Finite Group Databasebeta

← #122239

Subgroups — #122239

Order 512.

395 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 80 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#21820C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic
3#11459C211yesmaximal, characteristic
4#7055C214yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#432C411yescharacteristic
7C2×#431C2211yescharacteristic
8#2163—21no—
9#2162—21no—
10#1985—21no—
11#1985—21no—
12#1984—21no—
13#1984—21no—
14#1817C412yes—
15#1817C2212yes—
16#1322—21no—
17#1322—21no—
18#1314C2212yes—
19#1314C2212yes—
20C23×Q8C811yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×Q8—21no—
23C2×C4×Q8—21no—
24C2×C4×Q8—21no—
25C2×C4×Q8—41no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8—41no—
29C22×C4⋊C4C812yes—
30C22×C4⋊C4C811yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33#432—41no—
34#432—41no—
35#431—41no—
36#431—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8D411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8C2311yescharacteristic
43C2×#97—21no—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C4⋊C16—41no—
47C4⋊C16—41no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50#346—41no—
51C4×C16—41no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56C22×Q8—21no—
57C22×Q8—21no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8—44no—
61C23×C4C2×C811yescharacteristic
62C23×C4C2×C812yes—
63C23×C4C2×C811yescharacteristic
64C23×C4C2×C812yes—
65C23×C4C22×C411yescharacteristic
66C22×C8—22no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C4×Q8—42no—
75C4×Q8—42no—
76C4×Q8—42no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—42no—
96C2×C42—42no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42M1611yescharacteristic
100C2×C42M1611yescharacteristic
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109C22⋊C8—42no—
110#97—42no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C24C22×C811yescharacteristic
116C2×Q8—24no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—44no—
120C2×Q8—44no—
121C2×Q8—48no—
122C2×Q8—44no—
123C2×Q8—41no—
124C2×Q8—41no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C22⋊C811yescharacteristic
146C22×C4C22⋊C811yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C4⋊C4—44no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—44no—
173C42—42no—
174C42—42no—
175C42#9911yescharacteristic
176C42#9911yescharacteristic
177C42#10011yescharacteristic
178C42#10011yescharacteristic
179C16—82no—
180C16—82no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×C22⋊C811yescharacteristic
186C23#50412yes—
187Q8—44no—
188Q8—48no—
189Q8—44no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—84no—
218C2×C4#32311yescharacteristic
219C2×C4#32311yescharacteristic
220C2×C4#32311yescharacteristic
221C2×C4#32311yescharacteristic
222C2×C4C2×#9911yescharacteristic
223C2×C4C2×#10011yescharacteristic
224C8—41no—
225C8—41no—
226C8—41no—
227C8—41no—
228C8—82no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22C2×#32311yescharacteristic
237C22#211311yescharacteristic
238C22#211511yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—44no—
246C4—41no—
247C4—82no—
248C4#132012yes—
249C4#212311yescharacteristic
250C4#212311yescharacteristic
251C2—21no—
252C2—21no—
253C2—42no—
254C2#1141911yescharacteristic
255C2#1142811yescharacteristic
256C2C2×#132011yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutator, prime radicalC24C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#323
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#218202
#114602
#114592
#70552