Finite Group Databasebeta

← #391712

Subgroups — #391712

Order 512.

1799 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1799 conjugacy classes of subgroups — too many to draw. Only its 234 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#35659C211yesmaximal, characteristic
2#20593C212yesmaximal, complemented
3#11173C214yesmaximal, complemented
4#11171C212yesmaximal, complemented
5#10993C214yesmaximal, complemented
6#10968C212yesmaximal, complemented
7C22×2−1+4C2211yescharacteristic
8C2×#525C2212yes—
9C22×C22⋊Q8C2211yescharacteristic
10C2×#413C2214yes—
11C22×#101C2214yes—
12#2879C2214yes—
13C2×#407C2214yes—
14#2409—24no—
15#2407—24no—
16#2406C414yes—
17#2404C414yes—
18#2311C2212yes—
19#2310C2211yescharacteristic
20#1785—24no—
21#1784—24no—
22C2×#344C2218yes—
23C2×#343C2214yes—
24C2×2−1+4—44no—
25C2×2−1+4—48no—
26C22×#41C2×C414yes—
27C22×#41D414yes—
28C22×#41C2312yes—
29C23×Q8D414yes—
30C23×Q8C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16C2318yes—
33#525—48no—
34C2×#123C2×C414yes—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—44no—
38C2×#121—22no—
39C2×#120C2×C414yes—
40C2×C4×D4—24no—
41C2×#117C2312yes—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4C2×C412yes—
45#413—48no—
46C2×#101—28no—
47C2×#101—24no—
48C2×#101—48no—
49C2×#100—24no—
50#408—48no—
51#407—28no—
52#407—48no—
53C2×C22⋊C8—44no—
54#395—24no—
55#394—24no—
56#393—24no—
57#392—24no—
58#391—24no—
59#391—22no—
60#390—22no—
61#386—28no—
62#386C2×C418yes—
63#385—28no—
64#381C2×C414yes—
65#380—24no—
66#344—48no—
67#343—44no—
68C25C22×C411yescharacteristic
692−1+4—432no—
702−1+4—416no—
712−1+4—416no—
72C2×#41—28no—
73C2×#41—28no—
74C2×#41—28no—
75C2×#41—48no—
76C2×#41—416no—
77C2×#41—48no—
78C2×#41—48no—
79C2×#41—48no—
80C22×Q8—24no—
81C22×Q8—28no—
82C22×Q8—48no—
83C22×Q8—48no—
84C22×Q8—48no—
85C22×Q8C22⋊C414yes—
86C22×Q8C2×D412yes—
87C22×Q8C2×D414yes—
88C22×D4—24no—
89C22×D4C22⋊C414yes—
90C22×D4C22×C414yes—
91C22×D4C2×D412yes—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—28no—
95C23×C4C22⋊C418yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C412yes—
99C23×C4C2×D414yes—
100C23×C4C2411yescharacteristic
101C2×M16—28no—
102C2×M16—28no—
103C2×M16—28no—
104C2×M16—28no—
105C2×M16—28no—
106C2×M16—28no—
107C22×C8—28no—
108#123—48no—
109C22⋊Q8—416no—
110C22⋊Q8—48no—
111#121—48no—
112#120—48no—
113C4×D4—416no—
114#117—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—44no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4C22×C414yes—
127C2×C42—22no—
128#101—416no—
129#101—48no—
130#100—48no—
131C22⋊C8—416no—
132C22⋊C8—416no—
133#95—28no—
134#95—24no—
135#95—24no—
136C24—21no—
137C24—22no—
138C24—24no—
139C24—42no—
140C24—81no—
141C24C2×C22⋊C414yes—
142C24C23×C411yescharacteristic
143#41—464no—
144#41—464no—
145#41—416no—
146#41—48no—
147#41—48no—
148C2×Q8—28no—
149C2×Q8—24no—
150C2×Q8—28no—
151C2×Q8—24no—
152C2×Q8—28no—
153C2×Q8—24no—
154C2×Q8—48no—
155C2×Q8—44no—
156C2×Q8—464no—
157C2×Q8—48no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×D4—28no—
161C2×D4—28no—
162C2×D4—28no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—48no—
167C2×D4—44no—
168C2×D4—48no—
169C2×D4—48no—
170C2×D4—84no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—48no—
183C22×C4—48no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—82no—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C22⋊C414yes—
194C22×C4C22×D412yes—
195C22×C4H(5,𝔽2)14yes—
196C22×C4H(5,𝔽2)18yes—
197M16—48no—
198M16—48no—
199M16—48no—
200M16—48no—
201C2×C8—48no—
202C2×C8—48no—
203C2×C8—48no—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C22⋊C4—28no—
209C22⋊C4—48no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—88no—
213C42—44no—
214C23—22no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—41no—
220C23—41no—
221C23—44no—
222C23—44no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—44no—
227C23—42no—
228C23—41no—
229C23—44no—
230C23—88no—
231C23C22×C22⋊C411yescharacteristic
232C23#51812yes—
233C23#51814yes—
234Q8—416no—
235Q8—416no—
236Q8—48no—
237D4—416no—
238D4—416no—
239D4—48no—
240D4—816no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—432no—
247C2×C4—48no—
248C2×C4—416no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—88no—
258C2×C4—84no—
259C2×C4#53414yes—
260C2×C4#53414yes—
261C2×C4#53414yes—
262C2×C4#56018yes—
263C2×C4#56014yes—
264C2×C4H(3,𝔽4)14yes—
265C8—88no—
266C22—21no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—44no—
271C22—44no—
272C22—48no—
273C22—48no—
274C22—42no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—84no—
279C22—82no—
280C22—84no—
281C22—81no—
282C22—81no—
283C22#230812yes—
284C22#230811yescharacteristic
285C22#237514yes—
286C4—44no—
287C4—44no—
288C4—84no—
289C4—82no—
290C4—81no—
291C4#281218yes—
292C2—22no—
293C2—24no—
294C2—82no—
295C2—84no—
296C2#1465711yescharacteristic
297C2#2178011yescharacteristic
298C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radical#35659C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20593C2semidirect
3. #11173C2semidirect
4. #11171C2semidirect
5. #10993C2semidirect
6. #10968C2semidirect

Maximal subgroups

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

subgroupindex
#356592
#205932
#111732
#111712
#109932
#109682