Finite Group Databasebeta

← #173964

Subgroups — #173964

Order 512.

1135 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1135 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#59817C211yesmaximal, characteristic
2#29206C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4#7754C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×D4×Q8—22no—
8C2×#492—21no—
9C2×#461—22no—
10#2868C2211yescharacteristic
11#2683C2214yes—
12#2551—21no—
13#2538—22no—
14#2452—24no—
15#2409—22no—
16#2366—21no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2058C2218yes—
20#2013C2214yes—
21#1990—22no—
22#1956—21no—
23#1948—24no—
24#1844—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1271C2218yes—
28#1270C2214yes—
29C23×Q8—22no—
30C22×SD16—24no—
31D4×Q8—48no—
32D4×Q8—48no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8D412yes—
37C2×C4⋊Q8C2312yes—
38C2×#127D411yescharacteristic
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—28no—
41C2×C4×Q8—22no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4D412yes—
45#492—44no—
46#461—48no—
47C2×#106—28no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—21no—
50C2×#103—24no—
51C2×C4×C8—22no—
52C2×C4×C8—21no—
53#396—28no—
54#395—24no—
55#393—24no—
56#391—22no—
57#390—24no—
58#390—22no—
59#390C2312yes—
60#389—24no—
61#389—22no—
62#389C2314yes—
63#386—24no—
64#385—24no—
65#384—22no—
66#384—22no—
67#384D411yescharacteristic
68#384D412yes—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—24no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8—416no—
80C22×Q8—84no—
81C22×D4—22no—
82C22×D4C2×D412yes—
83C23×C4—24no—
84C23×C4—22no—
85C2×SD16—48no—
86C2×SD16—48no—
87C2×SD16—84no—
88C22×C8—22no—
89C22×C8—24no—
90C4⋊Q8—24no—
91C4⋊Q8—44no—
92C4⋊Q8—416no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—82no—
96C4⋊Q8SD1618yescomplemented
97#127—22no—
98#127SD1614yes—
99C22⋊Q8—416no—
100C22⋊Q8—416no—
101C22⋊Q8—416no—
102C4×Q8—44no—
103C4×Q8—44no—
104C4×D4—44no—
105C4×D4—416no—
106C4×D4—44no—
107C4×D4—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D414yes—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C42—24no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2411yescharacteristic
131#106—88no—
132C4⋊C8—28no—
133C4⋊C8—24no—
134#103—48no—
135C4×C8—42no—
136C4×C8—44no—
137#95—24no—
138#95—24no—
139#95—24no—
140C24—22no—
141C2×Q8—28no—
142C2×Q8—416no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—84no—
150C2×Q8—832no—
151C2×D4—22no—
152C2×D4—28no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4C2×SD1614yes—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—416no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—84no—
166C22×C4#12011yescharacteristic
167C22×C4#12012yes—
168C22×C4#12011yescharacteristic
169C22×C4C22×D412yes—
170C22×C4C22×D411yescharacteristic
171SD16—816no—
172C2×C8—24no—
173C2×C8—28no—
174C2×C8—44no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—44no—
178C4⋊C4—24no—
179C4⋊C4—28no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—416no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—82no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4C2×SD1618yes—
193C22⋊C4—416no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—82no—
197C22⋊C4—84no—
198C42—22no—
199C42—24no—
200C42—41no—
201C42—48no—
202C42C2×SD1612yes—
203C42C2×SD1614yes—
204C42AGL(1,ℤ/8)14yes—
205C42#13712yes—
206C23—24no—
207C23—44no—
208C23—42no—
209C23—44no—
210C23—42no—
211C23C2×#12011yescharacteristic
212Q8—416no—
213Q8—84no—
214Q8—84no—
215Q8—816no—
216D4—44no—
217D4—416no—
218D4—82no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—28no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—816no—
234C2×C4—88no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#44814yes—
239C2×C4#45014yes—
240C2×C4#45014yes—
241C2×C4#45312yes—
242C2×C4#45614yes—
243C2×C4C22×SD1612yes—
244C2×C4C22×SD1611yescharacteristic
245C2×C4C2×AGL(1,ℤ/8)12yes—
246C2×C4C2×#13711yescharacteristic
247C8—48no—
248C8—44no—
249C22—48no—
250C22—44no—
251C22—82no—
252C22—82no—
253C22—84no—
254C22C2×#45012yes—
255C22C2×#44811yescharacteristic
256C22C2×#45311yescharacteristic
257C22C2×#45612yes—
258C22#302711yescharacteristic
259C4—21no—
260C4—21no—
261C4—22no—
262C4—22no—
263C4—48no—
264C4—41no—
265C4—82no—
266C4—84no—
267C4—162no—
268C4#162914yes—
269C4#165118yes—
270C2—84no—
271C2#1838911yescharacteristic
272C2#1840212yes—
273C2C2×#162911yescharacteristic
274C2C2×#165112yes—
275C2#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. #29206C2semidirect
4. #7754C2semidirect
5. #7570C2semidirect
96. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598172
#292062
#284382
#77542
#75702
#75692