Finite Group Databasebeta

← #124413

Subgroups — #124413

Order 512.

286 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 98 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#21673C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic
3C4×C4⋊C16C211yesmaximal, characteristic
4#7118C212yesmaximal
5#7118C212yesmaximal
6C4×#125C411yescharacteristic
7C2×C4⋊C16—21no—
8C2×C4×C16—21no—
9#1939C2211yescharacteristic
10#1939C2211yescharacteristic
11#1916—21no—
12#1916—21no—
13#1831C411yescharacteristic
14#1771—21no—
15#1771—21no—
16#1770C411yescharacteristic
17#1770C411yescharacteristic
18C42×C8C2211yescharacteristic
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×#125C811yescharacteristic
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×#97—21no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#389C811yescharacteristic
39#389C811yescharacteristic
40#387—21no—
41#387—21no—
42#382C811yescharacteristic
43C4×C4⋊C4—21no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2×C411yescharacteristic
46C4×C4⋊C4C2×C411yescharacteristic
47C43C2×C411yescharacteristic
48C4⋊C16—24no—
49C4×C16—42no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62#125—42no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×C811yescharacteristic
68C2×C4⋊C4C2×C811yescharacteristic
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42C2×C811yescharacteristic
76C2×C42C2×C811yescharacteristic
77C2×C42M1611yescharacteristic
78C2×C42M1611yescharacteristic
79C2×C42C22×C411yescharacteristic
80C2×C16—24no—
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84C2×C16—42no—
85C2×C16—42no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—42no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91#97—42no—
92#97—42no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—22no—
96C4×C8—21no—
97C4×C8—21no—
98C4×C8—41no—
99C4×C8—41no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4C22⋊C811yescharacteristic
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C22⋊C811yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C22×C811yescharacteristic
116C22×C4C2×M1611yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8#10112yes—
132C2×C8#10412yes—
133C2×C8#13512yes—
134C2×C8AGL(1,ℤ/8)11yescharacteristic
135C2×C8#13711yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C42—22no—
145C42—22no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—22no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—41no—
158C16—44no—
159C16—82no—
160C23C2×C22⋊C811yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
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—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4#34912yes—
188C2×C4#35012yes—
189C2×C4#35012yes—
190C2×C4#41011yescharacteristic
191C2×C4C2×#10111yescharacteristic
192C2×C4#41611yescharacteristic
193C2×C4#41711yescharacteristic
194C2×C4C2×#10411yescharacteristic
195C2×C4#42111yescharacteristic
196C2×C4#50412yes—
197C8—22no—
198C8—21no—
199C8—21no—
200C8—22no—
201C8—22no—
202C8—81no—
203C8—81no—
204C22#146111yescharacteristic
205C22#147611yescharacteristic
206C22#147711yescharacteristic
207C22#211311yescharacteristic
208C22C2×#34911yescharacteristic
209C22C2×#35011yescharacteristic
210C22C2×#35011yescharacteristic
211C4—22no—
212C4—22no—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4#150911yescharacteristic
222C4#152111yescharacteristic
223C4#152811yescharacteristic
224C4#155211yescharacteristic
225C2#758511yescharacteristic
226C2#1149511yescharacteristic
227C2#1149611yescharacteristic
228C2#1161611yescharacteristic
229C2#1161711yescharacteristic
230C2#1162211yescharacteristic
231C2#1162611yescharacteristic
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
#216732
#128942
C4×C4⋊C162
#71182