Finite Group Databasebeta

← #482972

Subgroups — #482972

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic
2#33487C211yesmaximal, characteristic, complemented
3#22717C211yesmaximal, characteristic, complemented
4#11896C211yesmaximal, characteristic, complemented
5#11894C211yesmaximal, characteristic, complemented
6#11893C211yesmaximal, characteristic, complemented
7#11892C211yesmaximal, characteristic, complemented
8#11866C211yesmaximal, characteristic, complemented
9#11865C211yesmaximal, characteristic, complemented
10#11752C211yesmaximal, characteristic, complemented
11#11750C211yesmaximal, characteristic, complemented
12#11712C211yesmaximal, characteristic, complemented
13#11710C211yesmaximal, characteristic, complemented
14#11709C211yesmaximal, characteristic, complemented
15C2×#1350C211yesmaximal, characteristic, complemented
16#3520—21no—
17#3342C411yescharacteristic
18#3336C411yescharacteristic
19#3336C411yescharacteristic
20#3334C411yescharacteristic
21#3302—21no—
22#3300—21no—
23#3300—21no—
24#3298—21no—
25D4×Q16C411yescharacteristic
26#3285C411yescharacteristic
27#3285C411yescharacteristic
28#3283C411yescharacteristic
29#3248—21no—
30#3190—21no—
31#3190—21no—
32#3170C2211yescharacteristic
33#3168C2211yescharacteristic
34#3050C2211yescharacteristic
35#3050C2211yescharacteristic
36C2×#467C2211yescharacteristic
37C2×#467C2211yescharacteristic
38D4×M16—21no—
39C2×C8×D4C2211yescharacteristic
40#2168—21no—
41#2168—21no—
42#2168—21no—
43#2168—21no—
44#2163C2211yescharacteristic
45#2163C2211yescharacteristic
46#2112C2211yescharacteristic
47#2112C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49C2×C22⋊C16C2211yescharacteristic
50#1950—21no—
51#1855—21no—
52#1855—21no—
53#1837C2211yescharacteristic
54#1810C2211yescharacteristic
55#1810C2211yescharacteristic
56C4×#108C2211yescharacteristic
57C2×#341C2211yescharacteristic
58C2×#341C2211yescharacteristic
59#1350C2211yescharacteristic
60#1350C2211yescharacteristic
61#1350C2211yescharacteristic
62#1350C2211yescharacteristic
63#1350C2211yescharacteristic
64#1350C2211yescharacteristic
65#1350C2211yescharacteristic
66#1350C2211yescharacteristic
67#1350C2211yescharacteristic
68#1350C2211yescharacteristic
69#1350C2211yescharacteristic
70#1350C2211yescharacteristic
71#1349C2211yescharacteristic
72#1349C2211yescharacteristic
73#1349C2211yescharacteristic
74#1349C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C2×#135C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C22×Q16C2×C411yescharacteristic
79C22×M16—21no—
80C22×M16—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83#556—21no—
84#554—41no—
85#552—21no—
86#552—21no—
87#551—41no—
88#551—41no—
89D4×Q8—21no—
90#548—21no—
91#548—21no—
92#547—21no—
93#547—21no—
94#546—41no—
95#542—41no—
96#539—41no—
97#539—41no—
98#533—21no—
99#529—21no—
100#525—21no—
101#525—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104#520—41no—
105C2×C4×D4D411yescharacteristic
106C2×M32—21no—
107C2×M32—21no—
108C22×C16—21no—
109C22×C16—21no—
110#500C2×C411yescharacteristic
111#499C2×C411yescharacteristic
112#495C2×C411yescharacteristic
113#494C2×C411yescharacteristic
114#488—21no—
115#487—21no—
116#485—21no—
117#485—21no—
118#485—21no—
119#485—21no—
120#483—21no—
121#483—21no—
122#480—21no—
123#480—21no—
124#479—21no—
125#479—21no—
126#475—21no—
