Finite Group Databasebeta

← #481552

Subgroups — #481552

Order 512.

1033 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35693C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11731C214yesmaximal, complemented
5#11725C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#547C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2982—21no—
11C2×#410—21no—
12#2638C2212yes—
13#2628C2211yescharacteristic
14#2583C412yescomplemented
15#2567C411yescharacteristic, complemented
16#2536C411yescharacteristic, complemented
17#2470—22no—
18#2455—22no—
19#2422—21no—
20#2407—21no—
21#2406C412yescomplemented
22#2392C2211yescharacteristic
23#2388C2212yes—
24#2360—21no—
25#2357C411yescharacteristic, complemented
26D4×C4⋊C4—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1460—21no—
32#1460C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1351C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41C2×C412yes—
39C22×#41D411yescharacteristic
40#547—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#123—22no—
43C2×#123—22no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×#121—21no—
47#520—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117—22no—
58C2×#117D412yes—
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63C8×Q8—44no—
64#431—24no—
65#431—22no—
66#410—28no—
67C2×#99—24no—
68#400—21no—
69#399—21no—
70#398C2×C411yescharacteristic
71#398C2×C412yes—
72#397—22no—
73#397C2311yescharacteristic
74#396—21no—
75#395C2×C412yes—
76#395C2×C412yes—
77#395C2×C412yes—
78#394—22no—
79#394—22no—
80#394—21no—
81#394C2×C412yes—
82#394C2×C411yescharacteristic
83#393C2312yes—
84#392—22no—
85#392C2311yescharacteristic
86#391—21no—
87#390—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—21no—
92#388—22no—
93#388—22no—
94#387—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#384—21no—
102#382C2×C411yescharacteristic
103#381C2×C412yes—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—44no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C2×#41—44no—
122C2×#41—84no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22×C411yescharacteristic
129C22×D4C2×D412yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C4⋊Q8—42no—
138#123—48no—
139#123—84no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142#121—44no—
143C4×Q8—24no—
144C4×Q8—42no—
145C4×D4—24no—
146C4×D4—42no—
147C4×D4—44no—
148C4×D4—42no—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—84no—
152#117—28no—
153#117—42no—
154#117—44no—
155#117—44no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2411yescharacteristic
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—44no—
198C2×C42C22⋊C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C4⋊C8—44no—
203C4⋊C8—42no—
204#99—216no—
205C22⋊C8—24no—
206C4×C8—44no—
207C4×C8—42no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—44no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24—21no—
222C24C2×C22⋊C412yes—
223C24C22×D411yescharacteristic
224#41—48no—
225#41—48no—
226#41—88no—
227#41—88no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—82no—
232C2×Q8#10414yes—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—82no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4#13514yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C2×C8—44no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4#9914yes—
300C22⋊C4—24no—
301C22⋊C4—28no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—48no—
322C42—42no—
323C42—82no—
324C42—164no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C22×C22⋊C411yescharacteristic
342C23C2×#13512yes—
343Q8—28no—
344Q8—84no—
345D4—48no—
346D4—42no—
347D4—48no—
348D4—42no—
349D4—44no—
350D4—88no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—88no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—162no—
392C2×C4C2×#9914yes—
393C2×C4C2×#9912yes—
394C2×C4#41612yes—
395C2×C4#41612yes—
396C2×C4C2×#10414yes—
397C2×C4C2×#10412yes—
398C8—84no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—81no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22C22×#9911yescharacteristic
419C22C2×#41611yescharacteristic
420C22C22×#10411yescharacteristic
421C4—24no—
422C4—22no—
423C4—24no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—81no—
431C4—82no—
432C4—88no—
433C4—82no—
434C4—81no—
435C4#150114yes—
436C2—21no—
437C2—21no—
438C2—42no—
439C2—82no—
440C2—82no—
441C2C2×#150111yescharacteristic
442C2#3334811yescharacteristic
443C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35693C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20393C2semidirect
3. #17353C2semidirect
4. #11731C2semidirect
5. #11725C2semidirect
6. #11717C2semidirect
7. #11691C2semidirect
8. C2×#547C4semidirect
14. #2583C4semidirect
15. #2567C4semidirect
16. #2536C4semidirect
21. #2406C4semidirect
25. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#356932
#203932
#173532
#117312
#117252
#117172
#116912