Finite Group Databasebeta

← #237810

Subgroups — #237810

Order 512.

1124 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1124 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#35020C211yesmaximal, characteristic
2C2×#1824C212yesmaximal
3#13924C211yesmaximal, characteristic
4#10347C218yesmaximal, complemented
5#8023C212yesmaximal
6#7907C211yesmaximal, characteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2851—22no—
10#2646C2214yes—
11#2569—24no—
12#2442—24no—
13#2315—22no—
14C2×#380C2211yescharacteristic
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1824—24no—
18#1824C2218yes—
19C2×#340—24no—
20#1445C2214yes—
21C4⋊#95—22no—
22C22⋊#95—22no—
23#1294C2218yes—
24#1280C2214yes—
25C24×C4—21no—
26C24×C4C2311yescharacteristic
27C23×C8—22no—
28C2×#125—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D414yes—
36C22×C4⋊C4Q814yes—
37C22×C4⋊C4C2312yes—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4D412yes—
41C22×C42D412yes—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8C2314yes—
46#400—28no—
47#398—28no—
48#389—24no—
49#387—28no—
50#385—28no—
51#385—28no—
52#382—22no—
53#380—28no—
54#380—22no—
55#380—22no—
56#380—24no—
57#380D414yes—
58#380C2314yes—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C2×#95—22no—
62C2×#95—28no—
63C2×#95C2312yes—
64C2×#95C2312yes—
65#340—216no—
66C25C2×D411yescharacteristic
67C23×C4—22no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—44no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—84no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D414yes—
78C23×C4C2×D412yes—
79C23×C4C2×Q812yes—
80C23×C4#4114yes—
81C23×C4#4112yes—
82C23×C4C2411yescharacteristic
83C22×C8—24no—
84C22×C8—48no—
85C22×C8—44no—
86#125—84no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—48no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C2×D418yes—
105C2×C4⋊C4C2×Q814yes—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—44no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—44no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—44no—
122#95—28no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—88no—
129C24—42no—
130C24—42no—
131C24#12012yes—
132C24C22⋊Q812yes—
133C24#12412yes—
134C24#13514yes—
135C24#13714yes—
136C24C22×D411yescharacteristic
137C22×C4—24no—
138C22×C4—28no—
139C22×C4—24no—
140C22×C4—28no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—44no—
145C22×C4—48no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—48no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—88no—
161C22×C4—84no—
162C22×C4—84no—
163C22×C4—88no—
164C22×C4—88no—
165C22×C4#12011yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4C22⋊Q814yes—
168C22×C4C22⋊Q812yes—
169C22×C4C22⋊Q814yes—
170C22×C4#12414yes—
171C22×C4#12412yes—
172C22×C4#12414yes—
173C22×C4C22×D412yes—
174C22×C4C22×Q811yescharacteristic
175C22×C4C2×#4112yes—
176C22×C4C2×#4111yescharacteristic
177C2×C8—48no—
178C2×C8—44no—
179C2×C8—88no—
180C4⋊C4—28no—
181C4⋊C4—48no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—84no—
187C4⋊C4—88no—
188C4⋊C4—88no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—48no—
194C22⋊C4—84no—
195C42—44no—
196C42—82no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—48no—
206C23—48no—
207C23#39314yes—
208C23#45214yes—
209C23#45214yes—
210C23#47918yes—
211C23#48814yes—
212C23#49114yes—
213C23C2×#12011yescharacteristic
214C23C2×C22⋊Q811yescharacteristic
215C23C2×#12411yescharacteristic
216C23C2×#13512yes—
217C23C2×#13712yes—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—28no—
222C2×C4—28no—
223C2×C4—24no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—48no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—48no—
238C2×C4—84no—
239C2×C4—816no—
240C2×C4—84no—
241C2×C4—84no—
242C2×C4—82no—
243C2×C4—88no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—168no—
247C2×C4#39312yes—
248C2×C4#39312yes—
249C2×C4#45814yes—
250C2×C4C2×C22⋊Q812yes—
251C2×C4C2×#12412yes—
252C8—84no—
253C22—24no—
254C22—24no—
255C22—28no—
256C22—28no—
257C22—21no—
258C22—21no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—24no—
263C22—48no—
264C22—44no—
265C22—44no—
266C22#202618yes—
267C22C2×#39311yescharacteristic
268C22C2×#45212yes—
269C22#302112yes—
270C22C2×#45812yes—
271C22C2×#47912yes—
272C22C2×#48811yescharacteristic
273C22C2×#49111yescharacteristic
274C4—22no—
275C4—22no—
276C4—22no—
277C4—84no—
278C4—88no—
279C4—84no—
280C4—162no—
281C4#203014yes—
282C2—21no—
283C2—22no—
284C2—21no—
285C2—28no—
286C2#1820714yes—
287C2C2×#202611yescharacteristic
288C2#2346311yescharacteristic
289C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10347C2semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#18242
#139242
#103472
#80232
#79072