127#474—21no—
128#467—21no—
129#467—21no—
130#467—21no—
131#467—21no—
132#467—21no—
133#467C2×C411yescharacteristic
134#467C2×C411yescharacteristic
135#467C2×C411yescharacteristic
136#467C2×C411yescharacteristic
137#467C2×C411yescharacteristic
138#467C2×C411yescharacteristic
139#467C2311yescharacteristic
140#467C2311yescharacteristic
141#467C2311yescharacteristic
142#466C2×C411yescharacteristic
143#466C2×C411yescharacteristic
144#466C2311yescharacteristic
145#465—21no—
146#465—21no—
147#465—21no—
148#465—21no—
149#465—21no—
150#464—21no—
151#464—21no—
152#462—21no—
153#461—21no—
154#452—21no—
155#452—21no—
156#452—21no—
157#452—21no—
158#451—21no—
159#451—21no—
160#448—21no—
161#448—21no—
162C4×Q16—21no—
163C4×Q16C2×C411yescharacteristic
164C4×Q16C2×C411yescharacteristic
165C4×Q16C2×C411yescharacteristic
166C4×SD16—21no—
167C4×D8C2×C411yescharacteristic
168#436—41no—
169#435—41no—
170#435—41no—
171C8×D4—41no—
172C8×D4C2×C411yescharacteristic
173C8×D4C2×C411yescharacteristic
174C8×D4C2×C411yescharacteristic
175C8×D4C2×C411yescharacteristic
176C8×D4D411yescharacteristic
177C8×D4D411yescharacteristic
178C8×D4D411yescharacteristic
179C8×D4D411yescharacteristic
180C2×#108—21no—
181C2×#108—21no—
182C2×#108—21no—
183#427C2×C411yescharacteristic
184#427C2×C411yescharacteristic
185C2×#107C2×C411yescharacteristic
186C2×#107C2×C411yescharacteristic
187#423—21no—
188C2×C4⋊C8D411yescharacteristic
189#416—21no—
190#416—21no—
191C2×#103—21no—
192C2×#103—21no—
193#407—21no—
194#407—21no—
195C2×C22⋊C8D411yescharacteristic
196C2×C22⋊C8D411yescharacteristic
197C4×M16—21no—
198C2×C4×C8C2311yescharacteristic
199C4⋊C16—21no—
200C4⋊C16—21no—
201C22⋊C16—21no—
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205C22⋊C16C2311yescharacteristic
206C22⋊C16C2311yescharacteristic
207C22⋊C16C2311yescharacteristic
208C22⋊C16C2311yescharacteristic
209#346—21no—
210#346—21no—
211#341—21no—
212#341—21no—
213#341—21no—
214#341C2311yescharacteristic
215#341C2311yescharacteristic
216#341C2311yescharacteristic
217#341C2311yescharacteristic
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—41no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×D4SD1611yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226#135—81no—
227#135—81no—
228C2×Q16—21no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—81no—
232C2×Q16—81no—
233C2×Q16C22×C411yescharacteristic
234C2×Q16C22×C411yescharacteristic
235C2×Q16C22×C411yescharacteristic
236C2×SD16—21no—
237C2×SD16—21no—
238C2×SD16—21no—
239C2×D8C22×C411yescharacteristic
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—41no—
245C2×M16—41no—
246C2×M16—42no—
247C2×M16—42no—
248C2×M16—81no—
249C2×M16—81no—
250C22×C8—41no—
251C22×C8—41no—
252C22×C8—81no—
253C22×C8—81no—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C411yescharacteristic
257C22×C8C22×C411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2411yescharacteristic
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—41no—
269#126—41no—
270#126—41no—
271#126—41no—
272#126—41no—
273#126—41no—
274#125—21no—
275#125—41no—
276#125—41no—
277#125—41no—
278#124—21no—
279#124—41no—
280#124—41no—
281#124—41no—
282#123—41no—
283#123—41no—
284#123—41no—
285#123—41no—
286#123—41no—
287#123—41no—
288#123—41no—
289#123—41no—
290#123—41no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—41no—
