Finite Group Databasebeta

← #305039

Subgroups — #305039

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#29218C212yesmaximal, complemented
3#28438C211yesmaximal, characteristic
4#10352C214yesmaximal, complemented
5#9864C212yesmaximal, complemented
6#9064C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×#492C2212yes—
9C2×#461—22no—
10#2868C2211yescharacteristic
11#2804—22no—
12#2580—24no—
13#2571C2212yes—
14#2563—22no—
15#2487—22no—
16#2366C2212yes—
17D4×C4⋊C4—22no—
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#2100—24no—
21#2013—22no—
22#1990—22no—
23#1987C2218yes—
24#1956C2212yes—
25#1834—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C22×SD16—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2312yes—
36C2×#127C2311yescharacteristic
37C2×C4×Q8—22no—
38C2×C4×D4—24no—
39C2×C4×D4—22no—
40C2×C4×D4D412yes—
41C22×C4⋊C4—22no—
42#492—28no—
43#461—48no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2314yes—
46C2×#103—24no—
47C2×C4×C8—22no—
48C2×C4×C8C2312yes—
49#400—24no—
50#399—24no—
51#393—28no—
52#391—22no—
53#390—24no—
54#390—22no—
55#390D412yes—
56#389—24no—
57#389—22no—
58#389D414yes—
59#386—24no—
60#385—24no—
61#385—28no—
62#384—22no—
63#384—22no—
64#384D412yes—
65#384C2311yescharacteristic
66#382—24no—
67#380—22no—
68#380—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—28no—
73#340—28no—
74#336—22no—
75#336—21no—
76#331—24no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×D4—22no—
80C22×D4C2×D412yes—
81C23×C4—24no—
82C23×C4—22no—
83C2×SD16—48no—
84C2×SD16—48no—
85C2×SD16—84no—
86C22×C8—22no—
87C22×C8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—24no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—44no—
93C4⋊Q8—82no—
94#127—22no—
95#127#4114yes—
96C4×Q8—84no—
97C4×D4—44no—
98C4×D4—44no—
99C4×D4—44no—
100C4×D4—88no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—84no—
112C2×C4⋊C4C2×D414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D412yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C42—24no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2411yescharacteristic
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128#103—48no—
129C4×C8—24no—
130C4×C8—44no—
131#95—24no—
132#95—28no—
133#95—24no—
134#95—24no—
135C24—22no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×D4—22no—
144C2×D4—28no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—44no—
150C2×D4#12414yes—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—84no—
158C22×C4—88no—
159C22×C4#12011yescharacteristic
160C22×C4#12012yes—
161C22×C4#12011yescharacteristic
162C22×C4C22×D411yescharacteristic
163C22×C4C22×D412yes—
164SD16—816no—
165C2×C8—24no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—44no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—28no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—48no—
180C4⋊C4—82no—
181C4⋊C4—84no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4—82no—
186C22⋊C4—48no—
187C22⋊C4—82no—
188C22⋊C4—84no—
189C22⋊C4—88no—
190C42—24no—
191C42—22no—
192C42—41no—
193C42—48no—
194C42#13514yes—
195C42#13714yes—
196C42C2×#4112yes—
197C422−1+412yes—
198C23—24no—
199C23—44no—
200C23—44no—
201C23—42no—
202C23—42no—
203C23C2×#12011yescharacteristic
204Q8—88no—
205Q8—84no—
206Q8—84no—
207D4—44no—
208D4—48no—
209D4—48no—
210D4—82no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—88no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—168no—
230C2×C4#45214yes—
231C2×C4#45214yes—
232C2×C4#45312yes—
233C2×C4C2×#12411yescharacteristic
234C2×C4#53611yescharacteristic
235C2×C4#54814yes—
236C2×C4C2×#13512yes—
237C2×C4C2×#13712yes—
238C2×C4#57714yes—
239C8—44no—
240C8—84no—
241C22—44no—
242C22—48no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22#243611yescharacteristic
247C22C2×#45212yes—
248C22#300912yes—
249C22C2×#45311yescharacteristic
250C22#302111yescharacteristic
251C4—21no—
252C4—21no—
253C4—24no—
254C4—41no—
255C4—84no—
256C4—82no—
257C4—84no—
258C4—162no—
259C4#163014yes—
260C4#336014yes—
261C4#337214yes—
262C2—84no—
263C2C2×#163011yescharacteristic
264C2#1871511yescharacteristic
265C2#1891712yes—
266C2#2591811yescharacteristic
267C2#2593412yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29218C2semidirect
4. #10352C2semidirect
5. #9864C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#292182
#284382
#103522
#98642
#90642
C4×#3312