Finite Group Databasebeta

← #482163

Subgroups — #482163

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35703C211yesmaximal, characteristic
2#20689C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#11845C212yesmaximal, complemented
5#11844C212yesmaximal, complemented
6#11842C212yesmaximal, complemented
7#11839C212yesmaximal, complemented
8C2×#1351C212yesmaximal, complemented
9#11692C212yesmaximal, complemented
10C2×#548C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2972—21no—
13#2638C411yescharacteristic, complemented
14#2631C411yescharacteristic, complemented
15#2617C2211yescharacteristic
16#2610C2211yescharacteristic
17#2599C2211yescharacteristic
18#2561C2211yescharacteristic
19#2543C411yescharacteristic, complemented
20#2477—21no—
21#2472—21no—
22#2463—21no—
23#2459—21no—
24#2444—21no—
25#2410—21no—
26#2409C411yescharacteristic, complemented
27#2404C2211yescharacteristic
28#2398C411yescharacteristic, complemented
29#2388C411yescharacteristic, complemented
30C2×#393C2211yescharacteristic
31#2364—21no—
32#2358—21no—
33#2353C411yescharacteristic, complemented
34#2308—21no—
35#1820—21no—
36#1820C2212yes—
37#1819—21no—
38#1819C2212yes—
39C2×#342—21no—
40C2×#342C2212yes—
41#1460—21no—
42#1459—21no—
43#1459—21no—
44#1458C2212yes—
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1351C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41C2×C411yescharacteristic, complemented
52C23×Q8C2×C411yescharacteristic
53C23×D4D411yescharacteristic
54#548—44no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C411yescharacteristic
63C2×#120—21no—
64C2×#120—21no—
65#518—44no—
66C2×C4×Q8C2×C411yescharacteristic, complemented
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4C2311yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85C8×D4—44no—
86#431—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89#400—21no—
90#398C2×C411yescharacteristic, complemented
91#397—21no—
92#397—21no—
93#397C2×C411yescharacteristic
94#397C2×C411yescharacteristic, complemented
95#397C2311yescharacteristic
96#395C2×C411yescharacteristic, complemented
97#395C2×C411yescharacteristic, complemented
98#394—21no—
99#394—21no—
100#394—21no—
101#394—21no—
102#394—21no—
103#394C2×C411yescharacteristic, complemented
104#393—21no—
105#393—21no—
106#393C2×C411yescharacteristic
107#393C2×C411yescharacteristic
108#393C2×C411yescharacteristic, complemented
109#393C2×C411yescharacteristic, complemented
110#393C2311yescharacteristic
111#393C2311yescharacteristic
112#392—21no—
113#392—21no—
114#392C2×C411yescharacteristic
115#392C2×C411yescharacteristic, complemented
116#392C2311yescharacteristic
117#391—21no—
118#390—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—44no—
142#385—21no—
143#385—21no—
144#385—21no—
145#383C2×C411yescharacteristic, complemented
146#381—21no—
147#381—22no—
148#381C2×C411yescharacteristic, complemented
149#380C2×C411yescharacteristic
150#379—21no—
151#379—22no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4C2×C411yescharacteristic, complemented
157#376C2×C411yescharacteristic, complemented
158C2×#95—21no—
159C2×#95C2×C411yescharacteristic
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C25C2×D411yescharacteristic
169C25C2×D411yescharacteristic
170C2×#41—42no—
171C2×#41—82no—
172C22×Q8—21no—
173C22×Q8—42no—
174C22×Q8—82no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—42no—
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—42no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C22×C8—42no—
204C22×C8—42no—
205#124—42no—
206#124—44no—
207C22⋊Q8—44no—
208C22⋊Q8—42no—
209C22⋊Q8—82no—
210#121—42no—
211#121—44no—
212#121—44no—
213#120—44no—
214#120—82no—
215C4×Q8—42no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—44no—
225C4×D4—82no—
226#117—42no—
227#117—42no—
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⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C4⋊C8—42no—
291C22⋊C8—24no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C4×C8—42no—
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—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—44no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—41no—
323C24—81no—
324C24—81no—
325C24—81no—
326C24—81no—
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24#13514yes—
332C24C22×D411yescharacteristic
333#41—84no—
334#41—84no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4#9914yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—84no—
412C22×C4—162no—
413C22×C4—162no—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C2×C8—44no—
425C2×C8—162no—
426C2×C8—162no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—84no—
474C22⋊C4—162no—
475C22⋊C4—162no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—81no—
482C42—81no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—24no—
490C23—24no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—44no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—84no—
517C23—162no—
518C23—162no—
519C23—162no—
520C23C2×#9912yes—
521C23C2×#9912yes—
522C23#41612yes—
523C23#41612yes—
524C23#42112yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×#13512yes—
527Q8—42no—
528Q8—42no—
529Q8—84no—
530Q8—82no—
531D4—28no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—84no—
536D4—82no—
537D4—84no—
538D4—84no—
539D4—162no—
540D4—162no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—161no—
594C2×C4—161no—
595C2×C4—161no—
596C2×C4—161no—
597C2×C4—162no—
598C2×C4—168no—
599C2×C4C2×#9912yes—
600C8—84no—
601C22—21no—
602C22—22no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—84no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—84no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—164no—
637C22#151614yes—
638C22C22×#9911yescharacteristic
639C22C2×#41611yescharacteristic
640C22C2×#42111yescharacteristic
641C4—22no—
642C4—42no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—164no—
659C2—21no—
660C2—21no—
661C2—24no—
662C2—42no—
663C2—42no—
664C2—42no—
665C2—81no—
666C2—161no—
667C2C2×#151611yescharacteristic
668C2#3334811yescharacteristic
669C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35703C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20689C2semidirect
3. #17350C2semidirect
4. #11845C2semidirect
5. #11844C2semidirect
6. #11842C2semidirect
7. #11839C2semidirect
8. C2×#1351C2semidirect
9. #11692C2semidirect
10. C2×#548C4semidirect
13. #2638C4semidirect
14. #2631C4semidirect
19. #2543C4semidirect
26. #2409C4semidirect
28. #2398C4semidirect
29. #2388C4semidirect
33. #2353C4semidirect
51. C22×#41C2×C4semidirect
56. C2×#124C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
66. C2×C4×Q8C2×C4semidirect
90. #398C2×C4semidirect
94. #397C2×C4semidirect
96. #395C2×C4semidirect
97. #395C2×C4semidirect
103. #394C2×C4semidirect
108. #393C2×C4semidirect
109. #393C2×C4semidirect
115. #392C2×C4semidirect
145. #383C2×C4semidirect
148. #381C2×C4semidirect
156. C4×C22⋊C4C2×C4semidirect
157. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357032
#206892
#173502
#118452
#118442
#118422
#118392
C2×#13512
#116922