Subgroups — #129072

Order 512.

272 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#364	C₂	1	1	yes	maximal, characteristic
2	#12590	C₂	1	1	yes	maximal, characteristic
3	#11401	C₂	1	1	yes	maximal, characteristic
4	#11399	C₂	1	1	yes	maximal, characteristic
5	#8266	C₂	1	1	yes	maximal, characteristic
6	#7141	C₂	1	1	yes	maximal, characteristic
7	#7141	C₂	1	1	yes	maximal, characteristic
8	C₂×#364	C₄	1	2	yes	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1564	C₄	1	1	yes	characteristic
15	#1564	C₄	1	1	yes	characteristic
16	#1450	C₂²	1	1	yes	characteristic
17	#1450	C₂²	1	1	yes	characteristic
18	#1376	C₄	1	2	yes	—
19	#1374	C₄	1	2	yes	—
20	#1374	C₄	1	2	yes	—
21	#1312	—	2	1	no	—
22	#1312	—	2	1	no	—
23	#1311	—	2	1	no	—
24	C₈×C₁₆	—	2	1	no	—
25	#1276	C₄	1	1	yes	characteristic
26	#1276	C₄	1	1	yes	characteristic
27	C₂×M₃₂	C₂×C₄	1	2	yes	—
28	C₂×M₃₂	C₂×C₄	1	2	yes	—
29	C₂²×C₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₁₆	C₂×C₄	1	2	yes	—
31	#431	C₂×C₄	1	1	yes	characteristic
32	#431	C₂×C₄	1	1	yes	characteristic
33	#431	C₂×C₄	1	1	yes	characteristic
34	#431	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
41	C₄³	C₂×C₄	1	1	yes	characteristic
42	#364	—	2	4	no	—
43	#346	—	2	1	no	—
44	#346	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	D₄	1	1	yes	characteristic
50	C₄×C₁₆	D₄	1	1	yes	characteristic
51	C₄×C₁₆	Q₈	1	2	yes	—
52	#335	C₈	1	2	yes	—
53	#335	C₈	1	2	yes	—
54	#334	C₈	1	2	yes	—
55	#334	C₈	1	2	yes	—
56	#322	—	2	1	no	—
57	#322	—	2	1	no	—
58	C₈²	—	2	1	no	—
59	C₈²	—	2	1	no	—
60	C₂²×C₈	C₄²	1	1	yes	characteristic
61	C₂²×C₈	C₄²	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄²	—	2	1	no	—
65	C₂×C₄²	—	2	1	no	—
66	C₂×C₄²	C₄²	1	1	yes	characteristic
67	C₂×C₄²	C₄²	1	1	yes	characteristic
68	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
69	M₃₂	—	2	2	no	—
70	M₃₂	—	2	2	no	—
71	M₃₂	—	2	2	no	—
72	M₃₂	M₁₆	1	4	yes	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
81	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
82	C₄⋊C₈	C₂×C₈	1	2	yes	—
83	C₄⋊C₈	C₂×C₈	1	2	yes	—
84	C₄⋊C₈	C₂×C₈	1	2	yes	—
85	C₄⋊C₈	C₂×C₈	1	2	yes	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₂²⋊C₈	—	2	1	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂×C₈	1	1	yes	characteristic
99	C₄×C₈	C₂×C₈	1	1	yes	characteristic
100	C₄×C₈	C₂×C₈	1	1	yes	characteristic
101	C₄×C₈	C₂×C₈	1	1	yes	characteristic
102	C₄×C₈	C₂×D₄	1	1	yes	characteristic
103	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₄×C₈	#41	1	1	yes	characteristic
105	C₄×C₈	#41	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₂²×C₄	#131	1	1	yes	characteristic
110	C₂²×C₄	#131	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₄×C₈	1	1	yes	characteristic
120	C₂×C₈	C₄×C₈	1	1	yes	characteristic
121	C₂×C₈	C₄×C₈	1	1	yes	characteristic
122	C₂×C₈	C₄×C₈	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₈	#117	1	1	yes	characteristic
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
127	C₂×C₈	C₂×M₁₆	1	2	yes	—
128	C₂×C₈	#131	1	2	yes	—
129	C₂×C₈	#131	1	2	yes	—
130	C₂×C₈	#131	1	2	yes	—
131	C₄²	—	2	4	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	4	2	no	—
137	C₄²	C₄×D₄	1	1	yes	characteristic
138	C₄²	C₄×Q₈	1	1	yes	characteristic
139	C₄²	C₂²×C₈	1	1	yes	characteristic
140	C₄²	C₂²×C₈	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	2	4	no	—
144	C₁₆	—	4	2	no	—
145	C₁₆	—	4	2	no	—
146	C₂³	#405	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
158	C₂×C₄	#405	1	1	yes	characteristic
159	C₂×C₄	#405	1	1	yes	characteristic
160	C₂×C₄	#405	1	1	yes	characteristic
161	C₂×C₄	C₈×D₄	1	1	yes	characteristic
162	C₂×C₄	C₈×D₄	1	1	yes	characteristic
163	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
164	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₈	#424	1	2	yes	—
174	C₈	#433	1	2	yes	—
175	C₈	#435	1	2	yes	—
176	C₈	#446	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#1453	1	1	yes	characteristic
180	C₂²	#1454	1	1	yes	characteristic
181	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	C₈×M₁₆	1	1	yes	characteristic
187	C₄	#1450	1	1	yes	characteristic
188	C₄	#1768	1	1	yes	characteristic
189	C₄	#1771	1	1	yes	characteristic
190	C₄	#2170	1	2	yes	—
191	C₄	#2171	1	2	yes	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#8264	1	1	yes	characteristic
195	C₂	#12664	1	1	yes	characteristic
196	C₂	#12665	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×#364	2
#12590	2
#11401	2
#11399	2
#8266	2
#7141	2