301C22⋊Q8—41no—
302C22⋊Q8—41no—
303C22⋊Q8—41no—
304C22⋊Q8—41no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—41no—
308C22⋊Q8—41no—
309C22⋊Q8—41no—
310#121—21no—
311#121—21no—
312#121—21no—
313#121—41no—
314#121—41no—
315#121—41no—
316#120—41no—
317#120—41no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—41no—
322C4×Q8—41no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—41no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—41no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4C22⋊C411yescharacteristic
333C4×D4C22⋊C411yescharacteristic
334C4×D4C22⋊C411yescharacteristic
335C4×D4C22⋊C411yescharacteristic
336C4×D4D811yescharacteristic
337C4×D4D811yescharacteristic
338C4×D4D811yescharacteristic
339C4×D4D811yescharacteristic
340#117—21no—
341#117—21no—
342#117—41no—
343#117—41no—
344#117—41no—
345#117—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—41no—
351C2×C4⋊C4SD1611yescharacteristic
352C2×C22⋊C4—41no—
353C2×C22⋊C4—41no—
354C2×C22⋊C4SD1611yescharacteristic
355C2×C22⋊C4SD1611yescharacteristic
356C2×C42C2×D411yescharacteristic
357M32—81no—
358M32—81no—
359C2×C16—21no—
360C2×C16—21no—
361C2×C16—21no—
362C2×C16—21no—
363C2×C16—81no—
364C2×C16—81no—
365#108—42no—
366#108—81no—
367#108—81no—
368#107—21no—
369#107—21no—
370#107—21no—
371#107C22×C411yescharacteristic
372#107C22×C411yescharacteristic
373#107C22×C411yescharacteristic
374#107C22×C411yescharacteristic
375#106—21no—
376#106—21no—
377#106—21no—
378C4⋊C8—41no—
379C4⋊C8—41no—
380C4⋊C8C22⋊C411yescharacteristic
381C4⋊C8C22⋊C411yescharacteristic
382C4⋊C8C22⋊C411yescharacteristic
383C4⋊C8C22⋊C411yescharacteristic
384#103—21no—
385#103—21no—
386#103—21no—
387#103—21no—
388#103—21no—
389#103—21no—
390#103—21no—
391#103—21no—
392#103—21no—
393#103—21no—
394#103—21no—
395#103—21no—
396#102—21no—
397#102—21no—
398#102—21no—
399#102—21no—
400C22⋊C8—41no—
401C22⋊C8—41no—
402C22⋊C8—41no—
403C22⋊C8—41no—
404C22⋊C8C22⋊C411yescharacteristic
405C22⋊C8C22⋊C411yescharacteristic
406C22⋊C8C22⋊C411yescharacteristic
407C22⋊C8C22⋊C411yescharacteristic
408C22⋊C8C22⋊C411yescharacteristic
409C22⋊C8C22⋊C411yescharacteristic
410C22⋊C8C22⋊C411yescharacteristic
411C22⋊C8C22⋊C411yescharacteristic
412#97—41no—
413C4×C8—41no—
414C4×C8C22×C411yescharacteristic
415C4×C8C22×C411yescharacteristic
416C4×C8C2×D411yescharacteristic
417C4×C8C2×D411yescharacteristic
418C24C2×SD1611yescharacteristic
419C24C2×SD1611yescharacteristic
420#41—81no—
421#41—81no—
422#41—81no—
423#41—81no—
424#41—84no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—41no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×D4—21no—
436C2×D4—42no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4#10211yescharacteristic
443C2×D4#10211yescharacteristic
444C2×D4#10211yescharacteristic
445C2×D4#10211yescharacteristic
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×D811yescharacteristic
462C22×C4C2×D811yescharacteristic
463C22×C4C2×D811yescharacteristic
464C22×C4C2×D811yescharacteristic
465C22×C4C2×SD1611yescharacteristic
466C22×C4C2×SD1611yescharacteristic
467C22×C4C2×SD1611yescharacteristic
468C22×C4C2×SD1611yescharacteristic
469C22×C4C22×D411yescharacteristic
470Q16—41no—
471Q16—41no—
472Q16—41no—
473Q16—81no—
474Q16—81no—
475Q16—81no—
476SD16—81no—
477SD16—81no—
478SD16—81no—
479D8—41no—
