Finite Group Databasebeta

← #481393

Subgroups — #481393

Order 512.

733 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60695C211yesmaximal, characteristic
2#33421C211yesmaximal, characteristic, complemented
3#17798C211yesmaximal, characteristic, complemented
4#11682C211yesmaximal, characteristic, complemented
5#11680C211yesmaximal, characteristic, complemented
6#11678C211yesmaximal, characteristic, complemented
7#11677C211yesmaximal, characteristic, complemented
8#11667C211yesmaximal, characteristic, complemented
9#11666C211yesmaximal, characteristic, complemented
10#11665C211yesmaximal, characteristic, complemented
11#11663C211yesmaximal, characteristic, complemented
12#11656C211yesmaximal, characteristic, complemented
13#11654C211yesmaximal, characteristic, complemented
14#11652C211yesmaximal, characteristic, complemented
15#11651C211yesmaximal, characteristic, complemented
16#3460—21no—
17#3179—21no—
18#3179—21no—
19#3176C411yescharacteristic
20#3176C411yescharacteristic
21#3175C411yescharacteristic
22#3175C411yescharacteristic
23#3171—21no—
24#3171—21no—
25#3169C2211yescharacteristic
26#3169C2211yescharacteristic
27#3168C2211yescharacteristic
28#3168C2211yescharacteristic
29#3164C411yescharacteristic
30#3155C411yescharacteristic
31#3136—21no—
32#3062—21no—
33#3061C411yescharacteristic
34#3058C411yescharacteristic
35#2989—21no—
36C4×AGL(1,ℤ/8)—21no—
37C4×#135C2211yescharacteristic
38C2×C4×D8C2211yescharacteristic
39C4×#131C2211yescharacteristic
40#2164—21no—
41#2164—21no—
42#2163C2211yescharacteristic
43#2163C2211yescharacteristic
44#2163C2211yescharacteristic, complemented
45#2163C2211yescharacteristic, complemented
46#2151—21no—
47#2113—21no—
48C4×M32C2211yescharacteristic, complemented
49C2×#346C2211yescharacteristic, complemented
50#1569—21no—
51#1569—21no—
52#1568C2211yescharacteristic
53#1568C2211yescharacteristic, complemented
54#1566C2211yescharacteristic
55#1566C2211yescharacteristic, complemented
56#1565—21no—
57#1564C2211yescharacteristic, complemented
58C2×#334C2211yescharacteristic, complemented
59#1334C2211yescharacteristic, complemented
60#1334C2211yescharacteristic, complemented
61#1334C2211yescharacteristic, complemented
62#1334C2211yescharacteristic, complemented
63#1333C2211yescharacteristic
64#1333C2211yescharacteristic
65#1333C2211yescharacteristic
66#1333C2211yescharacteristic
67#1333C2211yescharacteristic
68#1333C2211yescharacteristic
69#1333C2211yescharacteristic
70#1333C2211yescharacteristic
71#1333C2211yescharacteristic, complemented
72#1333C2211yescharacteristic, complemented
73#1333C2211yescharacteristic, complemented
74#1333C2211yescharacteristic, complemented
75C2×AGL(1,ℤ/8)—21no—
76C2×#135C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131D411yescharacteristic
80#544—21no—
81#542—21no—
82#542—21no—
83#542—41no—
84#540—21no—
85#540—21no—
86#540—41no—
87#538—41no—
88#538—41no—
89#537—21no—
90#535—21no—
91#535—21no—
92#532—21no—
93#532—21no—
94#532—41no—
95#529—21no—
96#529—21no—
97#529—41no—
98#525—41no—
99#525—41no—
100C4×#41—21no—
101C4×#41D411yescharacteristic
102C2×C4×D4—21no—
103C2×M32—21no—
104C22×C16—21no—
105#501—21no—
106#501—21no—
107#500C2×C411yescharacteristic
108#500C2×C411yescharacteristic
109#499C2×C411yescharacteristic
110#499C2×C411yescharacteristic
111#496—21no—
112#496—21no—
