Finite Group Databasebeta

← #391694

Subgroups — #391694

Order 512.

2777 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 2777 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×#2613C211yesmaximal, characteristic
2#14335C212yesmaximal, complemented
3#11197C212yesmaximal, complemented
4#11196C214yesmaximal, complemented
5#10966C212yesmaximal, complemented
6#10965C214yesmaximal, complemented
7C24×D4C2211yescharacteristic
8C22×#123C2211yescharacteristic
9C22×#120C2212yes—
10C22×#99C2214yes—
11#2613—28no—
12C2×#394C414yescomplemented
13C2×#392C414yescomplemented
14C2×#381C2214yes—
15C2×#381C2211yescharacteristic
16C2×#379C2214yes—
17C2×#379C2214yes—
18C2×#379C2212yes—
19C2×#342C2214yes—
20C2×#342C2218yes—
21#1438—28no—
22C26C2312yes—
23C23×D4—44no—
24C23×D4—48no—
25C23×D4C2×C414yes—
26C23×D4C2×C414yes—
27C23×D4D418yes—
28C23×D4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C2×#123—24no—
31C2×#123—44no—
32C2×#120—24no—
33C2×#120—24no—
34C2×#120—48no—
35C22×C4⋊C4C2×C412yes—
36C22×C22⋊C4—28no—
37C22×C22⋊C4—28no—
38C22×C22⋊C4C2×C412yes—
39C22×C22⋊C4C2×C414yes—
40C22×C22⋊C4C2×C418yes—
41C22×C22⋊C4C2318yes—
42C22×C22⋊C4C2312yes—
43C22×C22⋊C4C2311yescharacteristic
44C2×#99—44no—
45C2×#99—48no—
46#394—216no—
47#392—216no—
48#381—24no—
49#381—48no—
50#379—24no—
51#379—24no—
52#379—48no—
53#379—48no—
54C2×#95—28no—
55C2×#95C2×C414yes—
56#342—48no—
57#342—44no—
58C25—28no—
59C25—28no—
60C25—28no—
61C25—88no—
62C25C22×C411yescharacteristic
63C25C22×C414yes—
64C25C2×D414yes—
65C25C2×D414yes—
66C25C2411yescharacteristic
67C22×D4—22no—
68C22×D4—28no—
69C22×D4—24no—
70C22×D4—28no—
71C22×D4—48no—
72C22×D4—416no—
73C22×D4—416no—
74C22×D4—416no—
75C22×D4—416no—
76C22×D4—48no—
77C22×D4—432no—
78C22×D4—416no—
79C22×D4—416no—
80C22×D4—88no—
81C22×D4C22⋊C4116yes—
82C23×C4—24no—
83C23×C4—28no—
84C23×C4—28no—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C412yes—
87C23×C4C22×C412yes—
88C23×C4C22×C414yes—
89C23×C4C2×D414yes—
90#123—416no—
91#120—416no—
92#120—416no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—44no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—28no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—28no—
108C2×C22⋊C4—216no—
109C2×C22⋊C4—48no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—416no—
112C2×C22⋊C4—416no—
113C2×C22⋊C4—816no—
114C2×C22⋊C4—816no—
115C2×C22⋊C4—88no—
116#99—416no—
117#95—28no—
118#95—416no—
119C24—28no—
120C24—28no—
121C24—28no—
122C24—28no—
123C24—28no—
124C24—28no—
125C24—28no—
126C24—28no—
127C24—28no—
128C24—21no—
129C24—22no—
130C24—24no—
131C24—44no—
132C24—48no—
133C24—416no—
134C24—416no—
135C24—48no—
136C24—44no—
137C24—416no—
138C24—48no—
139C24—48no—
140C24—832no—
141C24—832no—
142C24—832no—
143C24C2×C22⋊C418yes—
144C24C23×C411yescharacteristic
145C24C22×D412yes—
146C24H(5,𝔽2)18yes—
147C24H(5,𝔽2)14yes—
148C2×D4—416no—
149C2×D4—432no—
150C2×D4—464no—
151C2×D4—4128no—
152C2×D4—416no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—48no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—816no—
160C2×D4—832no—
161C22×C4—24no—
162C22×C4—28no—
163C22×C4—28no—
164C22×C4—28no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—432no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4C2×C22⋊C414yes—
183C4⋊C4—48no—
184C22⋊C4—48no—
185C22⋊C4—416no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—816no—
194C22⋊C4—832no—
195C22⋊C4—832no—
196C23—28no—
197C23—24no—
198C23—28no—
199C23—28no—
200C23—28no—
201C23—216no—
202C23—216no—
203C23—416no—
204C23—48no—
205C23—416no—
206C23—416no—
207C23—416no—
208C23—416no—
209C23—416no—
210C23—416no—
211C23—416no—
212C23—416no—
213C23—416no—
214C23—41no—
215C23—42no—
216C23—44no—
217C23—44no—
218C23—816no—
219C23—816no—
220C23—816no—
221C23—816no—
222C23—816no—
223C23—816no—
224C23—88no—
225C23—84no—
226C23—864no—
227C23—832no—
228C23—832no—
229C23C22×C22⋊C411yescharacteristic
230C23#51814yes—
231C23#51812yes—
232C23#53414yes—
233C23#53414yes—
234C23#53414yes—
235C23#56018yes—
236C23#56014yes—
237C23H(3,𝔽4)14yes—
238D4—464no—
239D4—832no—
240D4—88no—
241C2×C4—24no—
242C2×C4—28no—
243C2×C4—48no—
244C2×C4—416no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—44no—
252C2×C4—84no—
253C2×C4—88no—
254C2×C4—816no—
255C2×C4—816no—
256C2×C4—816no—
257C2×C4—88no—
258C22—28no—
259C22—28no—
260C22—28no—
261C22—28no—
262C22—24no—
263C22—48no—
264C22—44no—
265C22—44no—
266C22—48no—
267C22—48no—
268C22—48no—
269C22—416no—
270C22—416no—
271C22—832no—
272C22—832no—
273C22—832no—
274C22—81no—
275C22—84no—
276C22—82no—
277C22—164no—
278C22#230812yes—
279C22#230811yescharacteristic
280C22#237514yes—
281C22#281218yes—
282C22#288314yes—
283C4—44no—
284C4—81no—
285C4—84no—
286C4—82no—
287C4—168no—
288C2—28no—
289C2—24no—
290C2—88no—
291C2—161no—
292C2#1465711yescharacteristic
293C2#2174914yes—
294C2#2175012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC2×#2613C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×D4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14335C2semidirect
3. #11197C2semidirect
4. #11196C2semidirect
5. #10966C2semidirect
6. #10965C2semidirect
12. C2×#394C4semidirect
13. C2×#392C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26132
#143352
#111972
#111962
#109662
#109652