Finite Group Databasebeta

← #124331

Subgroups — #124331

Order 512.

284 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21341C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic
3#12653C211yesmaximal, characteristic
4#7128C212yesmaximal
5#7118C212yesmaximal
6#2295C411yescharacteristic
7C2×C4⋊C16—21no—
8C2×C4⋊C16—21no—
9C2×#346—21no—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1939C411yescharacteristic
13#1939C2211yescharacteristic
14#1916—21no—
15#1916—21no—
16#1768—21no—
17#1767C411yescharacteristic
18#1746—21no—
19C42×C8C2211yescharacteristic
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×Q8C2×C411yescharacteristic
24C22×C16—22no—
25C22×C16—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#97—21no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8D411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8C2311yescharacteristic
37#391C811yescharacteristic
38#391C811yescharacteristic
39#389C811yescharacteristic
40#389C811yescharacteristic
41#387—21no—
42#387—21no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4C2×C411yescharacteristic
45#376—21no—
46C43C2×C411yescharacteristic
47C4⋊C16—24no—
48C4⋊C16—42no—
49#346—42no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57C22×Q8C2×C811yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4×Q8—41no—
65C4×Q8—41no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C2×C811yescharacteristic
77C2×C42C2×C811yescharacteristic
78C2×C42M1611yescharacteristic
79C2×C42M1611yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C16—24no—
82C2×C16—42no—
83C2×C16—42no—
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92#97—42no—
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—22no—
96C4×C8—41no—
97C4×C8—41no—
98C4×C8—41no—
99C4×C8SD1612yes—
100C4×C8Q1612yes—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108C2×Q8—22no—
109C2×Q8—41no—
110C2×Q8—41no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4C22⋊C811yescharacteristic
116C22×C4C22⋊C811yescharacteristic
117C22×C4C22⋊C811yescharacteristic
118C22×C4C22⋊C811yescharacteristic
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C22×C811yescharacteristic
121C22×C4C2×M1611yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8#10312yes—
138C2×C8C2×SD1611yescharacteristic
139C2×C8C2×Q1611yescharacteristic
140C2×C8AGL(1,ℤ/8)11yescharacteristic
141C2×C8#13711yescharacteristic
142C4⋊C4—22no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—22no—
153C42—41no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—41no—
158C42—41no—
159C42—41no—
160C42#10312yes—
161C42#10412yes—
162C16—44no—
163C16—82no—
164C23C2×C22⋊C811yescharacteristic
165Q8—82no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4Q8⋊C812yes—
194C2×C4Q8⋊C812yes—
195C2×C4#41011yescharacteristic
196C2×C4#41311yescharacteristic
197C2×C4C2×#10311yescharacteristic
198C2×C4#41711yescharacteristic
199C2×C4C2×#10411yescharacteristic
200C2×C4#42111yescharacteristic
201C2×C4#50412yes—
202C8—22no—
203C8—21no—
204C8—21no—
205C8—21no—
206C8—21no—
207C8—41no—
208C8—81no—
209C8—81no—
210C22#145811yescharacteristic
211C22C2×Q8⋊C811yescharacteristic
212C22#147711yescharacteristic
213C22#211311yescharacteristic
214C22#211511yescharacteristic
215C22#211711yescharacteristic
216C22#211711yescharacteristic
217C4—22no—
218C4—22no—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—81no—
224C4—81no—
225C4—81no—
226C4—81no—
227C4#150911yescharacteristic
228C4#151811yescharacteristic
229C4#152911yescharacteristic
230C4#155011yescharacteristic
231C2#757711yescharacteristic
232C2#1146111yescharacteristic
233C2#1146311yescharacteristic
234C2#1160611yescharacteristic
235C2#1161211yescharacteristic
236C2#1162411yescharacteristic
237C2#1162711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socle, prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213412
#128792
#126532
#71282
#71182