113#495C2×C411yescharacteristic
114#495C2×C411yescharacteristic
115#493C2×C411yescharacteristic
116#493C2×C411yescharacteristic
117#490—21no—
118#490—21no—
119#489—21no—
120#489—21no—
121#488—21no—
122#488—21no—
123#486—21no—
124#486—21no—
125#469—21no—
126#469—21no—
127#469—21no—
128#469—21no—
129#468—21no—
130#468—21no—
131#468—21no—
132#468—21no—
133#467C2×C411yescharacteristic
134#467C2×C411yescharacteristic
135#466—21no—
136#466—21no—
137#466C2×C411yescharacteristic
138#466C2×C411yescharacteristic
139#466C2×C411yescharacteristic
140#466C2×C411yescharacteristic
141#466C2×C411yescharacteristic
142#466C2×C411yescharacteristic
143#465—21no—
144#465—21no—
145#446—41no—
146#442—21no—
147#442—21no—
148#440—21no—
149#440—21no—
150C4×Q16C2×C411yescharacteristic
151C4×Q16C2311yescharacteristic
152C4×SD16—21no—
153C4×SD16—21no—
154C4×SD16—21no—
155C4×SD16—21no—
156C4×D8—21no—
157C4×D8—21no—
158C4×D8—21no—
159C4×D8—21no—
160C4×D8C2×C411yescharacteristic
161C4×D8C2×C411yescharacteristic
162C4×D8C2×C411yescharacteristic
163C4×D8C2311yescharacteristic
164C4×D8C2311yescharacteristic
165C4×D8C2311yescharacteristic
166#436—41no—
167#435—41no—
168#435—41no—
169#432—21no—
170#432—21no—
171#431—21no—
172#428—21no—
173#427C2×C411yescharacteristic
174C2×#107C2×C411yescharacteristic
175#423—21no—
176#423—21no—
177C2×C4⋊C8—21no—
178#418—21no—
179#417—21no—
180#416—21no—
181C2×#102—21no—
182#405C2×C411yescharacteristic
183#405C2×C411yescharacteristic
184#405C2×C411yescharacteristic
185#405D411yescharacteristic
186#405D411yescharacteristic
187#405D411yescharacteristic
188C4×M16D411yescharacteristic
189C4×M16D411yescharacteristic
190C4×M16D411yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C22⋊C16—21no—
199C22⋊C16—21no—
200C22⋊C16—21no—
201C22⋊C16—21no—
202#346—21no—
203#346C2311yescharacteristic
204#346C2311yescharacteristic
205#346C2311yescharacteristic
206#346C2311yescharacteristic
207C4×C16—21no—
208#335—21no—
209#335—21no—
210#335—21no—
211#334—21no—
212#334—21no—
213#334—21no—
214#334C2311yescharacteristic
215#334C2311yescharacteristic
216#334C2311yescharacteristic
217#334C2311yescharacteristic
218C2×#41—21no—
219C2×#41—41no—
220C2×#41—41no—
221C2×#41C22⋊C411yescharacteristic
222C22×D4—21no—
223C23×C4—41no—
224AGL(1,ℤ/8)—44no—
225#135—42no—
226C2×Q16C22×C411yescharacteristic
227C2×SD16—21no—
228C2×SD16—21no—
229C2×SD16—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—42no—
234C2×D8C22×C411yescharacteristic
235C2×D8C22×C411yescharacteristic
236C2×D8C22×C411yescharacteristic
237#131—42no—
238#131—42no—
239C2×M16—41no—
240C2×M16—41no—
241C2×M16C22⋊C411yescharacteristic
242C2×M16C22⋊C411yescharacteristic
243C2×M16C22⋊C411yescharacteristic
244C2×M16C22⋊C411yescharacteristic
245C2×M16C22⋊C411yescharacteristic
246C2×M16C22⋊C411yescharacteristic
247C22×C8—41no—
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—41no—
257#127—21no—
258#127—21no—
259#127—41no—
260#126—41no—
261#126—41no—
262#126—41no—
263#126—41no—
264#125—21no—
265#125—21no—
266#125—41no—
267#124—21no—
268#124—21no—
269#124—41no—
270#124—41no—
271#124—41no—
272#124—41no—
273#124—41no—
274#123—41no—
275#123—41no—
276#123—41no—
277#123—41no—
278#123—41no—
279#123—41no—
280#123—41no—
281#123—41no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—41no—
