Finite Group Databasebeta

← #375661

Subgroups — #375661

Order 512.

1485 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1485 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×#2623C211yesmaximal, characteristic
2#13776C212yesmaximal, complemented
3#11197C212yesmaximal, complemented
4#10966C212yesmaximal
5#10960C214yesmaximal
6#10912C214yesmaximal, complemented
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×C4×C22⋊C4C2214yes—
20C2×#342C2218yes—
21C2×#342C2214yes—
22#1436—28no—
23C22⋊#95C2214yes—
24C23×D4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C24×C4C2312yes—
27C2×#123—24no—
28C2×#123—44no—
29C22×C4⋊C4—44no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C414yes—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D418yes—
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C22⋊C4C2×C414yes—
39C22×C22⋊C4C2311yescharacteristic
40C22×C22⋊C4C2312yes—
41C22×C22⋊C4C2318yes—
42C22×C42—24no—
43C2×#99—216no—
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—216no—
55C4×C22⋊C4—48no—
56C2×#95—28no—
57C2×#95—24no—
58C2×#95—28no—
59C2×#95C2×C412yes—
60C2×#95C2×C414yes—
61C2×#95C2×C414yes—
62C2×#95C2×C412yes—
63#342—216no—
64#342—44no—
65C25C22×C411yescharacteristic
66C25C2411yescharacteristic
67C22×D4—22no—
68C22×D4—28no—
69C22×D4—24no—
70C23×C4—28no—
71C23×C4—24no—
72C23×C4—28no—
73C23×C4—28no—
74C23×C4—84no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C414yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C414yes—
82C23×C4C2×D414yes—
83C23×C4C2×D414yes—
84C23×C4C2×D414yes—
85#123—416no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—48no—
92C2×C4⋊C4—48no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—84no—
96C2×C4⋊C4C22⋊C418yes—
97C2×C4⋊C4C22⋊C418yes—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—84no—
114C2×C22⋊C4#41116yes—
115C2×C42—48no—
116C2×C42—88no—
117#99—416no—
118#95—232no—
119#95—24no—
120#95—24no—
121#95—28no—
122#95—28no—
123#95—48no—
124#95—88no—
125C24—21no—
126C24—24no—
127C24—22no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24C23×C411yescharacteristic
133C24C22×D412yes—
134C24C2×#4118yes—
135C24H(5,𝔽2)14yes—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—48no—
141C2×D4—44no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—28no—
147C22×C4—28no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—216no—
160C22×C4—28no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—416no—
171C22×C4—416no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—816no—
179C22×C4C2×C22⋊C412yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4C2×C22⋊C418yes—
182C4⋊C4—48no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—88no—
187C22⋊C4—216no—
188C22⋊C4—216no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—88no—
197C42—816no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23—22no—
202C23—44no—
203C23—416no—
204C23—44no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—44no—
210C23—88no—
211C23—82no—
212C23—82no—
213C23C22×C22⋊C411yescharacteristic
214C23C4×#4114yes—
215C23#51812yes—
216C23#52514yes—
217C23#53214yes—
218C23#53414yes—
219C23#55114yes—
220C23#55318yes—
221C23#55914yes—
222D4—88no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—28no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—416no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—432no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—416no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—88no—
248C2×C4—88no—
249C2×C4—88no—
250C2×C4—816no—
251C2×C4—88no—
252C2×C4—88no—
253C2×C4—82no—
254C2×C4—82no—
255C22—28no—
256C22—28no—
257C22—28no—
258C22—24no—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—44no—
263C22—48no—
264C22—44no—
265C22—81no—
266C22—84no—
267C22—81no—
268C22—81no—
269C22—164no—
270C22#229712yes—
271C22#230811yescharacteristic
272C22#237414yes—
273C22#260218yes—
274C22#288312yes—
275C22#288412yes—
276C4—42no—
277C4—42no—
278C4—48no—
279C4—816no—
280C4—81no—
281C4—84no—
282C4—81no—
283C4—81no—
284C2—28no—
285C2—24no—
286C2—161no—
287C2#1416711yescharacteristic
288C2#2142314yes—
289C2#2175011yescharacteristic
290C2#2175111yescharacteristic
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. #13776C2semidirect
3. #11197C2semidirect
6. #10912C2semidirect
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
#137762
#111972
#109662
#109602
#109122