Finite Group Databasebeta

← #481553

Subgroups — #481553

Order 512.

1019 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35701C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11732C214yesmaximal, complemented
5#11726C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#554C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2972—21no—
11C2×#410—21no—
12#2633C2212yes—
13#2628C2211yescharacteristic
14#2611C412yescomplemented
15#2598C412yescomplemented
16#2542C411yescharacteristic, complemented
17#2535C411yescharacteristic, complemented
18#2463—22no—
19#2449—22no—
20#2423—21no—
21#2407—21no—
22#2397C2212yes—
23#2392C2211yescharacteristic
24#2361—21no—
25#2356C411yescharacteristic, complemented
26D4×C4⋊C4—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39#554—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125C2×C412yes—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×#121—21no—
46#520—44no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C8×D4—44no—
65#431—22no—
66#410—28no—
67C2×#99—24no—
68C2×C22⋊C8—24no—
69#400C2×C412yes—
70#399—22no—
71#399C2×C412yes—
72#399C2×C412yes—
73#399C2312yes—
74#398—22no—
75#398C2×C411yescharacteristic
76#397C2×C412yes—
77#397C2311yescharacteristic
78#396—21no—
79#395—21no—
80#394—22no—
81#394—21no—
82#394C2×C411yescharacteristic
83#393—21no—
84#392—22no—
85#392C2×C412yes—
86#392C2311yescharacteristic
87#391—21no—
88#390—21no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384—21no—
103#383C2×C412yes—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112#376C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C22×Q8—21no—
122C22×Q8C22⋊C411yescharacteristic
123C22×D4—21no—
124C22×D4—22no—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C22×D4C2×D412yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C22×C8—44no—
136C4⋊Q8—42no—
137#125—44no—
138C22⋊Q8—48no—
139C22⋊Q8—44no—
140C22⋊Q8—84no—
141#121—44no—
142C4×Q8—24no—
143C4×Q8—42no—
144C4×D4—24no—
145C4×D4—42no—
146C4×D4—44no—
147C4×D4—44no—
148C4×D4—42no—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—84no—
152C4×D4—84no—
153#117—28no—
154#117—42no—
155#117—44no—
156#117—44no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
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⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
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⋊C4—22no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C22⋊C4C2411yescharacteristic
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—44no—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C4⋊C8—42no—
204#99—216no—
205C22⋊C8—24no—
206C22⋊C8—44no—
207C4×C8—42no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95—22no—
212#95—22no—
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—21no—
222C24C2×C22⋊C412yes—
223C24C22×D411yescharacteristic
224#41—48no—
225#41—48no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—42no—
230C2×Q8—82no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—82no—
244C2×D4#10414yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4#13514yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—44no—
279C2×C8—164no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4#9914yes—
299C22⋊C4—24no—
300C22⋊C4—28no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C42—22no—
317C42—24no—
318C42—22no—
319C42—42no—
320C42—44no—
321C42—48no—
322C42—42no—
323C42—164no—
324C23—22no—
325C23—21no—
326C23—22no—
327C23—21no—
328C23—22no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C2×#10414yes—
340C23C22×C22⋊C411yescharacteristic
341C23C2×#13512yes—
342Q8—44no—
343Q8—162no—
344D4—28no—
345D4—48no—
346D4—42no—
347D4—48no—
348D4—42no—
349D4—88no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—164no—
391C2×C4—162no—
392C2×C4C2×#9914yes—
393C2×C4C2×#9912yes—
394C2×C4#41612yes—
395C2×C4#41612yes—
396C2×C4C2×#10412yes—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—24no—
402C22—42no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—41no—
407C22—42no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22C22×#9911yescharacteristic
418C22C2×#41611yescharacteristic
419C22C22×#10411yescharacteristic
420C4—24no—
421C4—22no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4—81no—
430C4—82no—
431C4—88no—
432C4—82no—
433C4—81no—
434C4#150114yes—
435C2—21no—
436C2—21no—
437C2—42no—
438C2—44no—
439C2—82no—
440C2C2×#150111yescharacteristic
441C2#3334911yescharacteristic
442C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35701C2
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. #17351C2semidirect
4. #11732C2semidirect
5. #11726C2semidirect
6. #11718C2semidirect
7. #11691C2semidirect
8. C2×#554C4semidirect
14. #2611C4semidirect
15. #2598C4semidirect
16. #2542C4semidirect
17. #2535C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357012
#203932
#173512
#117322
#117262
#117182
#116912