Finite Group Databasebeta

← #415267

Subgroups — #415267

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3#14159C212yesmaximal
4#14149C211yesmaximal, characteristic
5#11287C214yesmaximal
6#11222C214yesmaximal
7C4×#342C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9#2857—22no—
10#2659—22no—
11#2583—24no—
12#2502—22no—
13#2434—22no—
14#2315C2212yes—
15#2315C2212yes—
16C2×#385C2212yes—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1996—28no—
22C2×#342C2218yes—
23C2×#342C2212yes—
24C2×#342C2212yes—
25#1446—22no—
26#1445—22no—
27#1441—22no—
28#1436—22no—
29#1436—24no—
30C22⋊#95C2212yes—
31C22⋊#95C2218yes—
32C23×Q8D412yes—
33C24×C4C2312yes—
34C24×C4C2311yescharacteristic
35C2×C4⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4D412yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2311yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—22no—
54C22×C42D412yes—
55#398—28no—
56#395—28no—
57#391—24no—
58#389—24no—
59#389—24no—
60#385—24no—
61#385—24no—
62#385—24no—
63#385—24no—
64#385—28no—
65#384—22no—
66#380—24no—
67#380—24no—
68#380—22no—
69#380D414yes—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—28no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—28no—
85C2×#95D412yes—
86#342—216no—
87#342—24no—
88#342Q818yescomplemented
89C25C2411yescharacteristic
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—84no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—44no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4C2×D414yes—
105C23×C4C2×D414yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4#4112yes—
114C23×C4#4114yes—
115C4⋊Q8—84no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4—82no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4C2×D414yes—
160C2×C22⋊C4C2×Q818yes—
161C2×C22⋊C4C2×Q814yes—
162C2×C22⋊C4#4114yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—44no—
166C2×C42—82no—
167C2×C42—84no—
168#95—28no—
169#95—28no—
170#95—24no—
171#95—24no—
172#95—416no—
173#95—44no—
174#95—42no—
175#95—41no—
176#95—41no—
177#95—44no—
178#95—44no—
179#95—82no—
180#95—88no—
181C24—42no—
182C24—42no—
183C24C22×D412yes—
184C24C22×D411yescharacteristic
185C24C22×D411yescharacteristic
186C24C22×Q812yes—
187C24C2×#4111yescharacteristic
188C24C2×#4112yes—
189C24C2×#4112yes—
190C242−1+412yes—
191C242−1+412yes—
192C2×Q8—48no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—84no—
196C2×Q8—88no—
197C2×Q8—88no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—48no—
224C22×C4—48no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4—82no—
244C22×C4—88no—
245C22×C4—88no—
246C22×C4#12012yes—
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12112yes—
250C22×C4#12114yes—
251C22×C4#12114yes—
252C22×C4#12112yes—
253C22×C4#12314yes—
254C22×C4#12314yes—
255C22×C4#12414yes—
256C22×C4#12412yes—
257C22×C4#12412yes—
258C22×C4C4⋊Q814yes—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—88no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4—28no—
272C22⋊C4—28no—
273C22⋊C4—28no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C42—44no—
282C42—82no—
283C42—88no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—41no—
289C23—44no—
290C23—41no—
291C23—42no—
292C23—44no—
293C23—41no—
294C23—44no—
295C23—41no—
296C23—42no—
297C23—44no—
298C23C2×#12011yescharacteristic
299C23C2×#12112yes—
300C23C2×#12111yescharacteristic
301C23C2×#12312yes—
302C23C2×#12411yescharacteristic
303C23C2×#12411yescharacteristic
304C23C2×C4⋊Q811yescharacteristic
305C23#53314yes—
306C23#53611yescharacteristic
307C23#53611yescharacteristic
308C23#54114yes—
309C23#54712yes—
310C23#54814yes—
311C23D4×Q814yes—
312C23D4×Q812yes—
313C23#55612yes—
314C23#55712yes—
315Q8—88no—
316Q8—88no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—416no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—416no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—88no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—88no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4#39412yes—
368C2×C4#39414yes—
369C2×C4#39412yes—
370C22—22no—
371C22—22no—
372C22—28no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—21no—
381C22—22no—
382C22—28no—
383C22—22no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22C2×#39411yescharacteristic
391C22#242914yes—
392C22#243411yescharacteristic
393C22#243611yescharacteristic
394C22#244614yes—
395C22#254611yescharacteristic
396C22#254711yescharacteristic
397C22#264211yescharacteristic
398C22#286211yescharacteristic
399C22#302512yes—
400C22#302612yes—
401C4—22no—
402C4—24no—
403C4—22no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—88no—
408C4—84no—
409C4—84no—
410C4—84no—
411C2—21no—
412C2—24no—
413C2—21no—
414C2—21no—
415C2—24no—
416C2—21no—
417C2#1518811yescharacteristic
418C2#2351411yescharacteristic
419C2#2351511yescharacteristic
420C2#2585412yes—
421C2#2594912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
88. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#19962
#141592
#141492
#112872
#112222
C4×#3422