Finite Group Databasebeta

← #214732

Subgroups — #214732

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20725C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9469C212yesmaximal, complemented
5#9383C212yesmaximal, complemented
6#9202C212yesmaximal, complemented
7#9022C212yesmaximal, complemented
8#8760C212yesmaximal, complemented
9#8380C212yesmaximal, complemented
10C2×#554C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2972C2212yes—
13#2628C411yescharacteristic, complemented
14#2627C411yescharacteristic, complemented
15#2627C2211yescharacteristic
16#2581C411yescharacteristic, complemented
17#2568C411yescharacteristic, complemented
18#2564C411yescharacteristic, complemented
19#2564C2211yescharacteristic
20#2487C411yescharacteristic, complemented
21#2487C2211yescharacteristic
22#2453—21no—
23#2450—21no—
24#2450—21no—
25#2442—21no—
26#2438—21no—
27#2438—21no—
28#2396C411yescharacteristic, complemented
29C2×#400C411yescharacteristic, complemented
30C2×#400C2211yescharacteristic
31#2349—21no—
32#2349C2211yescharacteristic
33#2339—21no—
34#2313C2212yes—
35#1824C2214yes—
36#1820—21no—
37#1799C2212yes—
38#1799C2212yes—
39#1782—21no—
40C4×#99—21no—
41C4×#99—21no—
42#1520C2214yes—
43#1461—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#324—21no—
48C2×#324C2212yes—
49#1294C2214yes—
50#1289C2214yes—
51C22×#41D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8C2311yescharacteristic
59#520—44no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C8×D4—44no—
80#431—22no—
81C2×#99—22no—
82C2×#99—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85#400—21no—
86#400—21no—
87#400—21no—
88#400—21no—
89#400C2×C411yescharacteristic
90#400C2×C411yescharacteristic
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#400C2×C411yescharacteristic
98#399C2×C411yescharacteristic
99#399C2×C411yescharacteristic
100#399C2×C411yescharacteristic
101#398—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#396—21no—
107#395—21no—
108#395C2×C411yescharacteristic
109#395C2×C411yescharacteristic
110#395C2×C411yescharacteristic
111#393—21no—
112#393C2×C411yescharacteristic
113#391—21no—
114#391—21no—
115#391—21no—
116#391—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—44no—
127#386—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#382—21no—
140#382C2×C411yescharacteristic
141#381—21no—
142#381—21no—
143#381—22no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380C2×C411yescharacteristic
150#380C2314yes—
151#380C2311yescharacteristic
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—22no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95C2×C411yescharacteristic
161C2×#95C2311yescharacteristic
162#342—22no—
163#342—22no—
164#340—44no—
165#328—44no—
166#324—22no—
167#324—22no—
168#324—22no—
169#323—22no—
170C2×#41—24no—
171C2×#41—24no—
172C2×#41—42no—
173C22×Q8—21no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C2×D411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—24no—
193C22×C8—42no—
194C4⋊Q8—42no—
195#125—44no—
196C22⋊Q8—24no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—44no—
210C4×D4—44no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—42no—
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—42no—
274#99—44no—
275#99—44no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—42no—
279C4×C8—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—24no—
306#95—42no—
307#95—44no—
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311#95C22×C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C22×D411yescharacteristic
315#41—416no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—81no—
322C2×Q8—81no—
323C2×Q8—81no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4#13514yes—
332C2×D4#13714yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4H(5,𝔽2)14yes—
387C2×C8—44no—
388C2×C8—84no—
389C2×C8—162no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—84no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C42—22no—
437C42—22no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—162no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23#41612yes—
456C23#41912yes—
457C23C22×C22⋊C411yescharacteristic
458C23#53412yes—
459C23C2×#13512yes—
460C23C2×#13712yes—
461Q8—44no—
462Q8—82no—
463Q8—162no—
464D4—28no—
465D4—28no—
466D4—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—161no—
526C2×C4—161no—
527C2×C4—162no—
528C2×C4—162no—
529C2×C4—162no—
530C2×C4—162no—
531C2×C4#41612yes—
532C2×C4#41912yes—
533C2×C4#51812yes—
534C2×C4#53412yes—
535C8—84no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22#230811yescharacteristic
553C22C2×#41611yescharacteristic
554C22C2×#41911yescharacteristic
555C22#319014yes—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—164no—
577C2—21no—
578C2—24no—
579C2—21no—
580C2—42no—
581C2—42no—
582C2#2176211yescharacteristic
583C2#2177711yescharacteristic
584C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9469C2semidirect
5. #9383C2semidirect
6. #9202C2semidirect
7. #9022C2semidirect
8. #8760C2semidirect
9. #8380C2semidirect
13. #2628C4semidirect
14. #2627C4semidirect
16. #2581C4semidirect
17. #2568C4semidirect
18. #2564C4semidirect
20. #2487C4semidirect
28. #2396C4semidirect
29. C2×#400C4semidirect

Maximal subgroups

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

subgroupindex
#360312
#207252
#173512
#94692
#93832
#92022
#90222
#87602
#83802