Finite Group Databasebeta

← #481525

Subgroups — #481525

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35704C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11729C214yesmaximal, complemented
5#11725C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11693C212yesmaximal, complemented
8C2×#550C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2982—21no—
11C2×#410—21no—
12#2635C2212yes—
13#2619C2211yescharacteristic
14#2591C412yescomplemented
15#2581C411yescharacteristic, complemented
16#2548C411yescharacteristic, complemented
17#2476—22no—
18#2456—22no—
19#2445—21no—
20#2431—21no—
21#2406C412yescomplemented
22#2396C2211yescharacteristic
23#2388C2212yes—
24#2367—21no—
25#2361—21no—
26#2357C411yescharacteristic, complemented
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1460—21no—
32#1460C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1343C2218yes—
38C22×#41C2×C412yes—
39C22×#41D411yescharacteristic
40#550—44no—
41C2×#127—21no—
42C2×#127C2×C411yescharacteristic
43C2×#124—21no—
44C2×#123—22no—
45C2×#121—21no—
46C2×#121—22no—
47C2×#121C2×C412yes—
48#520—44no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117D412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63C8×Q8—44no—
64#431—24no—
65#431—22no—
66#410—28no—
67C2×#99—24no—
68#400C2×C412yes—
69#400C2×C411yescharacteristic
70#399C2311yescharacteristic
71#398—21no—
72#396—21no—
73#395C2×C412yes—
74#395C2×C411yescharacteristic
75#394—21no—
76#394—22no—
77#394—22no—
78#394C2×C412yes—
79#394C2×C412yes—
80#394C2×C412yes—
81#393C2311yescharacteristic
82#392—22no—
83#392—22no—
84#392—21no—
85#392C2312yes—
86#391—21no—
87#390—21no—
88#390—21no—
89#390—22no—
90#390—21no—
91#389—21no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#385—22no—
102#385—22no—
103#382C2×C411yescharacteristic
104#381C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—44no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110C4×C22⋊C4—22no—
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×C411yescharacteristic
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C412yes—
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139#127—42no—
140#127—44no—
141#124—44no—
142#123—84no—
143#121—48no—
144#121—44no—
145#121—44no—
146C4×Q8—24no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—84no—
155#117—28no—
156#117—42no—
157#117—44no—
158#117—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—24no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C22⋊C4C2×D412yes—
192C2×C22⋊C4C2411yescharacteristic
193C2×C42—21no—
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—21no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—44no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24—22no—
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—41no—
232C2×Q8#10414yes—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—82no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#13514yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—48no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4#9914yes—
301C22⋊C4—24no—
302C22⋊C4—28no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—81no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—48no—
324C42—42no—
325C42—42no—
326C42—82no—
327C42—164no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#13512yes—
346Q8—28no—
347Q8—42no—
348Q8—42no—
349D4—48no—
350D4—44no—
351D4—88no—
352D4—84no—
353D4—88no—
354D4—84no—
355D4—162no—
356D4—164no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—48no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—88no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—162no—
397C2×C4C2×#9914yes—
398C2×C4C2×#9912yes—
399C2×C4#41612yes—
400C2×C4#41612yes—
401C2×C4C2×#10414yes—
402C2×C4C2×#10412yes—
403C8—84no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22C22×#9911yescharacteristic
424C22C2×#41611yescharacteristic
425C22C22×#10411yescharacteristic
426C4—24no—
427C4—22no—
428C4—24no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—82no—
438C4—88no—
439C4—81no—
440C4#150114yes—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—82no—
445C2—162no—
446C2C2×#150111yescharacteristic
447C2#3334811yescharacteristic
448C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35704C2
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. #11729C2semidirect
5. #11725C2semidirect
6. #11715C2semidirect
7. #11693C2semidirect
8. C2×#550C4semidirect
14. #2591C4semidirect
15. #2581C4semidirect
16. #2548C4semidirect
21. #2406C4semidirect
26. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#357042
#203932
#173532
#117292
#117252
#117152
#116932