Finite Group Databasebeta

← #108323

Subgroups — #108323

Order 512.

766 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 190 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#34217C211yesmaximal, characteristic
2C2×#1534C211yesmaximal, characteristic
3C2×#1534C211yesmaximal, characteristic
4#7952C212yesmaximal
5#7930C212yesmaximal
6C22×#125C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9#2370—24no—
10#2345—24no—
11#2330—22no—
12#2316C412yes—
13C4×#125—22no—
14#2285C2211yescharacteristic
15#2274C412yes—
16C4×#117C2211yescharacteristic
17#1534—22no—
18#1534—22no—
19#1534—22no—
20#1534C414yescomplemented
21C2×#330—22no—
22C2×#330C414yescomplemented
23#1285—24no—
24#1282—24no—
25C2×#125—22no—
26C2×#125—22no—
27C2×#125—22no—
28C2×#125C2×C412yes—
29C2×#125C2×C412yes—
30C2×#125D412yes—
31C2×#125D412yes—
32C2×#125D412yes—
33C2×#125Q812yes—
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C412yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C412yes—
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432C2×C414yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×#97—22no—
53C2×#97C2×C414yes—
54#389—28no—
55#389—24no—
56#389—24no—
57#387—28no—
58#387—28no—
59#384—21no—
60#382—21no—
61#380—22no—
62#380—24no—
63C4×C4⋊C4—28no—
64C4×C4⋊C4—24no—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C4×C22⋊C4—24no—
69C4×C22⋊C4D414yes—
70#376—21no—
71C43—21no—
72#330—28no—
73#330—28no—
74C23×C4—22no—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D412yes—
77C22×C8—24no—
78C22×C8—42no—
79#125—24no—
80#125—24no—
81#125—24no—
82#125—24no—
83#125—24no—
84#125—24no—
85#125C22⋊C414yes—
86#125C22⋊C414yes—
87#125C4⋊C414yes—
88#125C4⋊C414yes—
89#117—24no—
90#117—42no—
91#117—44no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4C4214yes—
107C2×C4⋊C4C22⋊C414yes—
108C2×C4⋊C4C4⋊C414yes—
109C2×C22⋊C4—22no—
110C2×C22⋊C4C22⋊C412yes—
111C2×C22⋊C4C22⋊C412yes—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—28no—
115C2×C42—24no—
116C2×C42C22⋊C414yes—
117C2×C42C22×C411yescharacteristic
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×Q811yescharacteristic
124C2×C42#4112yes—
125C2×C42#4112yes—
126C4⋊C8—24no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C4⋊C8—42no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#97—24no—
135#97—24no—
136#97—24no—
137#97—42no—
138#95—28no—
139#95—28no—
140#95—24no—
141#95—24no—
142C24C2×C22⋊C411yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—82no—
152C22×C4—84no—
153C22×C4C2×C4211yescharacteristic
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4#11711yescharacteristic
158C22×C4#11711yescharacteristic
159C22×C4C4×D412yes—
160C22×C4C4×D412yes—
161C22×C4#12012yes—
162C22×C4#12012yes—
163C22×C4#12112yes—
164C22×C4C22⋊Q812yes—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—84no—
168C2×C8—84no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—28no—
174C4⋊C4—28no—
175C4⋊C4—28no—
176C4⋊C4—28no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—84no—
180C4⋊C4—88no—
181C4⋊C4#9518yes—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—44no—
185C22⋊C4#10114yes—
186C42—24no—
187C42—44no—
188C42—42no—
189C42—84no—
190C42—88no—
191C42C2×C22⋊C411yescharacteristic
192C42C2×C22⋊C411yescharacteristic
193C42C2×C4⋊C411yescharacteristic
194C42C2×C4⋊C411yescharacteristic
195C42C4×D412yes—
196C42C4×D412yes—
197C42#12112yes—
198C42C22⋊Q812yes—
199C42#12312yes—
200C42#12312yes—
201C23—42no—
202C23C4×C22⋊C411yescharacteristic
203C23#37911yescharacteristic
204C23#38011yescharacteristic
205C23C2×#10112yes—
206C23#41312yes—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4C2×#9511yescharacteristic
233C2×C4C4×C22⋊C411yescharacteristic
234C2×C4#38011yescharacteristic
235C2×C4#38111yescharacteristic
236C2×C4#38512yes—
237C2×C4#38512yes—
238C2×C4#38612yes—
239C2×C4#38612yes—
240C8—84no—
241C8—84no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22C22⋊#9511yescharacteristic
248C22C4×#10112yes—
249C22#178411yescharacteristic
250C22#178511yescharacteristic
251C22#180112yes—
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—48no—
262C4—84no—
263C4—82no—
264C2—21no—
265C2—21no—
266C2—22no—
267C2—22no—
268C2#1076311yescharacteristic
269C2#1078312yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
20. #1534C4semidirect
22. C2×#330C4semidirect

Maximal subgroups

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

subgroupindex
#342172
C2×#15342
#79522
#79302