Subgroups — #129074

Order 512.

250 conjugacy classes of subgroups.

The subgroup lattice has 193 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	#12591	C₂	1	1	yes	maximal, characteristic
3	#11406	C₂	1	1	yes	maximal, characteristic
4	#11405	C₂	1	1	yes	maximal, characteristic
5	#8266	C₂	1	1	yes	maximal, characteristic
6	#7140	C₂	1	1	yes	maximal, characteristic
7	#7140	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×#346	C₂²	1	1	yes	characteristic
12	C₂×#346	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	#1313	—	2	2	no	—
22	#1313	—	2	2	no	—
23	#1276	C₄	1	1	yes	characteristic
24	#1276	C₄	1	1	yes	characteristic
25	C₂×M₃₂	C₂×C₄	1	2	yes	—
26	C₂×M₃₂	C₂×C₄	1	2	yes	—
27	C₂²×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₁₆	C₂×C₄	1	2	yes	—
29	#431	C₂×C₄	1	1	yes	characteristic
30	#431	C₂×C₄	1	1	yes	characteristic
31	#431	C₂×C₄	1	1	yes	characteristic
32	#431	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	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×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄³	C₂×C₄	1	1	yes	characteristic
40	#364	—	2	4	no	—
41	#346	—	2	1	no	—
42	#346	—	2	1	no	—
43	#346	—	2	1	no	—
44	#346	—	2	1	no	—
45	#346	D₄	1	2	yes	—
46	#346	Q₈	1	2	yes	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	#335	—	2	1	no	—
50	#335	—	2	1	no	—
51	#334	—	2	1	no	—
52	#334	—	2	1	no	—
53	#322	—	2	1	no	—
54	#322	—	2	1	no	—
55	C₈²	—	2	1	no	—
56	C₈²	—	2	1	no	—
57	C₂²×C₈	C₄²	1	1	yes	characteristic
58	C₂²×C₈	C₄²	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄²	—	2	1	no	—
62	C₂×C₄²	—	2	1	no	—
63	C₂×C₄²	C₄²	1	1	yes	characteristic
64	C₂×C₄²	C₄²	1	1	yes	characteristic
65	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
66	M₃₂	—	2	2	no	—
67	M₃₂	—	2	2	no	—
68	M₃₂	—	2	2	no	—
69	M₃₂	M₁₆	1	4	yes	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
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	C₄⋊C₄	1	4	yes	—
78	C₄⋊C₈	—	2	1	no	—
79	C₄⋊C₈	—	2	1	no	—
80	C₄⋊C₈	—	2	1	no	—
81	C₄⋊C₈	—	2	1	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	2	1	no	—
85	C₂²⋊C₈	—	2	1	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	M₁₆	1	1	yes	characteristic
95	C₄×C₈	M₁₆	1	1	yes	characteristic
96	C₄×C₈	M₁₆	1	1	yes	characteristic
97	C₄×C₈	M₁₆	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
105	C₂²×C₄	#131	1	1	yes	characteristic
106	C₂²×C₄	#131	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	#97	1	1	yes	characteristic
116	C₂×C₈	#97	1	1	yes	characteristic
117	C₂×C₈	#97	1	1	yes	characteristic
118	C₂×C₈	#97	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈	#117	1	1	yes	characteristic
121	C₂×C₈	C₄×D₄	1	1	yes	characteristic
122	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
123	C₂×C₈	C₂×M₁₆	1	2	yes	—
124	C₂×C₈	#131	1	2	yes	—
125	C₂×C₈	#131	1	2	yes	—
126	C₂×C₈	#131	1	2	yes	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	4	2	no	—
132	C₄²	—	4	2	no	—
133	C₄²	C₄×D₄	1	1	yes	characteristic
134	C₄²	C₄×Q₈	1	1	yes	characteristic
135	C₄²	C₂×M₁₆	1	1	yes	characteristic
136	C₄²	C₂×M₁₆	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₁₆	—	2	4	no	—
139	C₁₆	—	4	2	no	—
140	C₁₆	—	4	2	no	—
141	C₂³	#405	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#97	1	1	yes	characteristic
152	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
153	C₂×C₄	#405	1	1	yes	characteristic
154	C₂×C₄	#405	1	1	yes	characteristic
155	C₂×C₄	#405	1	1	yes	characteristic
156	C₂×C₄	#436	1	1	yes	characteristic
157	C₂×C₄	#436	1	1	yes	characteristic
158	C₂×C₄	#446	1	1	yes	characteristic
159	C₂×C₄	#446	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	8	1	no	—
165	C₈	—	8	1	no	—
166	C₈	—	8	1	no	—
167	C₈	—	8	1	no	—
168	C₈	#424	1	2	yes	—
169	C₈	#433	1	2	yes	—
170	C₈	#435	1	2	yes	—
171	C₈	#446	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#1453	1	1	yes	characteristic
175	C₂²	#1454	1	1	yes	characteristic
176	C₂²	#1770	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	#1451	1	1	yes	characteristic
184	C₄	#1452	1	1	yes	characteristic
185	C₄	#1768	1	1	yes	characteristic
186	C₄	#1771	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#8265	1	1	yes	characteristic
190	C₂	#12667	1	1	yes	characteristic
191	C₂	#12667	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂³	#405
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ 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
#12591	2
#11406	2
#11405	2
#8266	2
#7140	2