Finite Group Databasebeta

← #327980

Subgroups — #327980

Order 512.

807 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 807 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×#2571C211yesmaximal, characteristic
2#33149C212yesmaximal
3#27905C211yesmaximal, characteristic
4#10577C214yesmaximal, complemented
5#10352C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic
7#8707C212yesmaximal
8C2×#483—22no—
9#2857C2211yescharacteristic
10#2571—22no—
11#2571—24no—
12#2571C2214yes—
13C2×#400—22no—
14C2×#389—22no—
15C2×#384C2211yescharacteristic
16C2×#382—22no—
17C2×C4×C4⋊C4C2212yes—
18#2096—22no—
19#1987C2214yes—
20#1956C2212yes—
21#1947C2214yes—
22#1921—24no—
23#1852—22no—
24#1844C2212yes—
25#1824—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C2×C4⋊Q8C2311yescharacteristic
31C2×C22⋊Q8—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C42—22no—
38C22×C42C2311yescharacteristic
39#483—48no—
40C2×#107—24no—
41C2×#106—24no—
42C2×C4⋊C8C2314yes—
43C2×#103—24no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—22no—
46C2×C4×C8C2312yes—
47#400—28no—
48#400—28no—
49#391—21no—
50#389—28no—
51#389—28no—
52#389—22no—
53#384—22no—
54#384D412yes—
55#384C2311yescharacteristic
56#384C2312yes—
57#382—28no—
58#382—28no—
59#380—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4D414yes—
63C4×C4⋊C4C2314yes—
64C2×#95—22no—
65C2×#95—24no—
66C2×#95—22no—
67#340—24no—
68#340—24no—
69#340—24no—
70#336—22no—
71#336—21no—
72C22×Q8—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—22no—
79C22×C8—24no—
80C4⋊Q8—24no—
81C22⋊Q8—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—84no—
99C2×C4⋊C4C2×D414yes—
100C2×C4⋊C4C2×D414yes—
101C2×C4⋊C4C2×D414yes—
102C2×C22⋊C4—41no—
103C2×C42—24no—
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—48no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D412yes—
109C2×C42C2411yescharacteristic
110#107—48no—
111#106—48no—
112C4⋊C8—28no—
113#103—48no—
114C22⋊C8—24no—
115C22⋊C8—44no—
116C4×C8—24no—
117#95—28no—
118#95—216no—
119#95—24no—
120#95—24no—
121#95—42no—
122C24#12011yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—41no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—48no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4#12011yescharacteristic
145C22×C4#12012yes—
146C22×C4#13514yes—
147C22×C4#13714yes—
148C22×C4C22×D411yescharacteristic
149C22×C4C22×D412yes—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—44no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—88no—
172C22⋊C4—82no—
173C42—22no—
174C42—24no—
175C42—48no—
176C42—48no—
177C422−1+414yes—
178C23—21no—
179C23—22no—
180C23—21no—
181C23#45214yes—
182C23#45312yes—
183C23C2×#12011yescharacteristic
184Q8—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—28no—
190C2×C4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—48no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—88no—
204C2×C4—88no—
205C2×C4—88no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4#45214yes—
212C2×C4#53612yes—
213C2×C4#53614yes—
214C2×C4C2×#13512yes—
215C2×C4C2×#13712yes—
216C2×C4#57514yes—
217C8—44no—
218C8—84no—
219C22—24no—
220C22—22no—
221C22—24no—
222C22—22no—
223C22#163014yes—
224C22#261411yescharacteristic
225C22C2×#45212yes—
226C22#300512yes—
227C22C2×#45311yescharacteristic
228C22#302011yescharacteristic
229C4—24no—
230C4—22no—
231C4—41no—
232C4—84no—
233C4—84no—
234C4—84no—
235C4—161no—
236C4—161no—
237C4#323214yes—
238C4#323614yes—
239C2—24no—
240C2C2×#163011yescharacteristic
241C2#1854711yescharacteristic
242C2#1878112yes—
243C2#2964312yes—
244C2#2965011yescharacteristic
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. #10577C2semidirect
5. #10352C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#331492
#279052
#105772
#103522
#87232
#87072