Subgroups — #128509

Order 512.

288 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2183	C₂	1	1	yes	maximal, characteristic
2	#11493	C₂	1	1	yes	maximal, characteristic
3	#11408	C₂	1	1	yes	maximal, characteristic
4	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×#101	C₂	1	1	yes	maximal, characteristic
6	#7209	C₂	1	1	yes	maximal, characteristic
7	#7209	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	#2183	—	2	1	no	—
10	#2183	—	2	1	no	—
11	#2183	—	2	1	no	—
12	#2183	—	2	1	no	—
13	#2162	C₂²	1	1	yes	characteristic
14	#2162	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#346	C₄	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₄×#101	C₄	1	1	yes	characteristic
20	#1461	C₄	1	1	yes	characteristic
21	C₈×M₁₆	C₂²	1	1	yes	characteristic
22	C₈×M₁₆	C₂²	1	1	yes	characteristic
23	#1322	C₄	1	4	yes	—
24	#1312	—	2	1	no	—
25	#1312	—	2	1	no	—
26	#1311	—	2	1	no	—
27	C₈×C₁₆	—	2	1	no	—
28	#1283	C₄	1	1	yes	characteristic
29	#1283	C₄	1	1	yes	characteristic
30	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₈×Q₈	—	2	1	no	—
35	C₈×Q₈	—	2	1	no	—
36	C₈×Q₈	—	2	1	no	—
37	C₈×Q₈	—	2	1	no	—
38	#431	C₂×C₄	1	1	yes	characteristic
39	#431	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×#101	C₈	1	2	yes	—
43	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄⋊C₁₆	—	2	2	no	—
48	C₄⋊C₁₆	—	2	2	no	—
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	—	2	1	no	—
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	—	2	1	no	—
53	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
54	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
55	#346	—	2	2	no	—
56	#346	—	2	2	no	—
57	C₄×C₁₆	—	2	1	no	—
58	C₄×C₁₆	—	2	1	no	—
59	C₄×C₁₆	D₄	1	1	yes	characteristic
60	C₄×C₁₆	D₄	1	1	yes	characteristic
61	C₄×C₁₆	D₄	1	1	yes	characteristic
62	C₄×C₁₆	D₄	1	1	yes	characteristic
63	#343	C₈	1	2	yes	—
64	#324	C₈	1	4	yes	—
65	#322	—	2	1	no	—
66	#322	—	2	1	no	—
67	C₈²	—	2	1	no	—
68	C₈²	—	2	1	no	—
69	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
70	C₂×M₁₆	C₂×C₈	1	2	yes	—
71	C₂×M₁₆	C₂×C₈	1	2	yes	—
72	C₂²×C₈	C₄²	1	2	yes	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	1	no	—
76	C₄×Q₈	—	2	1	no	—
77	C₄×Q₈	—	2	1	no	—
78	C₄×Q₈	—	2	1	no	—
79	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄²	C₄²	1	2	yes	—
82	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	4	1	no	—
88	C₂×C₁₆	—	4	2	no	—
89	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	#101	—	2	4	no	—
100	C₂²⋊C₈	C₂×C₈	1	2	yes	—
101	C₂²⋊C₈	C₂×C₈	1	2	yes	—
102	#97	—	2	1	no	—
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂²×C₄	C₄×C₈	1	2	yes	—
119	C₂²×C₄	C₄×C₈	1	2	yes	—
120	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
123	M₁₆	—	2	2	no	—
124	M₁₆	—	2	2	no	—
125	M₁₆	—	2	2	no	—
126	M₁₆	M₃₂	1	4	yes	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₈	#117	1	1	yes	characteristic
138	C₂×C₈	C₄×D₄	1	1	yes	characteristic
139	C₂×C₈	C₄×D₄	1	1	yes	characteristic
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₄×D₄	1	1	yes	characteristic
147	C₄²	C₄×D₄	1	1	yes	characteristic
148	C₄²	#131	1	1	yes	characteristic
149	C₄²	#131	1	1	yes	characteristic
150	C₁₆	—	2	1	no	—
151	C₁₆	—	2	1	no	—
152	C₁₆	—	2	1	no	—
153	C₁₆	—	4	2	no	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	4	2	no	—
156	C₁₆	#101	1	2	yes	complemented
157	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
158	Q₈	—	4	2	no	—
159	Q₈	—	4	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#405	1	1	yes	characteristic
170	C₂×C₄	C₈×D₄	1	1	yes	characteristic
171	C₂×C₄	C₈×D₄	1	1	yes	characteristic
172	C₂×C₄	C₈×D₄	1	1	yes	characteristic
173	C₂×C₄	C₈×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂×M₃₂	1	2	yes	—
175	C₂×C₄	#504	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	2	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₈	C₂×#101	1	1	yes	characteristic
185	C₈	#413	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²	C₄×M₃₂	1	1	yes	characteristic
190	C₂²	#2108	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	C₄×#101	1	1	yes	characteristic
197	C₄	#1757	1	1	yes	characteristic
198	C₄	#2168	1	2	yes	—
199	C₄	#2169	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	C₈×#101	1	1	yes	characteristic
203	C₂	#12612	1	1	yes	characteristic
204	C₂	#12613	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator, prime radical	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
156. C₁₆	#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2183	2
#11493	2
#11408	2
C₁₆×M₁₆	2
C₈×#101	2
#7209	2