Finite Group Databasebeta

← #298401

Subgroups — #298401

Order 512.

1091 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1091 conjugacy classes of subgroups — too many to draw. Only its 192 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#59827C211yesmaximal, characteristic
2#28511C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4C2×#1630C214yesmaximal
5#10237C214yesmaximal, complemented
6#9673C212yesmaximal, complemented
7C4×#331C211yesmaximal, characteristic
8C2×#559C2212yes—
9C2×#492C2212yes—
10C2×#477C2218yes—
11C2×#461—22no—
12#2868C2211yescharacteristic
13#2724—22no—
14#2582C2212yes—
15#2547—22no—
16#2486—22no—
17#2447—24no—
18D4×C4⋊C4—22no—
19C4×C4⋊Q8C2212yes—
20C4×#127C2211yescharacteristic
21#2095—24no—
22#2013—22no—
23#1956—22no—
24#1956C2212yes—
25#1845—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28#1630—216no—
29C2×#331C2212yes—
30C2×Q8⋊C8C2218yes—
31#1296C2212yes—
32C22×SD16—24no—
33#559—28no—
34#559—44no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8C2312yes—
39C2×#127C2311yescharacteristic
40C2×#124—24no—
41C2×#123—24no—
42C2×C4×Q8C2314yes—
43C2×C4×D4—24no—
44C2×C4×D4—22no—
45C2×C4×D4D412yes—
46C22×C4⋊C4—22no—
47#492—28no—
48#477—216no—
49#477—48no—
50#461—48no—
51C2×#107—28no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2314yes—
54C2×#103—28no—
55C2×#103—24no—
56C2×C4×C8—22no—
57C2×C4×C8C2312yes—
58#398—24no—
59#397—24no—
60#397—28no—
61#391—24no—
62#391D414yes—
63#390—22no—
64#390D412yes—
65#389—24no—
66#389—22no—
67#389—22no—
68#385—28no—
69#384—22no—
70#384—22no—
71#384D412yes—
72#384C2311yescharacteristic
73#381—24no—
74#380—24no—
75#380—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C43C2311yescharacteristic
79#336—22no—
80#336—21no—
81#331D418yes—
82Q8⋊C8—216no—
83C22×Q8—22no—
84C22×Q8C2×D414yes—
85C22×D4—22no—
86C22×D4C2×D412yes—
87C23×C4—24no—
88C23×C4—22no—
89C2×SD16—48no—
90C2×SD16—48no—
91C2×SD16—84no—
92C22×C8—22no—
93C22×C8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—44no—
99C4⋊Q8—82no—
100#127—22no—
101#127C2×D414yes—
102#124—416no—
103#123—48no—
104#123—48no—
105C4×Q8—28no—
106C4×Q8—28no—
107C4×D4—44no—
108C4×D4—416no—
109C4×D4—44no—
110C4×D4—44no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—84no—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D412yes—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C42—24no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135#107—416no—
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8C2×D418yes—
139#103—416no—
140#103—48no—
141C4×C8—24no—
142C4×C8—44no—
143#95—28no—
144#95—24no—
145#95—24no—
146#95—24no—
147C24—22no—
148C2×Q8—28no—
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—28no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×D4—22no—
157C2×D4—28no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—44no—
163C2×D4#12714yes—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4#12011yescharacteristic
174C22×C4#12012yes—
175C22×C4#12011yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C22×D412yes—
178SD16—816no—
179C2×C8—24no—
180C2×C8—28no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—44no—
184C2×C8—44no—
185C4⋊C4—28no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C4⋊C4—82no—
200C22⋊C4—416no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C22⋊C4—82no—
204C42—24no—
205C42—22no—
206C42—41no—
207C42—48no—
208C42#13514yes—
209C42#13714yes—
210C42C22×D412yes—
211C42H(5,𝔽2)12yes—
212C23—24no—
213C23—44no—
214C23—44no—
215C23—42no—
216C23—42no—
217C23C2×#12011yescharacteristic
218Q8—48no—
219Q8—48no—
220Q8—84no—
221Q8—84no—
222D4—44no—
223D4—48no—
224D4—48no—
225D4—82no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—88no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—168no—
245C2×C4#45214yes—
246C2×C4#45214yes—
247C2×C4#45312yes—
248C2×C4C2×#12711yescharacteristic
249C2×C4#53511yescharacteristic
250C2×C4D4214yes—
251C2×C4C2×#13512yes—
252C2×C4C2×#13712yes—
253C2×C4#57814yes—
254C8—44no—
255C8—48no—
256C22—44no—
257C22—48no—
258C22—82no—
259C22—82no—
260C22—84no—
261C22#243311yescharacteristic
262C22C2×#45212yes—
263C22#301112yes—
264C22C2×#45311yescharacteristic
265C22#302111yescharacteristic
266C4—21no—
267C4—21no—
268C4—24no—
269C4—41no—
270C4—82no—
271C4—84no—
272C4—84no—
273C4—162no—
274C4#163014yescomplemented
275C4#327614yes—
276C4#328314yes—
277C2—84no—
278C2C2×#163011yescharacteristic
279C2#1871911yescharacteristic
280C2#1889712yes—
281C2#2573911yescharacteristic
282C2#2575812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28511C2semidirect
5. #10237C2semidirect
6. #9673C2semidirect
274. C4#1630semidirect

Maximal subgroups

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

subgroupindex
#598272
#285112
#284382
C2×#16302
#102372
#96732
C4×#3312