Finite Group Databasebeta

← #173911

Subgroups — #173911

Order 512.

959 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 959 conjugacy classes of subgroups — too many to draw. Only its 176 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#29208C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4#7751C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×#559—21no—
8C2×#492—21no—
9C2×#461—22no—
10#2868C2211yescharacteristic
11#2724C2214yes—
12#2582—21no—
13#2547—22no—
14#2486—22no—
15#2447—24no—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2057C2218yes—
20#2013C2214yes—
21#1990—22no—
22#1956—21no—
23#1947—24no—
24#1845—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1271C2218yes—
28#1270C2214yes—
29C22×SD16—24no—
30#559—44no—
31#559—44no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2312yes—
36C2×#127D411yescharacteristic
37C2×#124—24no—
38C2×#123—24no—
39C2×C4×Q8—22no—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×C4×D4D412yes—
43C22×C4⋊C4—22no—
44#492—44no—
45#461—48no—
46C2×#107—28no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—21no—
49C2×#103—24no—
50C2×C4×C8—22no—
51C2×C4×C8—21no—
52#398—24no—
53#397—24no—
54#397—28no—
55#391—24no—
56#391—22no—
57#390—22no—
58#390C2312yes—
59#389—24no—
60#389—22no—
61#389C2314yes—
62#385—28no—
63#384—22no—
64#384—22no—
65#384D411yescharacteristic
66#384D412yes—
67#381—24no—
68#380—24no—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—22no—
77C22×Q8—22no—
78C22×D4—22no—
79C22×D4C2×D412yes—
80C23×C4—24no—
81C23×C4—22no—
82C2×SD16—48no—
83C2×SD16—48no—
84C2×SD16—84no—
85C22×C8—22no—
86C22×C8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—44no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—82no—
93#127—22no—
94#127D814yes—
95#124—416no—
96#123—48no—
97#123—48no—
98C4×Q8—44no—
99C4×Q8—44no—
100C4×D4—44no—
101C4×D4—416no—
102C4×D4—44no—
103C4×D4—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—84no—
115C2×C4⋊C4C2×D414yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D414yes—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—22no—
122C2×C42—24no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#107—88no—
129C4⋊C8—28no—
130C4⋊C8—24no—
131#103—48no—
132C4×C8—42no—
133C4×C8—44no—
134#95—28no—
135#95—24no—
136#95—24no—
137#95—24no—
138C24—22no—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×D4—22no—
147C2×D4—28no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4C2×D814yes—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4#12011yescharacteristic
163C22×C4#12012yes—
164C22×C4#12011yescharacteristic
165C22×C4C22×D411yescharacteristic
166C22×C4C22×D412yes—
167SD16—816no—
168C2×C8—24no—
169C2×C8—28no—
170C2×C8—44no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—44no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—48no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—82no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C4⋊C4—82no—
189C22⋊C4—416no—
190C22⋊C4—48no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—82no—
194C42—22no—
195C42—24no—
196C42—41no—
197C42—48no—
198C42C2×D812yes—
199C42#13514yes—
200C42AGL(1,ℤ/8)12yes—
201C42#13714yes—
202C23—24no—
203C23—44no—
204C23—42no—
205C23—44no—
206C23—42no—
207C23C2×#12011yescharacteristic
208Q8—84no—
209Q8—84no—
210Q8—88no—
211D4—44no—
212D4—416no—
213D4—82no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—88no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—168no—
233C2×C4#44714yes—
234C2×C4#45214yes—
235C2×C4#45214yes—
236C2×C4#45312yes—
237C2×C4#45614yes—
238C2×C4C22×D811yescharacteristic
239C2×C4C2×#13512yes—
240C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C4C2×#13712yes—
242C8—48no—
243C8—44no—
244C22—48no—
245C22—44no—
246C22—82no—
247C22—82no—
248C22—84no—
249C22C2×#44711yescharacteristic
250C22C2×#45212yes—
251C22C2×#45311yescharacteristic
252C22C2×#45612yes—
253C22#302711yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—22no—
258C4—41no—
259C4—82no—
260C4—84no—
261C4—84no—
262C4—162no—
263C4#163014yes—
264C4#164718yes—
265C2—84no—
266C2#1838711yescharacteristic
267C2#1839212yes—
268C2C2×#163011yescharacteristic
269C2C2×#164712yes—
270C2#1907811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29208C2semidirect
4. #7751C2semidirect
5. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#292082
#284382
#77512
#75702
#75692