Finite Group Databasebeta

← #238452

Subgroups — #238452

Order 512.

1324 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1324 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
1C2×#2659C211yesmaximal, characteristic
2#21765C212yesmaximal
3C2×#1508C212yesmaximal
4#13965C211yesmaximal, characteristic
5#10406C218yesmaximal, complemented
6#7909C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8C2×#407C2212yes—
9#2659—22no—
10#2659—24no—
11#2659C2214yes—
12C2×#398—22no—
13C2×#395—22no—
14C2×#385—22no—
15C2×#380—22no—
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1894—24no—
20#1872C2218yes—
21#1823—22no—
22#1787—22no—
23#1508—28no—
24C2×#328C2218yes—
25#1444C2214yes—
26C4⋊#95—22no—
27#1280C2214yes—
28C23×Q8D412yes—
29C24×C4D412yes—
30C24×C4C2311yescharacteristic
31C22×M16—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8D414yes—
39C2×C22⋊Q8C2314yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4C2312yes—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C42—21no—
50C2×#103—28no—
51#407—24no—
52#407—24no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8C2314yes—
55#398—28no—
56#398—28no—
57#395—28no—
58#395—28no—
59#385—48no—
60#380—28no—
61#380—28no—
62#380—22no—
63#380—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4D414yes—
66C2×#95—24no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95D412yes—
72C2×#95C2312yes—
73C2×#95C2312yes—
74#340—28no—
75#328—216no—
76C25C2×D411yescharacteristic
77C22×Q8—22no—
78C22×Q8—24no—
79C22×Q8—24no—
80C22×Q8—84no—
81C22×Q8C2×D414yes—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—28no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—44no—
90C23×C4—82no—
91C23×C4—82no—
92C23×C4C2×D414yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4#4114yes—
98C23×C4#4112yes—
99C23×C4C2411yescharacteristic
100C2×M16—44no—
101C2×M16—84no—
102C22×C8—24no—
103C22⋊Q8—28no—
104C22⋊Q8—28no—
105C22⋊Q8—24no—
106C22⋊Q8—24no—
107C22⋊Q8—24no—
108C22⋊Q8—24no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C2×D414yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—24no—
141C2×C42—82no—
142#103—88no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—44no—
146#95—216no—
147#95—24no—
148#95—24no—
149#95—22no—
150#95—48no—
151#95—42no—
152#95C2×D414yes—
153C24—24no—
154C24—22no—
155C24—22no—
156C24—42no—
157C24#12011yescharacteristic
158C24#12011yescharacteristic
159C24#12112yes—
160C24#12412yes—
161C24C22×D411yescharacteristic
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—48no—
165C2×Q8—44no—
166C2×Q8—84no—
167C2×Q8—88no—
168C2×Q8—88no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—84no—
201C22×C4—816no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#12012yes—
205C22×C4#12114yes—
206C22×C4#12114yes—
207C22×C4#12112yes—
208C22×C4#12314yes—
209C22×C4#12314yes—
210C22×C4#12412yes—
211C22×C4#13718yes—
212C22×C4C22×D412yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C2×#4112yes—
215C22×C4C2×#4111yescharacteristic
216M16—88no—
217C2×C8—48no—
218C2×C8—44no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—48no—
226C4⋊C4—88no—
227C4⋊C4—88no—
228C4⋊C4—816no—
229C22⋊C4—216no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—84no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—24no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—42no—
250C23—42no—
251C23—48no—
252C23—48no—
253C23#39412yes—
254C23#39412yes—
255C23#45312yes—
256C23#45612yes—
257C23C2×#12011yescharacteristic
258C23C2×#12111yescharacteristic
259C23C2×#12411yescharacteristic
260C23C2×#13714yes—
261Q8—48no—
262Q8—84no—
263Q8—88no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—416no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—84no—
291C2×C4—88no—
292C2×C4—88no—
293C2×C4—88no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#39414yes—
300C2×C4C2×#12112yes—
301C2×C4C2×#12312yes—
302C8—84no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—48no—
317C22—48no—
318C22—42no—
319C22—42no—
320C22#203914yes—
321C22C2×#39411yescharacteristic
322C22#300412yes—
323C22C2×#45311yescharacteristic
324C22C2×#45611yescharacteristic
325C22#302612yes—
326C22#303912yes—
327C22#313912yes—
328C4—22no—
329C4—22no—
330C4—24no—
331C4—48no—
332C4—48no—
333C4—84no—
334C4—84no—
335C4—162no—
336C2—22no—
337C2—22no—
338C2—24no—
339C2—42no—
340C2#1831812yes—
341C2#1832712yes—
342C2C2×#203911yescharacteristic
343C2#2350711yescharacteristic
344C2#2351511yescharacteristic
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
5. #10406C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#217652
C2×#15082
#139652
#104062
#79092