Finite Group Databasebeta

← #100789

Subgroups — #100789

Order 512.

508 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 80 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#14668C211yesmaximal, characteristic
3#9888C212yesmaximal
4#7629C211yesmaximal, characteristic
5#7036C212yesmaximal
6C2×#483—21no—
7C2×#451—21no—
8#2857C2211yescharacteristic
9C2×#391C2211yescharacteristic
10#2096C412yescomplemented
11#2016C412yescomplemented
12#1971—22no—
13#1749C2212yes—
14#1749C2211yescharacteristic
15#1446—21no—
16#1443—22no—
17#1442—22no—
18#1441—21no—
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×C42C2311yescharacteristic
30#483—42no—
31#483—42no—
32#451—42no—
33#451—42no—
34C2×#106—22no—
35C2×#103—22no—
36C2×#103—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—22no—
39C2×#97—21no—
40C2×#97—22no—
41#391—22no—
42#391—21no—
43#391C2×C412yes—
44#391D412yes—
45#389—22no—
46#384C2×C411yescharacteristic
47#380—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95D412yes—
57C2×#95D412yes—
58#336—22no—
59#336C2×C412yes—
60C22×Q8—21no—
61C22×Q8—21no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—81no—
66C22×Q8C22⋊C412yes—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D412yes—
73C2×Q16—44no—
74C2×Q16—44no—
75C2×Q16—82no—
76C22×C8—22no—
77C22×C8—42no—
78C4⋊Q8—81no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—82no—
92C2×C4⋊C4C22⋊C412yes—
93C2×C22⋊C4—41no—
94C2×C42—81no—
95C2×C42C22×C411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42#4112yes—
98#106—44no—
99#103—44no—
100#103—44no—
101C22⋊C8—22no—
102C22⋊C8—22no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105#97—81no—
106#97—82no—
107#95—24no—
108#95—24no—
109#95—42no—
110#95—42no—
111#95—44no—
112#95—42no—
113#95—82no—
114#95—82no—
115#95—81no—
116#95—81no—
117#95—81no—
118#95—82no—
119C24#12011yescharacteristic
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—84no—
132C2×Q8—84no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×D412yes—
169C22×C4#12112yes—
170C22×C4#12311yescharacteristic
171Q16—88no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—82no—
177C2×C8—82no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—81no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—84no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C42—41no—
190C42—42no—
191C42—41no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23#38611yescharacteristic
196C23#45311yescharacteristic
197C23#45311yescharacteristic
198C23#45512yes—
199C23#45612yes—
200Q8—44no—
201Q8—82no—
202Q8—82no—
203Q8—82no—
204Q8—84no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#42112yes—
250C8—44no—
251C8—84no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22#165911yescharacteristic
261C22#166011yescharacteristic
262C22#166112yes—
263C22#188212yes—
264C22#188711yescharacteristic
265C22#189711yescharacteristic
266C22#190512yes—
267C22#191311yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—162no—
275C4—162no—
276C4—161no—
277C4—161no—
278C2—21no—
279C2—22no—
280C2—21no—
281C2#900311yescharacteristic
282C2#932812yes—
283C2#934212yes—
284C2#941311yescharacteristic
285C2#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
10. #2096C4semidirect
11. #2016C4semidirect

Maximal subgroups

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

subgroupindex
#283132
#146682
#98882
#76292
#70362