Finite Group Databasebeta

← #375711

Subgroups — #375711

Order 512.

1582 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1582 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C2×#2639C211yesmaximal, characteristic
2#13796C212yesmaximal, complemented
3#11381C212yesmaximal, complemented
4#10980C212yesmaximal
5#10960C218yesmaximal, complemented
6C22×C22⋊Q8C2211yescharacteristic
7C22×#121C2211yescharacteristic
8C22×#99C2218yes—
9#2639—28no—
10C2×#400C412yescomplemented
11C2×#397C412yescomplemented
12C2×#394C412yescomplemented
13C2×#393C412yescomplemented
14C2×#386C2214yes—
15C2×C4×C22⋊C4C2214yes—
16C22×#95C2211yescharacteristic
17C2×#342C2218yes—
18#1434—28no—
19C22⋊#95C2214yes—
20C22⋊#95C2214yes—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C23×D4C2312yes—
24C24×C4C2311yescharacteristic
25C24×C4C2312yes—
26C2×C22⋊Q8—24no—
27C2×C22⋊Q8—44no—
28C2×#121—24no—
29C2×#121—44no—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C414yes—
35C22×C22⋊C4C2×C414yes—
36C22×C22⋊C4C2312yes—
37C22×C22⋊C4C2318yes—
38C22×C42—24no—
39C2×#99—232no—
40C2×#99—48no—
41#400—28no—
42#397—28no—
43#394—28no—
44#393—28no—
45#386—28no—
46#386—28no—
47C4×C22⋊C4—216no—
48C2×#95—24no—
49C2×#95—24no—
50C2×#95—24no—
51C2×#95—24no—
52C2×#95—24no—
53C2×#95C2×C414yes—
54C2×#95C2×C414yes—
55C2×#95C2×C412yes—
56C2×#95C2×C412yes—
57C2×#95D418yes—
58#342—48no—
59C25C22×C412yes—
60C25C2411yescharacteristic
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—82no—
64C22×D4—24no—
65C22×D4—216no—
66C22×D4—28no—
67C22×D4—22no—
68C22×D4—42no—
69C22×D4—82no—
70C23×C4—28no—
71C23×C4—24no—
72C23×C4—28no—
73C23×C4—82no—
74C23×C4—82no—
75C23×C4—82no—
76C23×C4C22×C412yes—
77C23×C4C22×C412yes—
78C23×C4C22×C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C414yes—
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D414yes—
83C23×C4C2×D414yes—
84C23×C4C2×D414yes—
85C22⋊Q8—416no—
86#121—416no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—44no—
93C2×C22⋊C4—216no—
94C2×C22⋊C4—216no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—84no—
103C2×C22⋊C4—84no—
104C2×C22⋊C4#41116yes—
105C2×C42—216no—
106C2×C42—84no—
107C2×C42—84no—
108#99—432no—
109#95—216no—
110#95—28no—
111#95—28no—
112#95—84no—
113#95—84no—
114#95—84no—
115#95—84no—
116#95C22⋊C418yes—
117#95C22⋊C418yes—
118C24—22no—
119C24—28no—
120C24—24no—
121C24—41no—
122C24—42no—
123C24—41no—
124C24—42no—
125C24—44no—
126C24—42no—
127C24C23×C411yescharacteristic
128C24C22×D412yes—
129C24C2×#4118yes—
130C24H(5,𝔽2)14yes—
131C2×Q8—44no—
132C2×Q8—84no—
133C2×Q8—88no—
134C2×D4—44no—
135C2×D4—48no—
136C2×D4—48no—
137C2×D4—48no—
138C2×D4—48no—
139C2×D4—48no—
140C2×D4—416no—
141C2×D4—84no—
142C2×D4—88no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—28no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—28no—
154C22×C4—24no—
155C22×C4—216no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—48no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—816no—
178C22×C4—88no—
179C22×C4C2×C22⋊C418yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4C2×C22⋊C412yes—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C22⋊C4—232no—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—416no—
189C22⋊C4—88no—
190C22⋊C4—88no—
191C42—816no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—48no—
200C23—44no—
201C23—48no—
202C23—44no—
203C23—42no—
204C23—42no—
205C23—44no—
206C23—42no—
207C23—44no—
208C23—48no—
209C23—84no—
210C23—84no—
211C23—88no—
212C23—88no—
213C23C22×C22⋊C411yescharacteristic
214C23C4×#4114yes—
215C23#51812yes—
216C23#52514yes—
217C23#53214yes—
218C23#53514yes—
219C23#55118yes—
220C23#55518yes—
221Q8—88no—
222D4—88no—
223D4—816no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—28no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—416no—
236C2×C4—416no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—416no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—48no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—88no—
247C2×C4—88no—
248C2×C4—88no—
249C2×C4—88no—
250C2×C4—88no—
251C2×C4—88no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—84no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—24no—
263C22—24no—
264C22—24no—
265C22—48no—
266C22—48no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—84no—
271C22—84no—
272C22—168no—
273C22#229712yes—
274C22#230911yescharacteristic
275C22#237214yes—
276C22#261018yes—
277C22#288312yes—
278C22#288311yescharacteristic
279C22#288411yescharacteristic
280C4—42no—
281C4—42no—
282C4—48no—
283C4—816no—
284C4—82no—
285C4—82no—
286C4—82no—
287C2—28no—
288C2—24no—
289C2—162no—
290C2#1416911yescharacteristic
291C2#2142212yes—
292C2#2142312yes—
293C2#2179511yescharacteristic
294C2#2179611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13796C2semidirect
3. #11381C2semidirect
5. #10960C2semidirect
10. C2×#400C4semidirect
11. C2×#397C4semidirect
12. C2×#394C4semidirect
13. C2×#393C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
#137962
#113812
#109802
#109602