Finite Group Databasebeta

← #98692

Subgroups — #98692

Order 512.

732 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 112 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#59827C211yesmaximal, characteristic
2#9644C211yesmaximal, characteristic
3#8590C212yesmaximal, complemented
4#7651C211yesmaximal, characteristic
5#7590C212yesmaximal, complemented
6C2×#559C412yes—
7#2724—22no—
8#2582C412yes—
9#2547—22no—
10#2486—22no—
11#2447—24no—
12D4×C4⋊C4—22no—
13C4×C4⋊Q8C2212yes—
14C4×#127C2211yescharacteristic
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1744C2211yescharacteristic
18C2×#333—22no—
19C2×D4⋊C8—22no—
20#1296—22no—
21#1272—24no—
22#1270—22no—
23#559—48no—
24C2×C4⋊Q8D412yes—
25C2×#127C2×C411yescharacteristic
26C2×#124—24no—
27C2×#123—24no—
28C2×C4×Q8C2×C414yes—
29C2×C4×D4—24no—
30C2×C4×D4—22no—
31C2×C4×D4D412yes—
32C22×C4⋊C4—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—21no—
36C2×C4×C8—21no—
37#398—24no—
38#397—24no—
39#397—28no—
40#391—24no—
41#391D414yes—
42#390—22no—
43#390D412yes—
44#389—24no—
45#389—22no—
46#389—22no—
47#385—28no—
48#384C2×C411yescharacteristic
49#384D412yes—
50#381—24no—
51#380—24no—
52#380—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C43C2311yescharacteristic
56#336—22no—
57#336—22no—
58#336—22no—
59#333—44no—
60D4⋊C8—44no—
61C22×Q8C22⋊C414yes—
62C22×D4—22no—
63C22×D4C22⋊C412yes—
64C23×C4—24no—
65C23×C4—22no—
66C22×C8—41no—
67C22×C8—42no—
68C22×C8—41no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73#127—24no—
74#124—416no—
75#123—48no—
76#123—48no—
77C4×Q8—48no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—416no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—82no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4C22⋊C414yes—
95C2×C4⋊C4C22⋊C412yes—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C42—24no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D412yes—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D412yes—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4×C8—22no—
112C4×C8—41no—
113#95—28no—
114#95—24no—
115#95—24no—
116#95—24no—
117C24—22no—
118C2×Q8—24no—
119C2×Q8—24no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×D4—28no—
123C2×D4—22no—
124C2×D4—44no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4#10014yes—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—84no—
139C22×C4—84no—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×C22⋊C412yes—
142C22×C4#12011yescharacteristic
143C22×C4#12012yes—
144C22×C4#12011yescharacteristic
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—81no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—48no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C22⋊C4—416no—
166C22⋊C4—48no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—82no—
170C42—22no—
171C42—21no—
172C42—21no—
173C42—41no—
174C42—42no—
175C42—48no—
176C42#13512yes—
177C42#13512yes—
178C42AGL(1,ℤ/8)12yes—
179C42#13712yes—
180C23—24no—
181C23—42no—
182C23—42no—
183C23—44no—
184C23—44no—
185C23#37911yescharacteristic
186Q8—84no—
187Q8—84no—
188D4—44no—
189D4—48no—
190D4—48no—
191D4—82no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4—168no—
212C2×C4C2×#10012yes—
213C2×C4#41612yes—
214C2×C4#41712yes—
215C2×C4#44912yes—
216C2×C4#44912yes—
217C2×C4#45212yes—
218C2×C4#45212yes—
219C2×C4#45311yescharacteristic
220C2×C4#45511yescharacteristic
221C8—82no—
222C8—162no—
223C8—161no—
224C22—44no—
225C22—48no—
226C22—82no—
227C22—84no—
228C22—82no—
229C22#178311yescharacteristic
230C22#178812yes—
231C22#179211yescharacteristic
232C22#179411yescharacteristic
233C22#179612yes—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—84no—
241C4—84no—
242C4—82no—
243C4#162011yescharacteristic
244C4#163011yescharacteristic
245C4#164012yes—
246C2—84no—
247C2#839312yes—
248C2#843512yes—
249C2#844511yescharacteristic
250C2#845611yescharacteristic
251C2#849211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8590C2semidirect
5. #7590C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#96442
#85902
#76512
#75902