C₈	1	1	yes	characteristic
21	C₂³×C₈	—	2	1	no	—
22	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
23	#431	—	2	2	no	—
24	#431	—	2	2	no	—
25	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₄⋊C₈	Q₈	1	1	yes	characteristic
27	C₂×C₂²⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₂²⋊C₈	Q₈	1	1	yes	characteristic
29	C₄×M₁₆	—	2	2	no	—
30	C₄×M₁₆	—	2	2	no	—
31	C₂×#97	C₈	1	1	yes	characteristic
32	C₂×#97	C₈	1	1	yes	characteristic
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	—	2	2	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	4	no	—
44	C₂²⋊C₁₆	—	2	4	no	—
45	C₄×C₁₆	—	2	4	no	—
46	#322	—	2	2	no	—
47	#322	—	2	2	no	—
48	C₈²	—	2	1	no	—
49	C₈²	—	2	1	no	—
50	C₈²	D₄	1	2	yes	—
51	C₈²	D₄	1	2	yes	—
52	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂³×C₄	M₁₆	1	1	yes	characteristic
55	C₂×M₁₆	—	2	2	no	—
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	—	2	2	no	—
58	C₂×M₁₆	—	2	2	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	C₁₆	1	4	yes	complemented
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	2	no	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	4	4	no	—
66	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	—	2	1	no	—
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄²	M₁₆	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₁₆	—	4	4	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	2	2	no	—
86	C₂²⋊C₈	—	2	2	no	—
87	#97	—	2	2	no	—
88	#97	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₈	C₂²⋊C₄	1	2	yes	—
100	C₄×C₈	M₁₆	1	2	yes	—
101	C₄×C₈	M₁₆	1	2	yes	—
102	C₄×C₈	C₂×D₄	1	2	yes	—
103	C₄×C₈	#41	1	2	yes	—
104	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	4	2	no	—
114	C₂²×C₄	—	4	1	no	—
115	C₂²×C₄	—	4	1	no	—
116	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₁₆	1	2	yes	—
120	C₂²×C₄	M₃₂	1	2	yes	—
121	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
124	M₁₆	—	2	8	no	—
125	M₁₆	—	2	8	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	C₂²⋊C₈	1	2	yes	—
146	C₂×C₈	C₂²⋊C₈	1	2	yes	—
147	C₂×C₈	C₂×C₁₆	1	2	yes	—
148	C₂×C₈	C₂×C₁₆	1	2	yes	—
149	C₂×C₈	M₃₂	1	2	yes	—
150	C₂×C₈	M₃₂	1	2	yes	—
151	C₂×C₈	#121	1	2	yes	—
152	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₄²	#117	1	1	yes	characteristic
161	C₄²	C₂×M₁₆	1	1	yes	characteristic
162	C₄²	C₂×M₁₆	1	1	yes	characteristic
163	C₁₆	—	8	4	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	4	1	no	—
167	C₂³	—	4	1	no	—
168	C₂³	C₄⋊C₁₆	1	2	yes	—
169	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
189	C₂×C₄	#380	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
191	C₂×C₄	#407	1	1	yes	characteristic
192	C₂×C₄	#423	1	1	yes	characteristic
193	C₂×C₄	#431	1	1	yes	characteristic
194	C₂×C₄	#432	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
196	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
197	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
198	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	1	no	—
208	C₈	—	4	1	no	—
209	C₈	C₂²⋊C₁₆	1	4	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	#1797	1	1	yes	characteristic
218	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
219	C₂²	#2150	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	#1812	1	1	yes	characteristic
226	C₄	#1813	1	1	yes	characteristic
227	C₄	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
228	C₄	#2112	1	1	yes	characteristic
229	C₄	#2162	1	1	yes	characteristic
230	C₄	#2163	1	1	yes	characteristic
231	C₄	#2165	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	C₂	#8602	1	1	yes	characteristic
236	C₂	#12783	1	1	yes	characteristic
237	C₂	#12791	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #12012	C₂	semidirect
4. #7145	C₂	semidirect
60. C₂×M₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×M₁₆	2
#12012	2
#11459	2
#7145	2