Finite Group Databasebeta

← #481371

Subgroups — #481371

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60695C211yesmaximal, characteristic
2#33420C211yesmaximal, characteristic, complemented
3#17799C211yesmaximal, characteristic, complemented
4#11675C211yesmaximal, characteristic, complemented
5#11674C211yesmaximal, characteristic, complemented
6#11673C211yesmaximal, characteristic, complemented
7#11671C211yesmaximal, characteristic, complemented
8#11661C211yesmaximal, characteristic, complemented
9#11660C211yesmaximal, characteristic, complemented
10#11659C211yesmaximal, characteristic, complemented
11#11657C211yesmaximal, characteristic, complemented
12#11649C211yesmaximal, characteristic, complemented
13#11648C211yesmaximal, characteristic, complemented
14#11647C211yesmaximal, characteristic, complemented
15#11645C211yesmaximal, 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#2162C2211yescharacteristic
43#2162C2211yescharacteristic
44#2162C2211yescharacteristic, complemented
45#2162C2211yescharacteristic, complemented
46#2151—21no—
47#2113—21no—
48C4×M32C2211yescharacteristic, complemented
49C2×C4×C16C2211yescharacteristic, complemented
50#1569—21no—
51#1569—21no—
52#1568C2211yescharacteristic
53#1568C2211yescharacteristic, complemented
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic, complemented
56#1565—21no—
57#1564C2211yescharacteristic, complemented
58C2×#335C2211yescharacteristic, complemented
59#1332C2211yescharacteristic, complemented
60#1332C2211yescharacteristic, complemented
61#1332C2211yescharacteristic, complemented
62#1332C2211yescharacteristic, complemented
63#1331C2211yescharacteristic
64#1331C2211yescharacteristic
65#1331C2211yescharacteristic
66#1331C2211yescharacteristic
67#1331C2211yescharacteristic
68#1331C2211yescharacteristic
69#1331C2211yescharacteristic
70#1331C2211yescharacteristic
71#1331C2211yescharacteristic, complemented
72#1331C2211yescharacteristic, complemented
73#1331C2211yescharacteristic, complemented
74#1331C2211yescharacteristic, 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—
203C4×C16—21no—
204C4×C16C2311yescharacteristic
205C4×C16C2311yescharacteristic
206C4×C16C2311yescharacteristic
207C4×C16C2311yescharacteristic
208#335—21no—
209#335—21no—
210#335—21no—
211#335C2311yescharacteristic
212#335C2311yescharacteristic
213#335C2311yescharacteristic
214#335C2311yescharacteristic
215#334—21no—
216#334—21no—
217#334—21no—
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—161no—
414#41—161no—
415#41—161no—
416#41—161no—
417C2×Q8—21no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8#10211yescharacteristic
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
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—41no—
437C2×D4—41no—
438C2×D4#10211yescharacteristic
439C2×D4#10211yescharacteristic
440C2×D4#10211yescharacteristic
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—161no—
450C22×C4—161no—
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×D811yescharacteristic
455C22×C4C2×D811yescharacteristic
456C22×C4C2×D811yescharacteristic
457C22×C4C2×SD1611yescharacteristic
458C22×C4C2×SD1611yescharacteristic
459C22×C4C2×SD1611yescharacteristic
460Q16—22no—
461SD16—42no—
462SD16—42no—
463SD16—42no—
464D8—22no—
465D8—22no—
466D8—22no—
467D8—42no—
468D8—42no—
469D8—42no—
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—161no—
477M16—161no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—161no—
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C2×C22⋊C411yescharacteristic
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C23×C411yescharacteristic
496C2×C8C22×D411yescharacteristic
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
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—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4#10211yescharacteristic
513C4⋊C4#10211yescharacteristic
514C4⋊C4#10211yescharacteristic
515C4⋊C4#10211yescharacteristic
516C4⋊C4#10211yescharacteristic
517C4⋊C4#10211yescharacteristic
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—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4#10211yescharacteristic
538C22⋊C4#10211yescharacteristic
539C22⋊C4#10211yescharacteristic
540C22⋊C4#10211yescharacteristic
541C22⋊C4#10211yescharacteristic
542C22⋊C4#10211yescharacteristic
543C42—41no—
544C42—41no—
545C42—41no—
546C42C2×C22⋊C411yescharacteristic
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C2×D811yescharacteristic
550C42C2×D811yescharacteristic
551C42C2×D811yescharacteristic
552C42C2×SD1611yescharacteristic
553C42C2×SD1611yescharacteristic
554C42C2×SD1611yescharacteristic
555C42C22×D411yescharacteristic
556C16—41no—
557C16—41no—
558C16—41no—
559C16—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—82no—
564C23—161no—
565C23—161no—
566C23C2×#10211yescharacteristic
567C23C2×#10211yescharacteristic
568C23C2×#10211yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—161no—
573Q8—161no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—82no—
584D4—82no—
585D4—161no—
586D4—161no—
587D4—161no—
588D4—161no—
589D4—161no—
590D4—161no—
591D4—161no—
592D4—161no—
593D4—161no—
594D4—161no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—82no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—161no—
623C2×C4—161no—
624C2×C4—161no—
625C2×C4—161no—
626C2×C4C2×#10211yescharacteristic
627C2×C4C2×#10211yescharacteristic
628C2×C4C2×#10211yescharacteristic
629C2×C4C2×#10211yescharacteristic
630C2×C4C2×#10211yescharacteristic
631C2×C4C2×#10211yescharacteristic
632C2×C4C2×#10211yescharacteristic
633C2×C4C2×#10211yescharacteristic
634C2×C4C2×#10211yescharacteristic
635C2×C4C22×C22⋊C411yescharacteristic
636C2×C4C22×D811yescharacteristic
637C2×C4C22×SD1611yescharacteristic
638C8—21no—
639C8—21no—
640C8—21no—
641C8—21no—
642C8—21no—
643C8—21no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8#42112yes—
649C8#56812yes—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—164no—
669C22C22×#10211yescharacteristic
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4#287911yescharacteristic
686C4C2×#42111yescharacteristic
687C4#341511yescharacteristic
688C4#341611yescharacteristic
689C2—41no—
690C2—41no—
691C2—41no—
692C2—161no—
693C2—161no—
694C2—161no—
695C2#1748511yescharacteristic
696C2#3338111yescharacteristic
697C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#60695C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #33420C2semidirect
3. #17799C2semidirect
4. #11675C2semidirect
5. #11674C2semidirect
6. #11673C2semidirect
7. #11671C2semidirect
8. #11661C2semidirect
9. #11660C2semidirect
10. #11659C2semidirect
11. #11657C2semidirect
12. #11649C2semidirect
13. #11648C2semidirect
14. #11647C2semidirect
15. #11645C2semidirect
44. #2162C22semidirect
45. #2162C22semidirect
48. C4×M32C22semidirect
49. C2×C4×C16C22semidirect
53. #1568C22semidirect
55. #1567C22semidirect
57. #1564C22semidirect
58. C2×#335C22semidirect
59. #1332C22semidirect
60. #1332C22semidirect
61. #1332C22semidirect
62. #1332C22semidirect
71. #1331C22semidirect
72. #1331C22semidirect
73. #1331C22semidirect
74. #1331C22semidirect

Maximal subgroups

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

subgroupindex
#606952
#334202
#177992
#116752
#116742
#116732
#116712
#116612
#116602
#116592
#116572
#116492
#116482
#116472
#116452