Finite Group Databasebeta

← #100779

Subgroups — #100779

Order 512.

580 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 84 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#28313C211yesmaximal, characteristic
2#13834C211yesmaximal, characteristic
3#9888C212yesmaximal
4#7629C211yesmaximal, characteristic
5#7036C212yesmaximal
6C2×#483—21no—
7C2×#451—21no—
8#2857C2211yescharacteristic
9C2×#391—21no—
10C2×#391C2211yescharacteristic
11#2096C412yescomplemented
12#2016C412yescomplemented
13#1971—22no—
14#1749C2212yes—
15#1749C2211yescharacteristic
16#1436—22no—
17#1436—21no—
18#1434—22no—
19#1434C2212yes—
20#1294—22no—
21#1294—22no—
22#1272—22no—
23C23×Q8D411yescharacteristic
24C22×Q16—22no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×C22⋊Q8—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32#483—42no—
33#483—42no—
34#451—42no—
35#451—42no—
36C2×#106—22no—
37C2×#103—22no—
38C2×#103—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—22no—
41C2×#97—21no—
42C2×#97—22no—
43#391—28no—
44#391—22no—
45#391—21no—
46#391C2×C412yes—
47#391D412yes—
48#389—22no—
49#384C2×C411yescharacteristic
50#380—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95D412yes—
58C2×#95D412yes—
59#336—22no—
60#336C2×C412yes—
61C22×Q8—24no—
62C22×Q8—21no—
63C22×Q8—21no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—81no—
67C22×Q8C22⋊C412yes—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C2×Q16—44no—
75C2×Q16—44no—
76C2×Q16—82no—
77C22×C8—22no—
78C22×C8—42no—
79C4⋊Q8—81no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4C22⋊C412yes—
95C2×C22⋊C4—41no—
96C2×C42—24no—
97C2×C42—81no—
98C2×C42—81no—
99C2×C42—81no—
100C2×C42—81no—
101C2×C42—82no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42#4112yes—
105#106—44no—
106#103—44no—
107#103—44no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112#97—81no—
113#97—82no—
114#95—28no—
115#95—28no—
116#95—24no—
117#95—24no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—82no—
122#95—81no—
123#95—82no—
124C24#12011yescharacteristic
125C2×Q8—24no—
126C2×Q8—22no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—84no—
136C2×Q8—84no—
137C2×Q8#10414yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C4×D412yes—
176C22×C4#12112yes—
177C22×C4#12311yescharacteristic
178Q16—88no—
179C2×C8—24no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—44no—
183C2×C8—82no—
184C2×C8—82no—
185C4⋊C4—42no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—81no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—84no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C42—41no—
197C42—42no—
198C42—41no—
199C42—84no—
200C42—84no—
201C23—21no—
202C23—22no—
203C23—21no—
204C23#38611yescharacteristic
205C23#45311yescharacteristic
206C23#45311yescharacteristic
207C23#45512yes—
208C23#45612yes—
209Q8—44no—
210Q8—82no—
211Q8—82no—
212Q8—82no—
213Q8—84no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4C2×#10412yes—
261C8—44no—
262C8—84no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22#165911yescharacteristic
272C22#166011yescharacteristic
273C22#166112yes—
274C22#188212yes—
275C22#188711yescharacteristic
276C22#189611yescharacteristic
277C22#190512yes—
278C22#191311yescharacteristic
279C4—22no—
280C4—42no—
281C4—42no—
282C4—41no—
283C4—44no—
284C4—84no—
285C4—84no—
286C4—82no—
287C4—161no—
288C4—161no—
289C2—21no—
290C2—22no—
291C2—21no—
292C2#900311yescharacteristic
293C2#932412yes—
294C2#934212yes—
295C2#940911yescharacteristic
296C2#942911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2096C4semidirect
12. #2016C4semidirect

Maximal subgroups

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

subgroupindex
#283132
#138342
#98882
#76292
#70362