Finite Group Databasebeta

← #100875

Subgroups — #100875

Order 512.

462 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 86 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#23347C211yesmaximal, characteristic
3#9903C212yesmaximal, complemented
4#7635C211yesmaximal, characteristic
5#7042C212yesmaximal, complemented
6C2×#497C412yes—
7C2×#462—21no—
8C2×#461—21no—
9#2868C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#2016—21no—
12#2013—21no—
13#1984—21no—
14#1960C412yes—
15#1923—22no—
16#1870—21no—
17#1851—22no—
18#1831C2212yes—
19#1767C2212yes—
20#1767C2211yescharacteristic
21#1296—22no—
22#1296—22no—
23#1272—21no—
24#1270—21no—
25C22×Q16—22no—
26C22×SD16—22no—
27C22×M16—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8C2×C412yes—
31C2×#124C2×C411yescharacteristic
32C2×C4×Q8D411yescharacteristic
33C2×C4×D4D411yescharacteristic
34#497—44no—
35#462—42no—
36#462—42no—
37#461—42no—
38#461—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D412yes—
43C2×C4⋊C8D412yes—
44C2×#103—22no—
45C2×#103—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—21no—
48C2×#97—22no—
49C2×#97C2×C412yes—
50C2×C4×C8—21no—
51#391D411yescharacteristic
52#390D411yescharacteristic
53#388—22no—
54#384—22no—
55#384—22no—
56#382C2×C411yescharacteristic
57C4×C22⋊C4—22no—
58C43C2311yescharacteristic
59#336—22no—
60#336—22no—
61#336—22no—
62#336—21no—
63#336—22no—
64C22×Q8—22no—
65C22×Q8C22⋊C411yescharacteristic
66C22×D4C22⋊C411yescharacteristic
67C23×C4—21no—
68C2×Q16—44no—
69C2×Q16—44no—
70C2×Q16—82no—
71C2×SD16—44no—
72C2×SD16—44no—
73C2×SD16—82no—
74C2×M16—81no—
75C2×M16—82no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—42no—
80C4⋊Q8—42no—
81C4⋊Q8—42no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—82no—
87#124—22no—
88#124—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C22⋊C411yescharacteristic
99C2×C4⋊C4C22⋊C411yescharacteristic
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C42—21no—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—24no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—81no—
116#103—44no—
117#103—44no—
118C22⋊C8—81no—
119C22⋊C8—82no—
120#97—42no—
121#97—82no—
122C4×C8—81no—
123#95—22no—
124#95—22no—
125C24—21no—
126C2×Q8—21no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8#10012yes—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D412yes—
150C22×C4#12011yescharacteristic
151C22×C4#12112yes—
152C22×C4#12311yescharacteristic
153Q16—88no—
154SD16—88no—
155M16—84no—
156C2×C8—24no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—81no—
171C4⋊C4—84no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—84no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C22⋊C4—44no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—41no—
190C42—41no—
191C42—41no—
192C23—42no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—41no—
197C23#38611yescharacteristic
198Q8—42no—
199Q8—81no—
200Q8—82no—
201Q8—82no—
202Q8—82no—
203Q8—82no—
204D4—42no—
205D4—81no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4C2×#10011yescharacteristic
229C2×C4#41311yescharacteristic
230C2×C4#45311yescharacteristic
231C2×C4#45311yescharacteristic
232C2×C4#45512yes—
233C2×C4#45612yes—
234C2×C4#47112yes—
235C2×C4#47312yes—
236C8—44no—
237C8—84no—
238C8—82no—
239C8—82no—
240C22—81no—
241C22—82no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22#188212yes—
246C22#188711yescharacteristic
247C22#190011yescharacteristic
248C22#190812yes—
249C22#191011yescharacteristic
250C4—22no—
251C4—22no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—81no—
258C4—81no—
259C4—162no—
260C4#165911yescharacteristic
261C4#166011yescharacteristic
262C4#166112yes—
263C2—161no—
264C2#900311yescharacteristic
265C2#935412yes—
266C2#937612yes—
267C2#943511yescharacteristic
268C2#945711yescharacteristic
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. #9903C2semidirect
5. #7042C2semidirect

Maximal subgroups

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

subgroupindex
#284492
#233472
#99032
#76352
#70422