Finite Group Databasebeta

← #238518

Subgroups — #238518

Order 512.

1138 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 202 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#35173C211yesmaximal, characteristic
2#21394C212yesmaximal
3C2×#1506C212yesmaximal
4#13856C211yesmaximal, characteristic
5#10406C218yesmaximal
6#7912C211yesmaximal, characteristic
7C2×#533C2212yes—
8C2×#432C2212yes—
9#2857—22no—
10#2659C2214yes—
11#2584—24no—
12#2567—22no—
13#2546—22no—
14#2316—22no—
15C2×C4×C4⋊C4—22no—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1988—22no—
20#1981—22no—
21#1872C2218yes—
22#1763—24no—
23#1506—28no—
24C2×#328C2218yes—
25#1444—22no—
26#1444C2214yes—
27#1280C2214yes—
28#533—24no—
29#533—24no—
30#533—24no—
31#533—24no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8D412yes—
35C2×#125—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8C2314yes—
38C2×C4×Q8—22no—
39C2×C4×Q8D414yes—
40C2×#117—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4C2312yes—
44C22×C42—21no—
45C22×C42—21no—
46C22×C42D412yes—
47C22×C42C2311yescharacteristic
48#432—24no—
49#432—24no—
50C2×C4⋊C8—22no—
51C2×#103—28no—
52C2×C22⋊C8C2314yes—
53C2×#97—22no—
54#398—28no—
55#395—28no—
56#391—24no—
57#389—28no—
58#389—28no—
59#389—28no—
60#384—22no—
61#380—28no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—48no—
67#376—22no—
68#376D414yes—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—24no—
72C2×#95C2312yes—
73C2×#95C2312yes—
74#340—28no—
75#328—216no—
76C22×Q8—22no—
77C22×Q8C2×D414yes—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D414yes—
83C23×C4C2411yescharacteristic
84C22×C8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—84no—
90#125—44no—
91#125—44no—
92C22⋊Q8—24no—
93C22⋊Q8—24no—
94C22⋊Q8—24no—
95C22⋊Q8—24no—
96C22⋊Q8—44no—
97C22⋊Q8—44no—
98C4×Q8—28no—
99C4×Q8—28no—
100C4×Q8—44no—
101C4×Q8—44no—
102#117—44no—
103#117—44no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—48no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4C2×D414yes—
119C2×C22⋊C4—24no—
120C2×C42—24no—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—24no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—44no—
127C2×C42—44no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42#4114yes—
134C2×C42#4112yes—
135C4⋊C8—44no—
136#103—416no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139#97—44no—
140#95—216no—
141#95—24no—
142#95—24no—
143#95—22no—
144#95—48no—
145#95—42no—
146#95C2×D414yes—
147C24C22×D411yescharacteristic
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—48no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—44no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4#12012yes—
175C22×C4#12011yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4#12112yes—
178C22×C4#12112yes—
179C22×C4#12412yes—
180C22×C4#12412yes—
181C22×C4#13718yes—
182C22×C4C22×D412yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C2×#4112yes—
185C22×C4C2×#4111yescharacteristic
186C2×C8—48no—
187C2×C8—44no—
188C4⋊C4—216no—
189C4⋊C4—28no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—44no—
193C4⋊C4—432no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—88no—
201C4⋊C4—88no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C42—216no—
205C42—48no—
206C42—44no—
207C42—82no—
208C42—82no—
209C42—82no—
210C42#12114yes—
211C42#12114yes—
212C42#12314yes—
213C42#12314yes—
214C23—21no—
215C23—21no—
216C23—22no—
217C23C2×#12011yescharacteristic
218C23C2×#12111yescharacteristic
219C23C2×#12411yescharacteristic
220C23C2×#13714yes—
221Q8—48no—
222Q8—84no—
223Q8—88no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—28no—
233C2×C4—28no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#39414yes—
256C2×C4#39412yes—
257C2×C4#39412yes—
258C2×C4#45312yes—
259C2×C4#45612yes—
260C2×C4C2×#12112yes—
261C2×C4C2×#12312yes—
262C8—84no—
263C22—24no—
264C22—24no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22C2×#39411yescharacteristic
270C22#300412yes—
271C22C2×#45311yescharacteristic
272C22C2×#45611yescharacteristic
273C22#302512yes—
274C22#303912yes—
275C22#313912yes—
276C4—22no—
277C4—24no—
278C4—22no—
279C4—22no—
280C4—48no—
281C4—48no—
282C4—84no—
283C4—84no—
284C4—84no—
285C4#203914yes—
286C2—22no—
287C2—22no—
288C2#1818012yes—
289C2#1818912yes—
290C2C2×#203911yescharacteristic
291C2#2350711yescharacteristic
292C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351732
#213942
C2×#15062
#138562
#104062
#79122