Finite Group Databasebeta

← #100757

Subgroups — #100757

Order 512.

477 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 96 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#28449C211yesmaximal, characteristic
2#23346C211yesmaximal, characteristic
3#9905C212yesmaximal, complemented
4#8579C211yesmaximal, characteristic
5#7049C212yesmaximal, complemented
6C2×#497—21no—
7C2×#462—21no—
8C2×#461—21no—
9#2868C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#2016C412yes—
12#2013C412yes—
13#1984—21no—
14#1960—21no—
15#1923—22no—
16#1869—21no—
17#1850—22no—
18#1831C2212yes—
19#1767C2212yes—
20#1767C2211yescharacteristic
21C2×#329—21no—
22#1296—22no—
23#1296—22no—
24#1295—21no—
25C22×Q16—22no—
26C22×SD16—22no—
27C23×C8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8C2×C412yes—
31C2×#124D411yescharacteristic
32C2×C4×Q8D411yescharacteristic
33C2×C4×D4D411yescharacteristic
34#497—42no—
35#497—42no—
36#462—42no—
37#462—42no—
38#461—42no—
39#461—42no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D412yes—
44C2×C4⋊C8D412yes—
45C2×#103—22no—
46C2×#103—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—21no—
49C2×#97—21no—
50C2×#97—22no—
51C2×C4×C8—21no—
52#391C2×C411yescharacteristic
53#390C2×C411yescharacteristic
54#388—22no—
55#384—22no—
56#384—22no—
57#382D411yescharacteristic
58C4×C22⋊C4—22no—
59C43C2311yescharacteristic
60#336—22no—
61#336—22no—
62#336—22no—
63#336—22no—
64#336C2×C412yes—
65#329—42no—
66C22×Q8—22no—
67C22×Q8C22⋊C411yescharacteristic
68C22×D4C22⋊C411yescharacteristic
69C23×C4—21no—
70C2×Q16—44no—
71C2×Q16—44no—
72C2×Q16—82no—
73C2×SD16—44no—
74C2×SD16—44no—
75C2×SD16—82no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—42no—
80C22×C8—81no—
81C22×C8—82no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—42no—
89C4⋊Q8—82no—
90#124—22no—
91#124D812yes—
92#124SD1612yes—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—41no—
96C4×D4—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C22⋊C411yescharacteristic
103C2×C4⋊C4C22⋊C411yescharacteristic
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C42—21no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42#4112yes—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—81no—
120#103—44no—
121#103—44no—
122C22⋊C8—81no—
123C22⋊C8—82no—
124#97—41no—
125#97—42no—
126#97—41no—
127#97—42no—
128C4×C8—81no—
129#95—22no—
130#95—22no—
131C24—21no—
132C2×Q8—21no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8#10212yes—
142C2×D4—21no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4#10212yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C4×D412yes—
156C22×C4#12011yescharacteristic
157C22×C4#12112yes—
158C22×C4#12311yescharacteristic
159Q16—88no—
160SD16—88no—
161C2×C8—24no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8—84no—
173C2×C8—82no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—81no—
177C4⋊C4—84no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—82no—
187C22⋊C4—81no—
188C42—21no—
189C42—21no—
190C42—22no—
191C42—22no—
192C42—41no—
193C42—42no—
194C42—41no—
195C42—41no—
196C42C2×D811yescharacteristic
197C42C2×SD1611yescharacteristic
198C42#13512yes—
199C23—22no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23#38611yescharacteristic
205Q8—42no—
206Q8—81no—
207Q8—82no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211D4—42no—
212D4—81no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4C2×#10211yescharacteristic
236C2×C4#41611yescharacteristic
237C2×C4#45311yescharacteristic
238C2×C4#45311yescharacteristic
239C2×C4#45512yes—
240C2×C4#45612yes—
241C2×C4#46512yes—
242C2×C4#46612yes—
243C8—44no—
244C8—84no—
245C8—82no—
246C8—82no—
247C22—44no—
248C22—42no—
249C22—81no—
250C22—82no—
251C22—81no—
252C22#188212yes—
253C22#188711yescharacteristic
254C22#189311yescharacteristic
255C22#190212yes—
256C22#191411yescharacteristic
257C4—22no—
258C4—22no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—162no—
267C4#165911yescharacteristic
268C4#166011yescharacteristic
269C4#166112yes—
270C2—82no—
271C2#900311yescharacteristic
272C2#931612yes—
273C2#933712yes—
274C2#939711yescharacteristic
275C2#942111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊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. #9905C2semidirect
5. #7049C2semidirect

Maximal subgroups

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

subgroupindex
#284492
#233462
#99052
#85792
#70492