480M16—22no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486M16—41no—
487M16—41no—
488M16—82no—
489M16—82no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—82no—
507C2×C8—82no—
508C2×C8C2×C22⋊C411yescharacteristic
509C2×C8C2×C22⋊C411yescharacteristic
510C2×C8C2×C22⋊C411yescharacteristic
511C2×C8C2×C22⋊C411yescharacteristic
512C2×C8C2×C22⋊C411yescharacteristic
513C2×C8C2×C22⋊C411yescharacteristic
514C2×C8C23×C411yescharacteristic
515C2×C8C22×D411yescharacteristic
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4#10211yescharacteristic
540C4⋊C4#10211yescharacteristic
541C4⋊C4#10211yescharacteristic
542C4⋊C4#10211yescharacteristic
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4#10211yescharacteristic
561C22⋊C4#10211yescharacteristic
562C22⋊C4#10211yescharacteristic
563C22⋊C4#10211yescharacteristic
564C22⋊C4#10211yescharacteristic
565C22⋊C4#10211yescharacteristic
566C22⋊C4#10211yescharacteristic
567C22⋊C4#10211yescharacteristic
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42C2×C22⋊C411yescharacteristic
574C42C2×C22⋊C411yescharacteristic
575C42C2×D811yescharacteristic
576C42C2×D811yescharacteristic
577C16—41no—
578C16—41no—
579C16—41no—
580C16—41no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23C2×#10211yescharacteristic
589C23C2×#10211yescharacteristic
590C23C2×#10211yescharacteristic
591C23C2×#10211yescharacteristic
592C23C22×SD1611yescharacteristic
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597Q8—41no—
598Q8—41no—
599Q8—82no—
600Q8—161no—
601Q8—161no—
602D4—22no—
603D4—22no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—82no—
609D4—82no—
610D4—82no—
611D4—161no—
612D4—161no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—161no—
642C2×C4—161no—
643C2×C4—161no—
644C2×C4—161no—
645C2×C4C2×#10211yescharacteristic
646C2×C4C2×#10211yescharacteristic
647C2×C4C2×#10211yescharacteristic
648C2×C4C2×#10211yescharacteristic
649C2×C4C2×#10211yescharacteristic
650C2×C4C2×#10211yescharacteristic
651C2×C4C2×#10211yescharacteristic
652C2×C4C2×#10211yescharacteristic
653C2×C4C22×C22⋊C411yescharacteristic
654C2×C4C22×D811yescharacteristic
655C8—21no—
656C8—21no—
657C8—21no—
658C8—21no—
659C8—21no—
660C8—21no—
661C8—21no—
662C8—21no—
663C8—42no—
664C8—81no—
665C8—81no—
666C8—81no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—82no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22#214312yes—
687C22C22×#10211yescharacteristic
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4#288311yescharacteristic
705C4#290111yescharacteristic
706C2—22no—
707C2—41no—
708C2—41no—
709C2—41no—
710C2—41no—
711C2—161no—
712C2#1761311yescharacteristic
713C2C2×#214311yescharacteristic
714C2#3339311yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62252C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33487C2semidirect
3. #22717C2semidirect
4. #11896C2semidirect
5. #11894C2semidirect
6. #11893C2semidirect
7. #11892C2semidirect
8. #11866C2semidirect
9. #11865C2semidirect
10. #11752C2semidirect
11. #11750C2semidirect
12. #11712C2semidirect
13. #11710C2semidirect
14. #11709C2semidirect
15. C2×#1350C2semidirect

Maximal subgroups

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

subgroupindex
#622522
#334872
#227172
#118962
#118942
#118932
#118922
#118662
#118652
#117522
#117502
#117122
#117102
#117092
C2×#13502