Subgroups — #152197

Order 512.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₃₂	C₂	1	2	yes	maximal
4	#7465	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃₂	C₂²	1	4	yes	—
9	C₂×C₄×C₁₆	—	2	4	no	—
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄⋊C₃₂	—	2	4	no	—
15	C₂²⋊C₃₂	—	2	4	no	—
16	C₂³×C₈	C₈	1	1	yes	characteristic
17	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄²	C₈	1	1	yes	characteristic
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	—	2	2	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	C₂²×C₁₆	—	2	2	no	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	D₄	1	2	yes	—
28	C₂²×C₁₆	Q₈	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₈	—	2	2	no	—
31	C₂×C₄×C₈	—	2	2	no	—
32	C₂×C₄×C₈	C₈	1	1	yes	characteristic
33	C₂×C₄×C₈	C₈	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃₂	—	2	4	no	—
37	C₂×C₃₂	—	2	4	no	—
38	C₂×C₃₂	—	2	4	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	—	2	2	no	—
42	C₄×C₁₆	D₄	1	4	yes	—
43	C₂³×C₄	C₁₆	1	1	yes	characteristic
44	C₂³×C₄	C₁₆	1	1	yes	characteristic
45	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	C₄⋊C₄	1	2	yes	—
54	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₈	M₁₆	1	2	yes	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄²	—	2	2	no	—
60	C₂×C₄²	—	2	2	no	—
61	C₂×C₄²	C₁₆	1	1	yes	characteristic
62	C₂×C₄²	C₁₆	1	1	yes	characteristic
63	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
65	C₂×C₁₆	—	2	8	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	—	2	4	no	—
74	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
75	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
76	C₂×C₁₆	M₁₆	1	2	yes	—
77	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
78	C₂×C₁₆	C₂×D₄	1	2	yes	—
79	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₁₆	#41	1	2	yes	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	C₂²⋊C₄	1	2	yes	—
86	C₄×C₈	M₁₆	1	2	yes	—
87	C₃₂	—	4	4	no	—
88	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₃₂	1	4	yes	complemented
99	C₂²×C₄	C₄⋊C₈	1	2	yes	—
100	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
104	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
105	C₂²×C₄	M₃₂	1	2	yes	—
106	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₈	—	2	8	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₂²⋊C₈	1	2	yes	—
119	C₂×C₈	C₄⋊C₈	1	2	yes	—
120	C₂×C₈	M₃₂	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	#117	1	1	yes	characteristic
124	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
127	C₄²	—	2	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂²⋊C₈	1	2	yes	—
132	C₄²	M₃₂	1	2	yes	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	2	2	no	—
135	C₁₆	—	2	2	no	—
136	C₁₆	#121	1	2	yes	—
137	C₁₆	C₂²⋊Q₈	1	2	yes	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	1	no	—
140	C₂³	—	2	1	no	—
141	C₂³	—	2	1	no	—
142	C₂³	C₄⋊C₁₆	1	2	yes	—
143	C₂³	C₂×C₃₂	1	2	yes	—
144	C₂³	M₆₄	1	2	yes	—
145	C₂³	C₂²×C₁₆	1	1	yes	characteristic
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
161	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
162	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
163	C₂×C₄	C₂×C₃₂	1	2	yes	—
164	C₂×C₄	C₂×C₃₂	1	2	yes	—
165	C₂×C₄	M₆₄	1	2	yes	—
166	C₂×C₄	M₆₄	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
169	C₂×C₄	#431	1	1	yes	characteristic
170	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
171	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
172	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	#380	1	1	yes	characteristic
177	C₈	#407	1	1	yes	characteristic
178	C₈	#423	1	1	yes	characteristic
179	C₈	#432	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₃₂	1	4	yes	complemented
187	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
188	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
189	C₂²	#2162	1	1	yes	characteristic
190	C₂²	C₂²×C₃₂	1	1	yes	characteristic
191	C₂²	C₂×M₆₄	1	1	yes	characteristic
192	C₂²	C₂×M₆₄	1	1	yes	characteristic
193	C₂²	C₂×M₆₄	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	C₂²⋊C₃₂	1	4	yes	complemented
199	C₄	#1797	1	1	yes	characteristic
200	C₄	#2112	1	1	yes	characteristic
201	C₄	#2150	1	1	yes	characteristic
202	C₄	#2163	1	1	yes	characteristic
203	C₂	—	2	4	no	—
204	C₂	#12783	1	1	yes	characteristic
205	C₂	C₂×C₂²⋊C₃₂	1	1	yes	characteristic
206	C₂	#13553	1	1	yes	characteristic
207	C₂	C₂×C₄⋊C₃₂	1	1	yes	characteristic
208	C₂	#13627	1	1	yes	characteristic
209	C₂	#13639	1	1	yes	characteristic
210	C₂	#13640	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator	C₂²	C₂²×C₃₂
socle	C₂³	C₂²×C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂×C₄⋊C₃₂	C₂	semidirect
4. #7465	C₂	semidirect
98. C₂²×C₄	C₃₂	semidirect
186. C₂²	C₄⋊C₃₂	semidirect
198. C₄	C₂²⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₆	2
C₂×C₄⋊C₃₂	2
C₂×C₂²⋊C₃₂	2
#7465	2