Finite Group Databasebeta

← #164366

Subgroups — #164366

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36509C211yesmaximal, characteristic
2#20559C212yesmaximal
3#7681C212yesmaximal, complemented
4#7677C212yesmaximal, complemented
5#7675C212yesmaximal, complemented
6#7639C211yesmaximal, characteristic, complemented
7#7634C212yesmaximal, complemented
8#7571C211yesmaximal, characteristic, complemented
9#7566C212yesmaximal, complemented
10C2×#435—22no—
11#2864C2211yescharacteristic
12#2855C2212yes—
13#2590—22no—
14#2536C411yescharacteristic
15#2536C411yescharacteristic
16#2515—22no—
17#2496C412yes—
18#2466—22no—
19#2429C412yes—
20#2373—22no—
21#2356C411yescharacteristic
22#2351C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25C42×D4C2211yescharacteristic
26#1771—22no—
27C4×C4⋊C8C2212yes—
28#1753—22no—
29C8×C22⋊C4—22no—
30#1749C2212yes—
31C4×C22⋊C8C2212yes—
32#1745C2212yes—
33#1728C2212yes—
34C2×#336C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2211yescharacteristic
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C22×M16—22no—
46C23×C8—22no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#126—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C22×C42C2312yes—
57#435—216no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×#97—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#400—22no—
66#398—21no—
67#398—21no—
68#398—22no—
69#397—22no—
70#396—22no—
71#394—21no—
72#394—21no—
73#391C2×C412yes—
74#391C2311yescharacteristic
75#391C2311yescharacteristic
76#389—22no—
77#389—22no—
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2312yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#384C2×C411yescharacteristic
95#383—21no—
96#382—21no—
97#382C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#381—22no—
100#380C2×C412yes—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112#376—21no—
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121C22×Q8C22×C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×D4C22⋊C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C2×M16—28no—
128C2×M16—82no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—28no—
135C22×C8—82no—
136C22×C8—82no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139#126—48no—
140C22⋊Q8—24no—
141C22⋊Q8—42no—
142C22⋊Q8—82no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—82no—
146C4×D4—28no—
147C4×D4—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—82no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—28no—
185C22⋊C8—28no—
186C22⋊C8—28no—
187#97—28no—
188#97—82no—
189C4×C8—82no—
190C4×C8—82no—
191#95—21no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—21no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95C22⋊C411yescharacteristic
206#95C22⋊C412yes—
207#95C22⋊C412yes—
208#95C22⋊C411yescharacteristic
209C24C2×C22⋊C412yes—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8#13112yes—
218C2×D4—24no—
219C2×D4—41no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246M16—88no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—88no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C22⋊C4#10418yes—
285C42—21no—
286C42—21no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—42no—
292C42—42no—
293C42—41no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—44no—
298C42—82no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23C2×#10414yes—
306C23C22×C22⋊C411yescharacteristic
307Q8—42no—
308Q8—42no—
309Q8—81no—
310Q8—81no—
311D4—44no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#41312yes—
352C2×C4C2×#10414yes—
353C2×C4C2×#10414yes—
354C2×C4C2×#13111yescharacteristic
355C2×C4#56811yescharacteristic
356C2×C4#57712yes—
357C2×C4#57812yes—
358C8—44no—
359C8—44no—
360C8—44no—
361C8—44no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22#288011yescharacteristic
370C22#288311yescharacteristic
371C22C2×#41311yescharacteristic
372C22#289812yes—
373C22C22×#10412yes—
374C4—21no—
375C4—21no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—24no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—81no—
386C4—82no—
387C4—82no—
388C4—81no—
389C4—81no—
390C2—44no—
391C2#1742411yescharacteristic
392C2#1751912yes—
393C2#1754812yes—
394C2#1763811yescharacteristic
395C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7681C2semidirect
4. #7677C2semidirect
5. #7675C2semidirect
6. #7639C2semidirect
7. #7634C2semidirect
8. #7571C2semidirect
9. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#365092
#205592
#76812
#76772
#76752
#76392
#76342
#75712
#75662