Finite Group Databasebeta

← #109668

Subgroups — #109668

Order 512.

1028 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 178 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#13966C211yesmaximal, characteristic
2#13749C211yesmaximal, characteristic
3C2×#1297C212yesmaximal
4#8019C212yesmaximal
5#7909C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#380C2211yescharacteristic
8C2×C4×C22⋊C4—22no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2212yes—
11C22×#95C2211yescharacteristic
12#1442C414yescomplemented
13#1434—22no—
14C22⋊#95—22no—
15C4×#95—24no—
16#1297C418yescomplemented
17#1294—24no—
18#1280—22no—
19C24×C4D412yes—
20C24×C4C2311yescharacteristic
21C22×M16—22no—
22C22×C4⋊C4C2×C412yes—
23C22×C22⋊C4—22no—
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C42—24no—
27C22×C42—22no—
28C22×C42—21no—
29#407—22no—
30#407—22no—
31#407—22no—
32#407C2×C414yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35#380—22no—
36#380—22no—
37C4×C22⋊C4—216no—
38C4×C22⋊C4—22no—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C4×C22⋊C4C2×C414yes—
42C4×C22⋊C4C2×C414yes—
43C4×C22⋊C4Q814yes—
44C2×#95—24no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—24no—
48C2×#95—21no—
49C2×#95—24no—
50C2×#95C2×C412yes—
51C2×#95C2×C412yes—
52C2×#95D412yes—
53C2×#95D414yes—
54C25C2×D411yescharacteristic
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—28no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—44no—
62C23×C4—82no—
63C23×C4—82no—
64C23×C4C22⋊C414yes—
65C23×C4C22×C411yescharacteristic
66C23×C4C22×C412yes—
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q812yes—
70C23×C4#4112yes—
71C23×C4#4112yes—
72C23×C4#4112yes—
73C2×M16—24no—
74C2×M16—42no—
75C2×M16—84no—
76C22×C8—42no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—82no—
79C2×C22⋊C4—216no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C42—216no—
89C2×C42—24no—
90C2×C42—24no—
91C2×C42—24no—
92C2×C42—84no—
93C2×C42—82no—
94C2×C42—84no—
95C2×C42—84no—
96C22⋊C8—24no—
97C22⋊C8—24no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#95—216no—
101#95—22no—
102#95—24no—
103#95—24no—
104#95—24no—
105#95—48no—
106#95—42no—
107#95—84no—
108#95C4214yescomplemented
109#95C4⋊C418yes—
110C24—24no—
111C24—22no—
112C24—22no—
113C24—42no—
114C24C2×C22⋊C411yescharacteristic
115C24C4×D412yes—
116C24#12011yescharacteristic
117C24C22⋊Q812yes—
118C24#12311yescharacteristic
119C22×C4—24no—
120C22×C4—216no—
121C22×C4—24no—
122C22×C4—28no—
123C22×C4—24no—
124C22×C4—28no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—48no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4—82no—
150C22×C4—816no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4#10418yes—
154C22×C4C2×C4211yescharacteristic
155C22×C4C2×C4⋊C412yes—
156C22×C4#11712yes—
157C22×C4#11711yescharacteristic
158C22×C4C4×D412yes—
159C22×C4C4×D412yes—
160C22×C4C4×D412yes—
161C22×C4C4×Q812yes—
162C22×C4C4×Q812yes—
163C22×C4#12112yes—
164C22×C4C22⋊Q812yes—
165C22×C4#12312yes—
166C22×C4#12411yescharacteristic
167C22×C4#12512yes—
168C22×C4#12612yes—
169C22×C4C4⋊Q811yescharacteristic
170M16—84no—
171M16—84no—
172C2×C8—44no—
173C2×C8—84no—
174C4⋊C4—84no—
175C22⋊C4—216no—
176C22⋊C4—216no—
177C22⋊C4—48no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C42—84no—
181C42—84no—
182C42—84no—
183C42—816no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—24no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—42no—
198C23—42no—
199C23—48no—
200C23—48no—
201C23C4×C22⋊C411yescharacteristic
202C23#38512yes—
203C23#38611yescharacteristic
204C23#38712yes—
205C23#39111yescharacteristic
206C23C2×#10414yes—
207C23#45412yes—
208C23#45512yes—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—216no—
212C2×C4—28no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—48no—
228C2×C4—416no—
229C2×C4—48no—
230C2×C4—416no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—84no—
240C2×C4—88no—
241C2×C4—84no—
242C2×C4—84no—
243C2×C4—88no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4C4×C4⋊C412yes—
248C2×C4#38712yes—
249C2×C4#38812yes—
250C2×C4#38912yes—
251C8—84no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—21no—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—48no—
266C22—48no—
267C22—42no—
268C22—42no—
269C22#143611yescharacteristic
270C22C4×#10412yes—
271C22#186014yes—
272C22#188511yescharacteristic
273C22#188811yescharacteristic
274C22#189612yes—
275C22#191312yes—
276C22#199412yes—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—22no—
281C4—416no—
282C4—48no—
283C4—84no—
284C4—162no—
285C2—24no—
286C2—22no—
287C2—22no—
288C2—42no—
289C2#940712yes—
290C2#941012yes—
291C2#1087811yescharacteristic
292C2#1089211yescharacteristic
293C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1442C4semidirect
16. #1297C4semidirect
108. #95C42semidirect

Maximal subgroups

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

subgroupindex
#139662
#137492
C2×#12972
#80192
#79092