Finite Group Databasebeta

← #481412

Subgroups — #481412

Order 512.

715 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61433C211yesmaximal, characteristic
2#33479C211yesmaximal, characteristic, complemented
3#17836C211yesmaximal, characteristic, complemented
4#11676C211yesmaximal, characteristic, complemented
5#11674C211yesmaximal, characteristic, complemented
6#11673C211yesmaximal, characteristic, complemented
7#11662C211yesmaximal, characteristic, complemented
8#11659C211yesmaximal, characteristic, complemented
9#11658C211yesmaximal, characteristic, complemented
10#11648C211yesmaximal, characteristic, complemented
11#11647C211yesmaximal, characteristic, complemented
12#11646C211yesmaximal, characteristic, complemented
13#11594C211yesmaximal, characteristic, complemented
14#11592C211yesmaximal, characteristic, complemented
15C2×#1332C211yesmaximal, characteristic, complemented
16#3482—21no—
17#3393C411yescharacteristic
18#3383C411yescharacteristic
19#3364—21no—
20#3336C411yescharacteristic
21#3336C411yescharacteristic
22#3334C411yescharacteristic
23#3300—21no—
24#3300—21no—
25#3298—21no—
26D4×Q16C411yescharacteristic
27#3285C411yescharacteristic
28#3285C411yescharacteristic
29#3170C2211yescharacteristic
30#3168C2211yescharacteristic
31#3168C2211yescharacteristic
32#3108—21no—
33#3108—21no—
34#3107—21no—
35#2990—21no—
36C8×#41C2211yescharacteristic
37C4×#135C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39C2×C4×Q16C2211yescharacteristic
40#2183—21no—
41#2169—21no—
42#2168—21no—
43#2168—21no—
44#2162C2211yescharacteristic
45#2162C2211yescharacteristic
46#2162C2211yescharacteristic
47C4×M32C2211yescharacteristic
48C4×M32C2211yescharacteristic
49C2×C4×C16C2211yescharacteristic
50#1574—21no—
51#1573—21no—
52#1573—21no—
53#1568C2211yescharacteristic
54#1568C2211yescharacteristic
55#1567C2211yescharacteristic
56#1564C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1332C2211yescharacteristic
62#1332C2211yescharacteristic
63#1332C2211yescharacteristic
64#1332C2211yescharacteristic
65#1332C2211yescharacteristic
66#1332C2211yescharacteristic
67#1332C2211yescharacteristic
68#1332C2211yescharacteristic
69#1332C2211yescharacteristic
70#1332C2211yescharacteristic
71#1331C2211yescharacteristic
72#1331C2211yescharacteristic
73#1331C2211yescharacteristic
74#1331C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C22×Q16C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78#557—21no—
79#556—21no—
80#556—41no—
81#552—21no—
82#552—21no—
83#552—41no—
84#552—41no—
85D4×Q8—21no—
86#548—21no—
87#548—21no—
88#547—21no—
89#547—41no—
90#533—21no—
91#533—41no—
92#529—21no—
93#529—21no—
94#529—41no—
95#529—41no—
96C4×#41—21no—
97C4×#41—41no—
98C4×#41D411yescharacteristic
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×M32—21no—
102C2×M32—21no—
103C22×C16—21no—
104#500C2×C411yescharacteristic
105#499C2×C411yescharacteristic
106#499C2×C411yescharacteristic
107#495C2×C411yescharacteristic
108#495C2×C411yescharacteristic
109#494C2×C411yescharacteristic
110#488—21no—
111#488—21no—
112#487—21no—
113#485—21no—
114#485—21no—
115#483—21no—
116#483—21no—
117#480—21no—
118#480—21no—
119#479—21no—
120#479—21no—
121#479—21no—
122#479—21no—
123#475—21no—
124#474—21no—
125#467—21no—
126#467C2×C411yescharacteristic
127#467C2×C411yescharacteristic
128#467C2×C411yescharacteristic
129#467C2×C411yescharacteristic
130#466C2×C411yescharacteristic
131#466C2×C411yescharacteristic
132#465—21no—
133#465—21no—
134#464—21no—
135#464—21no—
136#464—21no—
137#464—21no—
138#462—21no—
139#461—21no—
140#452—21no—
141#452—21no—
142#451—21no—
143#451—21no—
144#448—21no—
145#448—21no—
146#446—41no—
147C8×Q8C2×C411yescharacteristic
148C8×Q8D411yescharacteristic
149C4×Q16—21no—
150C4×Q16—21no—
151C4×Q16—21no—
152C4×Q16—21no—
153C4×Q16—21no—
154C4×Q16C2×C411yescharacteristic
155C4×Q16C2×C411yescharacteristic
156C4×Q16C2×C411yescharacteristic
