Finite Group Databasebeta

← #391730

Subgroups — #391730

Order 512.

1381 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1381 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2623C211yesmaximal, characteristic
2#14339C212yesmaximal, complemented
3#11203C214yesmaximal, complemented
4#11197C212yesmaximal, complemented
5#11043C214yesmaximal
6#10966C212yesmaximal
7C22×#123C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×#99C2214yes—
10#2623—28no—
11C2×#400C412yescomplemented
12C2×#399C412yescomplemented
13C2×#398C412yescomplemented
14C2×#397C412yescomplemented
15C2×#386C2212yes—
16C2×#385C2212yes—
17C2×#381C2214yes—
18C2×#381C2211yescharacteristic
19C2×#380C2214yes—
20C2×#342C2218yes—
21C2×#342C2214yes—
22#1441—28no—
23C22⋊#95C2214yes—
24C23×D4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C24×C4C2312yes—
27C2×#123—24no—
28C2×#123—44no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—44no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C414yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D418yes—
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4C2×C414yes—
40C22×C22⋊C4C2318yes—
41C22×C22⋊C4C2311yescharacteristic
42C22×C22⋊C4C2312yes—
43C2×#99—48no—
44C2×#99—44no—
45#400—28no—
46#399—28no—
47#398—28no—
48#397—28no—
49#386—24no—
50#386—24no—
51#385—24no—
52#385—24no—
53#381—24no—
54#381—48no—
55#380—48no—
56C2×#95—24no—
57C2×#95—28no—
58C2×#95—28no—
59C2×#95C2×C412yes—
60C2×#95C2×C414yes—
61C2×#95C2×C414yes—
62C2×#95C2×C412yes—
63#342—44no—
64#342—48no—
65C25C22×C411yescharacteristic
66C25C2411yescharacteristic
67C22×D4—22no—
68C22×D4—24no—
69C22×D4—24no—
70C22×D4—24no—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—28no—
75C23×C4—28no—
76C23×C4—84no—
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C22×C414yes—
83C23×C4C22×C414yes—
84C23×C4C2×D414yes—
85C23×C4C2×D414yes—
86C23×C4C2×D414yes—
87#123—416no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—48no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4—88no—
99C2×C4⋊C4—84no—
100C2×C4⋊C4C22⋊C418yes—
101C2×C4⋊C4C22⋊C418yes—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—84no—
120#99—416no—
121#95—24no—
122#95—24no—
123#95—28no—
124#95—28no—
125#95—48no—
126#95—416no—
127#95—88no—
128C24—21no—
129C24—22no—
130C24—24no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24—42no—
135C24C23×C411yescharacteristic
136C24C22×D412yes—
137C24H(5,𝔽2)14yes—
138C24H(5,𝔽2)14yes—
139C242−1+414yes—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—48no—
146C2×D4—44no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—28no—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—28no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—416no—
180C22×C4—416no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4—816no—
185C22×C4C2×C22⋊C412yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C418yes—
188C4⋊C4—48no—
189C4⋊C4—48no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—88no—
193C4⋊C4—816no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C22⋊C4—48no—
204C22⋊C4—88no—
205C23—24no—
206C23—22no—
207C23—24no—
208C23—22no—
209C23—44no—
210C23—48no—
211C23—44no—
212C23—48no—
213C23—41no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—44no—
218C23—88no—
219C23—82no—
220C23—82no—
221C23C22×C22⋊C411yescharacteristic
222C23#51812yes—
223C23#52014yes—
224C23#53414yes—
225C23#53512yes—
226C23#53612yes—
227C23#53714yes—
228C23#56014yes—
229C23#56218yes—
230C23#56314yes—
231D4—84no—
232D4—84no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—48no—
241C2×C4—48no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—88no—
257C2×C4—88no—
258C2×C4—88no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—88no—
262C2×C4—816no—
263C2×C4—816no—
264C2×C4—88no—
265C2×C4—88no—
266C22—24no—
267C22—28no—
268C22—28no—
269C22—24no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—48no—
276C22—44no—
277C22—81no—
278C22—84no—
279C22—81no—
280C22—81no—
281C22—164no—
282C22#230811yescharacteristic
283C22#231112yes—
284C22#237614yes—
285C22#281314yes—
286C22#281414yes—
287C22#288312yes—
288C22#288412yes—
289C4—42no—
290C4—42no—
291C4—48no—
292C4—81no—
293C4—84no—
294C4—81no—
295C4—81no—
296C4—168no—
297C2—24no—
298C2—24no—
299C2—24no—
300C2—161no—
301C2#1465811yescharacteristic
302C2#2175011yescharacteristic
303C2#2175111yescharacteristic
304C2#2187714yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14339C2semidirect
3. #11203C2semidirect
4. #11197C2semidirect
11. C2×#400C4semidirect
12. C2×#399C4semidirect
13. C2×#398C4semidirect
14. C2×#397C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
#143392
#112032
#111972
#110432
#109662