Finite Group Databasebeta

← #482680

Subgroups — #482680

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35701C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17368C211yesmaximal, characteristic, complemented
4#11901C212yesmaximal, complemented
5#11875C212yesmaximal, complemented
6#11827C212yesmaximal, complemented
7#11821C212yesmaximal, complemented
8#11760C212yesmaximal, complemented
9#11754C212yesmaximal, complemented
10C2×#554C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2973—21no—
13C2×#410—21no—
14#2633C411yescharacteristic, complemented
15#2633C2211yescharacteristic
16#2628C411yescharacteristic, complemented
17#2611C411yescharacteristic, complemented
18#2611C2211yescharacteristic
19#2598C411yescharacteristic, complemented
20#2598C2211yescharacteristic
21#2542C2211yescharacteristic
22#2535C2211yescharacteristic
23#2463—21no—
24#2463—21no—
25#2449—21no—
26#2449—21no—
27#2423—21no—
28#2407—21no—
29#2397C411yescharacteristic, complemented
30#2397C2211yescharacteristic
31#2392C411yescharacteristic, complemented
32#2361—21no—
33#2356C411yescharacteristic, complemented
34D4×C4⋊C4—21no—
35#2316—21no—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40#1740C2212yes—
41#1467—21no—
42#1465C2212yes—
43#1464C2212yes—
44#1463—21no—
45#1459C2212yes—
46C2×#324—21no—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C411yescharacteristic
55C2×#125C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×#121—21no—
62#520—44no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×#117—22no—
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87#435—44no—
88#432—22no—
89#410—44no—
90C2×#99—22no—
91C2×#99—22no—
92#407—22no—
93C2×C22⋊C8—22no—
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#399—21no—
97#399—21no—
98#399C2×C411yescharacteristic
99#399C2×C411yescharacteristic
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2311yescharacteristic
103#399C2311yescharacteristic
104#398—21no—
105#398—21no—
106#398C2×C411yescharacteristic
107#397C2×C411yescharacteristic
108#397C2×C411yescharacteristic
109#397C2311yescharacteristic
110#396—21no—
111#395—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#394C2×C411yescharacteristic
116#393—21no—
117#392—21no—
118#392—21no—
119#392C2×C411yescharacteristic
120#392C2×C411yescharacteristic
121#392C2311yescharacteristic
122#391—21no—
123#390—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—44no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#384—21no—
151#383C2×C411yescharacteristic
152#383C2×C411yescharacteristic
153#382C2×C411yescharacteristic
154#380—21no—
155#380—21no—
156#380—22no—
157#380—22no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C22⋊C4—21no—
161#376C2×C411yescharacteristic
162#376C2×C411yescharacteristic
163#342—22no—
164#342C2314yes—
165#324—22no—
166#324—22no—
167#324—22no—
168#323C2314yes—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8—21no—
174C22×Q8C22⋊C411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C22×D4C2×D411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C2×M16—42no—
195C22×C8—42no—
196C4⋊Q8—42no—
197#125—42no—
198#125—42no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202C22⋊Q8—82no—
203C22⋊Q8—82no—
204#121—42no—
205#121—42no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—82no—
219C4×D4—82no—
220C4×D4—82no—
221C4×D4—82no—
222#117—42no—
223#117—42no—
224#117—42no—
225#117—44no—
226#117—44no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22×C411yescharacteristic
282C2×C22⋊C4C22×C411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2411yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C4⋊C8—42no—
296#99—44no—
297#99—44no—
298C22⋊C8—24no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301#97—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—42no—
318#95—42no—
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323C24—21no—
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C24C22×D411yescharacteristic
327#41—48no—
328#41—48no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4AGL(1,ℤ/8)12yes—
357C2×D4#13712yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409M16—162no—
410C2×C8—44no—
411C2×C8—162no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—162no—
440C4⋊C4—162no—
441C4⋊C4#9914yes—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—41no—
473C42—41no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—84no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23#41712yes—
499C23C22×C22⋊C411yescharacteristic
500C23C2×AGL(1,ℤ/8)11yescharacteristic
501C23C2×#13711yescharacteristic
502Q8—44no—
503Q8—162no—
504D4—42no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—48no—
511D4—42no—
512D4—82no—
513D4—82no—
514D4—84no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—162no—
572C2×C4—161no—
573C2×C4—161no—
574C2×C4—162no—
575C2×C4C2×#9912yes—
576C2×C4C2×#9912yes—
577C2×C4C2×#9912yes—
578C2×C4#41712yes—
579C2×C4#42112yes—
580C8—84no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C22×#9911yescharacteristic
608C22C2×#41711yescharacteristic
609C22C2×#42111yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—164no—
630C2—21no—
631C2—21no—
632C2—41no—
633C2—41no—
634C2—42no—
635C2—81no—
636C2—82no—
637C2#1759211yescharacteristic
638C2#3334911yescharacteristic
639C2#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. #20634C2semidirect
3. #17368C2semidirect
4. #11901C2semidirect
5. #11875C2semidirect
6. #11827C2semidirect
7. #11821C2semidirect
8. #11760C2semidirect
9. #11754C2semidirect
10. C2×#554C4semidirect
14. #2633C4semidirect
16. #2628C4semidirect
17. #2611C4semidirect
19. #2598C4semidirect
29. #2397C4semidirect
31. #2392C4semidirect
33. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357012
#206342
#173682
#119012
#118752
#118272
#118212
#117602
#117542