Finite Group Databasebeta

← #97502

Subgroups — #97502

Order 512.

768 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 136 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#9870C212yesmaximal, complemented
3#9638C211yesmaximal, characteristic
4#8586C212yesmaximal, complemented
5#8585C211yesmaximal, characteristic
6C2×#559—21no—
7#2724C414yes—
8#2582—21no—
9#2547—22no—
10#2486—22no—
11#2447—24no—
12D4×C4⋊C4—22no—
13C4×C4⋊Q8C2212yes—
14C4×#127C2211yescharacteristic
15#1842C2212yes—
16#1842C2211yescharacteristic
17C42×C8C2211yescharacteristic
18C2×#332—22no—
19C2×#331—21no—
20C2×D4⋊C8—22no—
21#1296—21no—
22#1296—22no—
23#1279—24no—
24#559—44no—
25#559—42no—
26#559—42no—
27C2×C4⋊Q8D412yes—
28C2×#127D411yescharacteristic
29C2×#124—24no—
30C2×#123—24no—
31C2×C4×Q8—22no—
32C2×C4×D4—24no—
33C2×C4×D4—22no—
34C2×C4×D4D412yes—
35C22×C4⋊C4—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×C4×C8—21no—
42C2×C4×C8—22no—
43#398—24no—
44#397—24no—
45#397—28no—
46#391—24no—
47#391—22no—
48#390—22no—
49#390C2×C412yes—
50#389—24no—
51#389—22no—
52#389C2×C414yes—
53#385—28no—
54#384D411yescharacteristic
55#384D412yes—
56#381—24no—
57#380—24no—
58#380—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4D414yes—
61C43C2311yescharacteristic
62#332—44no—
63#331—42no—
64D4⋊C8—44no—
65C22×Q8—22no—
66C22×D4—22no—
67C22×D4C22⋊C412yes—
68C23×C4—24no—
69C23×C4—22no—
70C22×C8—41no—
71C22×C8—42no—
72C22×C8—41no—
73C4⋊Q8—22no—
74C4⋊Q8—22no—
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77#127—22no—
78#127D812yes—
79#127SD1612yes—
80#124—48no—
81#124—48no—
82#123—48no—
83#123—48no—
84C4×Q8—42no—
85C4×Q8—42no—
86C4×Q8—42no—
87C4×Q8—42no—
88C4×D4—42no—
89C4×D4—42no—
90C4×D4—48no—
91C4×D4—48no—
92C4×D4—42no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—42no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4C22⋊C414yes—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C414yes—
107C2×C22⋊C4—28no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C42—24no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C4⋊C8—42no—
118C4⋊C8—44no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C4×C8—22no—
126C4×C8—24no—
127C4×C8—42no—
128C4×C8—41no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133C24—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×D4—22no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4#10214yes—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4C2×C22⋊C412yes—
162C22×C4#12011yescharacteristic
163C22×C4#12012yes—
164C22×C4#12011yescharacteristic
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—82no—
168C2×C8—82no—
169C2×C8—82no—
170C2×C8—81no—
171C2×C8—82no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—84no—
187C22⋊C4—416no—
188C22⋊C4—48no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—82no—
192C42—21no—
193C42—22no—
194C42—21no—
195C42—22no—
196C42—41no—
197C42—44no—
198C42—44no—
199C42C2×D811yescharacteristic
200C42C2×SD1611yescharacteristic
201C42#13512yes—
202C42#13512yes—
203C42AGL(1,ℤ/8)12yes—
204C42AGL(1,ℤ/8)12yes—
205C42#13712yes—
206C23—24no—
207C23—42no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23#37911yescharacteristic
212Q8—82no—
213Q8—82no—
214Q8—82no—
215Q8—82no—
216D4—44no—
217D4—44no—
218D4—44no—
219D4—44no—
220D4—44no—
221D4—82no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—84no—
239C2×C4—168no—
240C2×C4C2×#10211yescharacteristic
241C2×C4#41612yes—
242C2×C4#41712yes—
243C2×C4#41811yescharacteristic
244C2×C4#44712yes—
245C2×C4#44912yes—
246C2×C4#44912yes—
247C2×C4#45012yes—
248C2×C4#45212yes—
249C2×C4#45212yes—
250C2×C4#45311yescharacteristic
251C2×C4#45412yes—
252C2×C4#45511yescharacteristic
253C2×C4#45612yes—
254C8—44no—
255C8—161no—
256C8—162no—
257C22—44no—
258C22—48no—
259C22—82no—
260C22—84no—
261C22—82no—
262C22#178611yescharacteristic
263C22#178812yes—
264C22#179111yescharacteristic
265C22#179211yescharacteristic
266C22#179312yes—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—84no—
273C4—82no—
274C4—84no—
275C4#162011yescharacteristic
276C4#162512yes—
277C4#163011yescharacteristic
278C4#163512yes—
279C4#164012yes—
280C4#164712yes—
281C4#165412yes—
282C2—84no—
283C2#835311yescharacteristic
284C2#835712yes—
285C2#844511yescharacteristic
286C2#844612yes—
287C2#853111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
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
2. #9870C2semidirect
4. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#98702
#96382
#85862
#85852