285C22⋊Q8—41no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288C22⋊Q8—41no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—41no—
296#121—41no—
297#121—41no—
298#121—41no—
299#121—41no—
300#121—41no—
301#121—41no—
302#121—41no—
303#121—41no—
304#121—41no—
305#121—41no—
306#120—41no—
307#120—41no—
308#120—41no—
309#120—41no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8D811yescharacteristic
313C4×Q8SD1611yescharacteristic
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—41no—
321C4×D4—41no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4—41no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4D811yescharacteristic
329C4×D4D811yescharacteristic
330C4×D4D811yescharacteristic
331C4×D4SD1611yescharacteristic
332C4×D4SD1611yescharacteristic
333C4×D4SD1611yescharacteristic
334#117—21no—
335#117—41no—
336#117—41no—
337#117—41no—
338#117C22⋊C411yescharacteristic
339#117C22⋊C411yescharacteristic
340#117C22⋊C411yescharacteristic
341C2×C4⋊C4—21no—
342C2×C22⋊C4—41no—
343C2×C42—41no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347M32—42no—
348C2×C16—21no—
349C2×C16—21no—
350C2×C16—21no—
351C2×C16—21no—
352C2×C16—42no—
353#107—21no—
354#107—21no—
355#107—21no—
356#107C22×C411yescharacteristic
357#107C22×C411yescharacteristic
358#107C22×C411yescharacteristic
359#107C22×C411yescharacteristic
360#106—21no—
361#106—21no—
362#106—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—41no—
368C4⋊C8—41no—
369C4⋊C8—41no—
370#103—21no—
371#103—21no—
372#103—21no—
373#103—21no—
374#102—21no—
375#102—21no—
376#102—21no—
377#102—21no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—21no—
385#102—21no—
386C22⋊C8—41no—
387C22⋊C8—41no—
388C22⋊C8—41no—
389#97—41no—
390#97—41no—
391#97C22⋊C411yescharacteristic
392#97C22⋊C411yescharacteristic
393#97C22⋊C411yescharacteristic
394#97C22⋊C411yescharacteristic
395#97C22⋊C411yescharacteristic
396#97C22⋊C411yescharacteristic
397C4×C8—41no—
398C4×C8C22×C411yescharacteristic
399C4×C8C22×C411yescharacteristic
400C4×C8C22×C411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2411yescharacteristic
408C24—41no—
409#41—41no—
410#41—41no—
411#41—42no—
412#41—42no—
413#41—162no—
414#41—162no—
415C2×Q8—21no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8#10211yescharacteristic
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4#10211yescharacteristic
437C2×D4#10211yescharacteristic
438C2×D4#10211yescharacteristic
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—162no—
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×D811yescharacteristic
452C22×C4C2×D811yescharacteristic
453C22×C4C2×D811yescharacteristic
454C22×C4C2×SD1611yescharacteristic
455C22×C4C2×SD1611yescharacteristic
456C22×C4C2×SD1611yescharacteristic
457Q16—22no—
458SD16—42no—
459SD16—42no—
460SD16—42no—
461D8—22no—
462D8—22no—
463D8—22no—
464D8—42no—
465D8—42no—
466D8—42no—
467M16—22no—
468M16—22no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—161no—
474M16—161no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—161no—
486C2×C8C2×C22⋊C411yescharacteristic
487C2×C8C2×C22⋊C411yescharacteristic
488C2×C8C2×C22⋊C411yescharacteristic
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C23×C411yescharacteristic
493C2×C8C22×D411yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4#10211yescharacteristic
510C4⋊C4#10211yescharacteristic
511C4⋊C4#10211yescharacteristic
