Subgroups — #152201

Order 512.

308 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#504	C₂	1	1	yes	maximal, characteristic
2	#13627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1401	C₂	1	2	yes	maximal
5	#7470	C₂	1	2	yes	maximal, complemented
6	C₂×#504	C₄	1	1	yes	characteristic
7	C₂×#504	C₂²	1	1	yes	characteristic
8	C₄×#131	C₄	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	2	yes	—
10	C₂²×C₃₂	C₂²	1	2	yes	—
11	#2108	—	2	1	no	—
12	#2108	—	2	1	no	—
13	#2108	C₄	1	1	yes	characteristic
14	#2108	C₂²	1	1	yes	characteristic
15	C₄×M₃₂	—	2	1	no	—
16	C₄×M₃₂	C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₆	—	2	1	no	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₄⋊C₃₂	—	2	1	no	—
20	C₄⋊C₃₂	—	2	1	no	—
21	C₄⋊C₃₂	—	2	1	no	—
22	C₄⋊C₃₂	—	2	1	no	—
23	#1401	—	2	4	no	—
24	C₂×#131	C₈	1	1	yes	characteristic
25	C₂×#131	C₂×C₄	1	1	yes	characteristic
26	C₄×#41	C₈	1	1	yes	characteristic
27	#504	—	2	2	no	—
28	#504	—	2	2	no	—
29	#504	—	2	1	no	—
30	#504	—	2	1	no	—
31	#504	D₄	1	2	yes	—
32	#504	Q₈	1	2	yes	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
36	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	—	2	1	no	—
39	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	#405	—	2	1	no	—
42	#405	—	2	1	no	—
43	#405	C₈	1	1	yes	characteristic
44	#405	C₂×C₄	1	1	yes	characteristic
45	C₄×M₁₆	—	2	1	no	—
46	C₄×M₁₆	C₈	1	1	yes	characteristic
47	C₂×C₄×C₈	—	2	1	no	—
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	M₆₄	—	2	2	no	—
50	M₆₄	—	2	2	no	—
51	C₂×C₃₂	—	2	2	no	—
52	C₂×C₃₂	—	2	2	no	—
53	C₂×C₃₂	—	2	2	no	—
54	C₂×C₃₂	—	2	2	no	—
55	#346	—	2	1	no	—
56	#346	—	2	1	no	—
57	#346	—	2	1	no	—
58	C₄×C₁₆	—	2	1	no	—
59	C₄×C₁₆	—	2	1	no	—
60	C₄×C₁₆	—	2	1	no	—
61	C₄×C₁₆	D₄	1	1	yes	characteristic
62	C₄×C₁₆	D₄	1	1	yes	characteristic
63	C₄×C₁₆	D₄	1	1	yes	characteristic
64	C₄×C₁₆	D₄	1	1	yes	characteristic
65	C₂×#41	C₂×C₈	1	1	yes	characteristic
66	#131	—	2	2	no	—
67	#131	—	2	2	no	—
68	#131	—	2	1	no	—
69	#131	—	2	1	no	—
70	#131	C₄⋊C₄	1	2	yes	—
71	#131	M₁₆	1	2	yes	—
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
75	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₄×Q₈	C₁₆	1	1	yes	characteristic
81	C₄×Q₈	C₁₆	1	1	yes	characteristic
82	C₄×D₄	—	2	1	no	—
83	C₄×D₄	—	2	1	no	—
84	C₄×D₄	C₁₆	1	1	yes	characteristic
85	C₄×D₄	C₁₆	1	1	yes	characteristic
86	#117	—	2	1	no	—
87	#117	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
90	M₃₂	—	2	4	no	—
91	M₃₂	—	2	1	no	—
92	M₃₂	—	2	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	2	no	—
95	M₃₂	C₄⋊C₄	1	2	yes	—
96	M₃₂	M₁₆	1	2	yes	—
97	C₂×C₁₆	—	2	4	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
107	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
108	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
109	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₁₆	#41	1	1	yes	characteristic
111	C₂×C₁₆	#41	1	1	yes	characteristic
112	#97	—	2	1	no	—
113	#97	—	2	1	no	—
114	#97	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₈	M₁₆	1	1	yes	characteristic
121	C₄×C₈	M₁₆	1	1	yes	characteristic
122	C₃₂	—	4	2	no	—
123	C₃₂	—	4	2	no	—
124	#41	—	2	1	no	—
125	#41	—	2	1	no	—
126	#41	C₄⋊C₈	1	2	yes	—
127	#41	M₃₂	1	2	yes	—
128	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	M₃₂	1	1	yes	characteristic
135	C₂²×C₄	M₃₂	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	—	2	1	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	—	2	2	no	—
142	M₁₆	C₄⋊C₈	1	2	yes	—
143	M₁₆	M₃₂	1	2	yes	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₈	#117	1	1	yes	characteristic
156	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
158	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
162	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	C₂²⋊C₈	1	1	yes	characteristic
170	C₄²	C₂²⋊C₈	1	1	yes	characteristic
171	C₄²	C₂×C₁₆	1	1	yes	characteristic
172	C₄²	C₂×C₁₆	1	1	yes	characteristic
173	C₁₆	—	2	2	no	—
174	C₁₆	—	2	1	no	—
175	C₁₆	—	2	2	no	—
176	C₁₆	—	2	1	no	—
177	C₁₆	#121	1	1	yes	characteristic
178	C₁₆	#121	1	1	yes	characteristic
179	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
180	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂³	—	2	1	no	—
182	C₂³	C₂×M₃₂	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₄⋊C₁₆	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₄⋊C₁₆	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
199	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
200	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
204	C₂×C₄	#431	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
206	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
207	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	#380	1	1	yes	characteristic
213	C₈	#407	1	1	yes	characteristic
214	C₈	#423	1	1	yes	characteristic
215	C₈	#432	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
222	C₂²	#2150	1	1	yes	characteristic
223	C₂²	#2163	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	#1401	1	2	yes	complemented
231	C₄	#1797	1	1	yes	characteristic
232	C₄	#2112	1	1	yes	characteristic
233	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
234	C₄	#2162	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	—	4	2	no	—
238	C₂	#12783	1	1	yes	characteristic
239	C₂	C₂×#1401	1	1	yes	characteristic
240	C₂	#13558	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13627	C₂	semidirect
3. C₂×C₄⋊C₃₂	C₂	semidirect
5. #7470	C₂	semidirect
230. C₄	#1401	semidirect

Maximal subgroups

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

subgroup	index
C₄×#504	2
#13627	2
C₂×C₄⋊C₃₂	2
C₂×#1401	2
#7470	2