Subgroups — #483365

Order 512.

506 conjugacy classes of subgroups.

The subgroup lattice has 209 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	2	yes	maximal
2	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#12017	C₂	1	8	yes	maximal, complemented
4	#12012	C₂	1	4	yes	maximal, complemented
5	C₂×#431	C₄	1	1	yes	characteristic
6	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2163	—	2	2	no	—
9	#2163	C₂²	1	4	yes	—
10	#2162	—	2	2	no	—
11	#2162	C₂²	1	4	yes	—
12	#2150	C₂²	1	4	yes	—
13	#2112	—	2	2	no	—
14	C₄×M₃₂	—	2	2	no	—
15	C₈×M₁₆	C₄	1	4	yes	—
16	C₈×M₁₆	C₂²	1	4	yes	—
17	C₂×#322	C₄	1	2	yes	—
18	C₂×C₈²	C₂²	1	2	yes	—
19	#1370	C₂²	1	8	yes	complemented
20	#1367	C₂²	1	8	yes	complemented
21	C₂²×M₁₆	C₈	1	2	yes	—
22	C₂³×C₈	D₄	1	1	yes	characteristic
23	C₂³×C₈	Q₈	1	1	yes	characteristic
24	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	4	no	—
26	#431	—	2	2	no	—
27	#431	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₂²⋊C₈	—	2	1	no	—
30	C₄×M₁₆	C₈	1	4	yes	—
31	C₄×M₁₆	C₂×C₄	1	4	yes	—
32	C₂×#97	C₈	1	1	yes	characteristic
33	C₂×#97	C₈	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈	Q₈	1	2	yes	—
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	4	no	—
45	C₄⋊C₁₆	C₂³	1	8	yes	—
46	C₂²⋊C₁₆	—	2	4	no	—
47	C₂²⋊C₁₆	—	2	4	no	—
48	#346	—	2	4	no	—
49	C₄×C₁₆	—	2	4	no	—
50	#322	C₂×C₄	1	8	yes	—
51	C₈²	C₂×C₄	1	4	yes	—
52	C₈²	C₂³	1	4	yes	—
53	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂³×C₄	M₁₆	1	1	yes	characteristic
56	C₂×M₁₆	—	2	8	no	—
57	C₂×M₁₆	C₂×C₈	1	8	yes	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	—	4	2	no	—
61	C₂²×C₈	C₄⋊C₄	1	2	yes	—
62	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₄⋊C₄	1	2	yes	—
64	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×C₈	1	2	yes	—
66	C₂²×C₈	C₂×C₈	1	2	yes	—
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	2	yes	—
69	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×Q₈	1	2	yes	—
71	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄²	C₄⋊C₄	1	2	yes	—
73	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄²	C₂×C₈	1	2	yes	—
75	C₂×C₄²	M₁₆	1	1	yes	characteristic
76	C₂×C₄²	M₁₆	1	2	yes	—
77	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄²	C₂²×C₄	1	2	yes	—
79	M₃₂	—	4	8	no	—
80	C₂×C₁₆	—	2	8	no	—
81	C₄⋊C₈	—	2	4	no	—
82	C₂²⋊C₈	—	2	4	no	—
83	#97	C₂×C₈	1	4	yes	—
84	#97	C₂×C₈	1	4	yes	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	C₄⋊C₄	1	2	yes	—
87	C₄×C₈	C₄⋊C₄	1	2	yes	—
88	C₄×C₈	C₄⋊C₄	1	2	yes	—
89	C₄×C₈	C₂×C₈	1	4	yes	—
90	C₄×C₈	C₂²×C₄	1	4	yes	—
91	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²×C₄	1	2	yes	—
93	C₄×C₈	C₂²×C₄	1	4	yes	—
94	C₄×C₈	C₂×D₄	1	2	yes	—
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	C₂×Q₈	1	2	yes	—
97	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×C₈	C₂⁴	1	1	yes	characteristic
99	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	4	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	4	2	no	—
104	C₂²×C₄	C₄⋊C₈	1	2	yes	—
105	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	C₄⋊C₈	1	2	yes	—
107	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊C₈	1	2	yes	—
109	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
111	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₈	1	2	yes	—
113	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
114	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×M₁₆	1	2	yes	—
116	M₁₆	—	2	16	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	M₃₂	1	8	yes	complemented
129	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₈	1	4	yes	—
134	C₂×C₈	C₂²×C₈	1	4	yes	—
135	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
137	C₄²	C₄⋊C₈	1	2	yes	—
138	C₄²	C₄⋊C₈	1	2	yes	—
139	C₄²	C₄⋊C₈	1	2	yes	—
140	C₄²	C₂×C₄⋊C₄	1	2	yes	—
141	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₄²	C₂²×C₈	1	1	yes	characteristic
143	C₄²	C₂²×C₈	1	2	yes	—
144	C₄²	C₂×M₁₆	1	1	yes	characteristic
145	C₄²	C₂×M₁₆	1	2	yes	—
146	C₄²	C₂³×C₄	1	1	yes	characteristic
147	C₁₆	—	4	8	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	4	1	no	—
150	C₂³	—	4	1	no	—
151	C₂³	C₂×C₄⋊C₈	1	2	yes	—
152	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
167	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
168	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×M₃₂	1	4	yes	—
173	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
175	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	2	2	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	4	no	—
182	C₈	#430	1	2	yes	—
183	C₈	C₂×M₃₂	1	8	yes	—
184	C₈	#575	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	#2150	1	2	yes	—
192	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	2	no	—
198	C₄	#2150	1	2	yes	—
199	C₄	#2153	1	2	yes	—
200	C₄	#2911	1	1	yes	characteristic
201	C₄	C₂×#430	1	1	yes	characteristic
202	C₄	C₂²×M₃₂	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	#17787	1	1	yes	characteristic
206	C₂	C₂×#2150	1	1	yes	characteristic
207	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×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
3. #12017	C₂	semidirect
4. #12012	C₂	semidirect
19. #1370	C₂²	semidirect
20. #1367	C₂²	semidirect
128. C₂×C₈	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂×C₈×M₁₆	2
#12017	2
#12012	2