Finite Group Databasebeta

← #327148

Subgroups — #327148

Order 512.

828 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 148 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#37641C211yesmaximal, characteristic
2#29691C212yesmaximal
3#27975C211yesmaximal, characteristic
4#10438C214yesmaximal, complemented
5#10237C214yesmaximal, complemented
6#8729C211yesmaximal, characteristic
7#8314C212yesmaximal
8C2×#533C2212yes—
9C2×#477C2214yes—
10#2857C2211yescharacteristic
11#2822—22no—
12#2625—24no—
13#2621—22no—
14#2582C2214yes—
15#2568—22no—
16#2546—22no—
17#2285C2211yescharacteristic
18#2074—22no—
19#2057—22no—
20#2043—22no—
21#2036—24no—
22#1956C2212yes—
23#1873—22no—
24#1834C2212yes—
25#1803C2214yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×Q8⋊C8C2218yes—
29#1296C2214yes—
30#533—48no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8D412yes—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C2×#125—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8D414yes—
39C2×C4×Q8—22no—
40C2×C4×Q8C2314yes—
41C2×#117—22no—
42C2×#117D412yes—
43C22×C42C2311yescharacteristic
44#477—48no—
45C2×#107—24no—
46C2×C4⋊C8C2314yes—
47C2×#103—24no—
48C2×#103—24no—
49C2×#103—24no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—22no—
52C2×C4×C8C2312yes—
53#400—28no—
54#398—28no—
55#398—28no—
56#397—28no—
57#391—24no—
58#391—24no—
59#391—24no—
60#391—21no—
61#389—24no—
62#389—24no—
63#389—22no—
64#384C2311yescharacteristic
65#384C2312yes—
66#382—22no—
67#382D412yes—
68#380—22no—
69#380—22no—
70C4×C4⋊C4—22no—
71#340—24no—
72#340—24no—
73#336—22no—
74#336—21no—
75Q8⋊C8—48no—
76C22×Q8—22no—
77C22×Q8—21no—
78C22×Q8C2×D414yes—
79C23×C4C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C22×C8—22no—
82C22×C8—24no—
83C4⋊Q8—24no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—84no—
87#125—48no—
88C22⋊Q8—44no—
89C22⋊Q8—44no—
90C22⋊Q8—48no—
91C22⋊Q8—44no—
92C4×Q8—28no—
93C4×Q8—28no—
94C4×Q8—48no—
95#117—48no—
96#117—82no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D414yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—41no—
114C2×C42—24no—
115C2×C42—22no—
116C2×C42—24no—
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2411yescharacteristic
120#107—48no—
121C4⋊C8—28no—
122#103—48no—
123#103—84no—
124#103—84no—
125C22⋊C8—42no—
126C22⋊C8—82no—
127C4×C8—24no—
128#95—28no—
129#95—28no—
130#95—28no—
131#95—24no—
132#95—42no—
133C24#12011yescharacteristic
134C2×Q8—24no—
135C2×Q8—24no—
136C2×Q8—24no—
137C2×Q8—24no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—44no—
145C2×Q8—42no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4#12012yes—
159C22×C4#12011yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D412yes—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—44no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—48no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—88no—
184C4⋊C4—82no—
185C22⋊C4—48no—
186C22⋊C4—82no—
187C22⋊C4—82no—
188C42—24no—
189C42—22no—
190C42—44no—
191C42—48no—
192C42H(5,𝔽2)14yes—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23C2×#12011yescharacteristic
199Q8—44no—
200Q8—44no—
201Q8—44no—
202Q8—44no—
203Q8—88no—
204Q8—84no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#45412yes—
230C2×C4#53414yes—
231C2×C4#53512yes—
232C2×C4#57712yes—
233C2×C4#57814yes—
234C2×C4#57812yes—
235C8—82no—
236C8—84no—
237C22—41no—
238C22—41no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22#261311yescharacteristic
243C22#301112yes—
244C22#301312yes—
245C22C2×#45411yescharacteristic
246C22#301911yescharacteristic
247C4—22no—
248C4—22no—
249C4—42no—
250C4—84no—
251C4—84no—
252C4—84no—
253C4—161no—
254C4—161no—
255C4#320114yes—
256C4#320214yes—
257C2—81no—
258C2#1869811yescharacteristic
259C2#1873211yescharacteristic
260C2#1893912yes—
261C2#2960512yes—
262C2#2961711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10438C2semidirect
5. #10237C2semidirect

Maximal subgroups

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

subgroupindex
#376412
#296912
#279752
#104382
#102372
#87292
#83142