Finite Group Databasebeta

← #174044

Subgroups — #174044

Order 512.

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

normal subgroups only

This group has 883 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#59864C211yesmaximal, characteristic
2#29141C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4#7755C218yesmaximal, complemented
5#7571C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×#492—21no—
8C2×#461—22no—
9#2868C2211yescharacteristic
10#2804C2214yes—
11#2580—24no—
12#2571—21no—
13#2563—22no—
14#2487—22no—
15#2366—21no—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2024C2218yes—
20#2013C2214yes—
21#1956—21no—
22#1956—22no—
23#1834—22no—
24#1803—24no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1272C2218yes—
28#1270C2214yes—
29C22×SD16—24no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8D412yes—
33C2×C4⋊Q8C2312yes—
34C2×#127D411yescharacteristic
35C2×C4×Q8D414yes—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4D412yes—
39C22×C4⋊C4—22no—
40#492—44no—
41#461—48no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×#103—24no—
45C2×#103—28no—
46C2×C4×C8—22no—
47C2×C4×C8—21no—
48#400—24no—
49#399—24no—
50#393—28no—
51#391C2314yes—
52#390—24no—
53#390—22no—
54#390C2312yes—
55#389—24no—
56#389—22no—
57#389—22no—
58#386—24no—
59#385—24no—
60#385—28no—
61#384—22no—
62#384—22no—
63#384D411yescharacteristic
64#384D412yes—
65#382—24no—
66#380—22no—
67#380—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C43C2311yescharacteristic
71#340—28no—
72#336C2312yes—
73#336C2314yes—
74C22×Q8—22no—
75C22×Q8C2×D414yes—
76C22×D4—22no—
77C22×D4C2×D412yes—
78C23×C4—24no—
79C23×C4—22no—
80C2×SD16—48no—
81C2×SD16—48no—
82C2×SD16—84no—
83C22×C8—22no—
84C22×C8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—44no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—82no—
91#127—22no—
92#127SD1614yes—
93C4×Q8—48no—
94C4×D4—44no—
95C4×D4—44no—
96C4×D4—44no—
97C4×D4—88no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D414yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—22no—
116C2×C42—24no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2411yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—44no—
124#103—48no—
125#103—88no—
126C4×C8—42no—
127C4×C8—44no—
128#95—24no—
129#95—28no—
130#95—24no—
131#95—24no—
132C24—22no—
133C2×Q8—24no—
134C2×Q8—24no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×D4—22no—
142C2×D4—28no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4C2×SD1614yes—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—84no—
155C22×C4—88no—
156C22×C4#12011yescharacteristic
157C22×C4#12012yes—
158C22×C4#12011yescharacteristic
159C22×C4C22×D411yescharacteristic
160C22×C4C22×D412yes—
161SD16—816no—
162C2×C8—24no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—44no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—84no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—82no—
181C22⋊C4—44no—
182C22⋊C4—44no—
183C22⋊C4—82no—
184C22⋊C4—84no—
185C22⋊C4—88no—
186C42—22no—
187C42—24no—
188C42—41no—
189C42—48no—
190C42C2×SD1612yes—
191C42#13514yes—
192C42#13712yes—
193C42#13714yes—
194C23—24no—
195C23—44no—
196C23—42no—
197C23—44no—
198C23—42no—
199C23C2×#12011yescharacteristic
200Q8—48no—
201Q8—84no—
202Q8—84no—
203Q8—84no—
204D4—44no—
205D4—416no—
206D4—82no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—88no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—168no—
226C2×C4#44814yes—
227C2×C4#45214yes—
228C2×C4#45214yes—
229C2×C4#45312yes—
230C2×C4#45514yes—
231C2×C4C22×SD1611yescharacteristic
232C2×C4C2×#13512yes—
233C2×C4C2×#13711yescharacteristic
234C2×C4C2×#13712yes—
235C8—44no—
236C8—84no—
237C22—48no—
238C22—44no—
239C22—82no—
240C22—82no—
241C22—84no—
242C22C2×#44811yescharacteristic
243C22C2×#45212yes—
244C22C2×#45311yescharacteristic
245C22C2×#45512yes—
246C22#302711yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—82no—
253C4—84no—
254C4—84no—
255C4—162no—
256C4#163014yes—
257C4#164318yes—
258C2—84no—
259C2#1838911yescharacteristic
260C2#1840612yes—
261C2C2×#163011yescharacteristic
262C2C2×#164312yes—
263C2#1907411yescharacteristic
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. #29141C2semidirect
4. #7755C2semidirect
5. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#291412
#284382
#77552
#75712
#75692