Finite Group Databasebeta

← #135235

Subgroups — #135235

Order 512.

323 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 78 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#20754C211yesmaximal, characteristic
2#8730C211yesmaximal, characteristic
3#8730C211yesmaximal, characteristic
4#7190C211yesmaximal, characteristic
5#7190C211yesmaximal, characteristic
6#7188C211yesmaximal, characteristic
7#7188C211yesmaximal, characteristic
8C2×#424C411yescharacteristic
9C2×#424C411yescharacteristic
10C4×#117C2211yescharacteristic
11#1832—21no—
12#1832—21no—
13#1831—21no—
14#1831—21no—
15#1729C2211yescharacteristic
16#1729C2211yescharacteristic
17#1728C411yescharacteristic
18C2×#336C411yescharacteristic
19#1458C2211yescharacteristic
20#1458C2211yescharacteristic
21#1458C2211yescharacteristic
22#1458C2211yescharacteristic
23#1276—21no—
24#1276—21no—
25#1274—21no—
26#1274—21no—
27C22×M16C2×C411yescharacteristic
28C22×M16C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C23×C8C2×C411yescharacteristic
31C2×#117C2×C411yescharacteristic
32C2×#117C2311yescharacteristic
33C22×C42C2×C411yescharacteristic
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#424—42no—
39#424—42no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4—21no—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48#376D411yescharacteristic
49#376D411yescharacteristic
50C43D411yescharacteristic
51C43D411yescharacteristic
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#324—21no—
61#324—21no—
62#324—21no—
63#324—21no—
64#323—21no—
65#323—21no—
66#323—21no—
67#323—21no—
68C23×C4—21no—
69C23×C4C22×C411yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86#117—22no—
87#117—22no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4#4111yescharacteristic
92C2×C4⋊C4#4111yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4#4111yescharacteristic
95C2×C22⋊C4#4111yescharacteristic
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8—41no—
124C4×C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129C24#11711yescharacteristic
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4#11711yescharacteristic
143C22×C4#12311yescharacteristic
144C22×C4#12311yescharacteristic
145C22×C4#12311yescharacteristic
146C22×C4#12311yescharacteristic
147C22×C4#13112yes—
148C22×C4#13112yes—
149M16—82no—
150M16—82no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—81no—
169C4⋊C4—24no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C22⋊C4—24no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42—81no—
189C42—81no—
190C42#10012yes—
191C42#10112yes—
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23#38111yescharacteristic
197C23#43312yes—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—84no—
225C2×C4#40811yescharacteristic
226C2×C4#40811yescharacteristic
227C2×C4#41011yescharacteristic
228C2×C4#41011yescharacteristic
229C2×C4C2×#10011yescharacteristic
230C2×C4C2×#10111yescharacteristic
231C2×C4#43312yes—
232C8—81no—
233C8—81no—
234C8—81no—
235C8—81no—
236C8—161no—
237C8—161no—
238C8—161no—
239C8—161no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22#181811yescharacteristic
248C22#182011yescharacteristic
249C22#182111yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4#146911yescharacteristic
260C4#146911yescharacteristic
261C4#147011yescharacteristic
262C4#147011yescharacteristic
263C2—21no—
264C2—21no—
265C2—42no—
266C2#870311yescharacteristic
267C2#1286411yescharacteristic
268C2#1286511yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1820
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#207542
#87302
#71902
#71882