512C4⋊C4#10211yescharacteristic
513C4⋊C4#10211yescharacteristic
514C4⋊C4#10211yescharacteristic
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4#10211yescharacteristic
535C22⋊C4#10211yescharacteristic
536C22⋊C4#10211yescharacteristic
537C22⋊C4#10211yescharacteristic
538C22⋊C4#10211yescharacteristic
539C22⋊C4#10211yescharacteristic
540C42—41no—
541C42—41no—
542C42—41no—
543C42C2×C22⋊C411yescharacteristic
544C42C2×C22⋊C411yescharacteristic
545C42C2×C22⋊C411yescharacteristic
546C42C2×D811yescharacteristic
547C42C2×D811yescharacteristic
548C42C2×D811yescharacteristic
549C42C2×SD1611yescharacteristic
550C42C2×SD1611yescharacteristic
551C42C2×SD1611yescharacteristic
552C42C22×D411yescharacteristic
553C16—41no—
554C16—41no—
555C16—41no—
556C16—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—82no—
561C23—161no—
562C23—161no—
563C23C2×#10211yescharacteristic
564C23C2×#10211yescharacteristic
565C23C2×#10211yescharacteristic
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—161no—
570Q8—161no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—82no—
581D4—82no—
582D4—161no—
583D4—161no—
584D4—161no—
585D4—161no—
586D4—161no—
587D4—161no—
588D4—161no—
589D4—161no—
590D4—161no—
591D4—161no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—161no—
614C2×C4—161no—
615C2×C4—161no—
616C2×C4—161no—
617C2×C4C2×#10211yescharacteristic
618C2×C4C2×#10211yescharacteristic
619C2×C4C2×#10211yescharacteristic
620C2×C4C2×#10211yescharacteristic
621C2×C4C2×#10211yescharacteristic
622C2×C4C2×#10211yescharacteristic
623C2×C4C2×#10211yescharacteristic
624C2×C4C2×#10211yescharacteristic
625C2×C4C2×#10211yescharacteristic
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4C22×D811yescharacteristic
628C2×C4C22×SD1611yescharacteristic
629C8—21no—
630C8—21no—
631C8—21no—
632C8—21no—
633C8—21no—
634C8—21no—
635C8—81no—
636C8—81no—
637C8—81no—
638C8—81no—
639C8#42112yes—
640C8#56812yes—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—164no—
660C22C22×#10211yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—81no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4#287911yescharacteristic
678C4C2×#42111yescharacteristic
679C2—41no—
680C2—41no—
681C2—41no—
682C2—161no—
683C2—161no—
684C2—161no—
685C2#1748511yescharacteristic
686C2#3338811yescharacteristic
687C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#60695C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33421C2semidirect
3. #17798C2semidirect
4. #11682C2semidirect
5. #11680C2semidirect
6. #11678C2semidirect
7. #11677C2semidirect
8. #11667C2semidirect
9. #11666C2semidirect
10. #11665C2semidirect
11. #11663C2semidirect
12. #11656C2semidirect
13. #11654C2semidirect
14. #11652C2semidirect
15. #11651C2semidirect
44. #2163C22semidirect
45. #2163C22semidirect
48. C4×M32C22semidirect
49. C2×#346C22semidirect
53. #1568C22semidirect
55. #1566C22semidirect
57. #1564C22semidirect
58. C2×#334C22semidirect
59. #1334C22semidirect
60. #1334C22semidirect
61. #1334C22semidirect
62. #1334C22semidirect
71. #1333C22semidirect
72. #1333C22semidirect
73. #1333C22semidirect
74. #1333C22semidirect

Maximal subgroups

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

subgroupindex
#606952
#334212
#177982
#116822
#116802
#116782
#116772
#116672
#116662
#116652
#116632
#116562
#116542
#116522
#116512