157C4×Q16C2×C411yescharacteristic
158C4×Q16C2×C411yescharacteristic
159C4×Q16C2311yescharacteristic
160C4×Q16C2311yescharacteristic
161C4×Q16C2311yescharacteristic
162C4×SD16—21no—
163C4×SD16—21no—
164C4×SD16—21no—
165C4×SD16—21no—
166C4×D8C2×C411yescharacteristic
167C4×D8C2311yescharacteristic
168#435—41no—
169#435—41no—
170C8×D4—41no—
171C8×D4C2×C411yescharacteristic
172C8×D4C2×C411yescharacteristic
173C8×D4C2×C411yescharacteristic
174C8×D4D411yescharacteristic
175C8×D4D411yescharacteristic
176C8×D4D411yescharacteristic
177#432—21no—
178#432—21no—
179#431—21no—
180#431D411yescharacteristic
181#431D411yescharacteristic
182#431D411yescharacteristic
183#427C2×C411yescharacteristic
184C2×#107C2×C411yescharacteristic
185C2×#107C2×C411yescharacteristic
186#423—21no—
187C2×C4⋊C8—21no—
188C2×C4⋊C8—21no—
189#416—21no—
190C2×#103—21no—
191C2×#103—21no—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C22⋊C16—21no—
199C22⋊C16—21no—
200C22⋊C16—21no—
201#346—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×#41SD1611yescharacteristic
221C22×Q8—21no—
222C22×Q8—21no—
223#135—22no—
224#135—41no—
225C2×Q16—22no—
226C2×Q16—22no—
227C2×Q16—21no—
228C2×Q16—21no—
229C2×Q16—21no—
230C2×Q16—41no—
231C2×Q16—41no—
232C2×Q16C22×C411yescharacteristic
233C2×Q16C22×C411yescharacteristic
234C2×Q16C22×C411yescharacteristic
235C2×SD16—21no—
236C2×SD16—21no—
237C2×SD16—21no—
238C2×D8C22×C411yescharacteristic
239C2×M16—41no—
240C22×C8—41no—
241C22×C8—41no—
242C22×C8C22×C411yescharacteristic
243C22×C8C22×C411yescharacteristic
244C22×C8C22×C411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—41no—
251C4⋊Q8—41no—
252#126—41no—
253#126—41no—
254#126—41no—
255#126—41no—
256#126—41no—
257#126—41no—
258#125—21no—
259#125—21no—
260#125—41no—
261#125—41no—
262#125—41no—
263#125—41no—
264#125—41no—
265#124—21no—
266#124—21no—
267#124—41no—
268#124—41no—
269#123—41no—
270#123—41no—
271#123—41no—
272#123—41no—
273#123—41no—
274#123—41no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—41no—
280C22⋊Q8—41no—
281C22⋊Q8—41no—
282C22⋊Q8—41no—
283C22⋊Q8—41no—
284C22⋊Q8—41no—
285C22⋊Q8—41no—
286#121—21no—
287#121—21no—
288#121—41no—
289#121—41no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—41no—
299C4×Q8—41no—
300C4×Q8—41no—
301C4×Q8—41no—
302C4×Q8—41no—
303C4×Q8—41no—
304C4×Q8C22⋊C411yescharacteristic
305C4×Q8D811yescharacteristic
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4—41no—
315C4×D4C22⋊C411yescharacteristic
316C4×D4C22⋊C411yescharacteristic
317C4×D4C22⋊C411yescharacteristic
318C4×D4D811yescharacteristic
319C4×D4D811yescharacteristic
320C4×D4D811yescharacteristic
321#117—21no—
322#117—41no—
323#117—41no—
324#117—41no—
325#117—41no—
326#117—41no—
327#117—41no—
328#117—41no—
329#117SD1611yescharacteristic
330#117SD1611yescharacteristic
331#117SD1611yescharacteristic
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—41no—
336C2×C42—41no—
337C2×C42—41no—
338C2×C42—41no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342M32—42no—
343M32—42no—
344C2×C16—21no—
345C2×C16—21no—
346C2×C16—21no—
347C2×C16—21no—
348C2×C16—42no—
349#107—21no—
350#107—21no—
351#107—21no—
352#107C22×C411yescharacteristic
353#107C22×C411yescharacteristic
354#107C22×C411yescharacteristic
355#107C22×C411yescharacteristic
356#106—21no—
357#106—21no—
358#106—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—41no—
364C4⋊C8—41no—
365C4⋊C8—41no—
366C4⋊C8C22⋊C411yescharacteristic
367C4⋊C8C22⋊C411yescharacteristic
368C4⋊C8C22⋊C411yescharacteristic
369C4⋊C8C22⋊C411yescharacteristic
370C4⋊C8C22⋊C411yescharacteristic
371C4⋊C8C22⋊C411yescharacteristic
372#103—21no—
373#103—21no—
374#103—21no—
375#103—21no—
376#103—21no—
377#103—21no—
378#103—21no—
379#103—21no—
380#103—21no—
381#103—21no—
382#103—21no—
383#103—21no—
384#102—21no—
385#102—21no—
386#102—21no—
387#102—21no—
388C22⋊C8—41no—
389C22⋊C8—41no—
390C22⋊C8—41no—
391C22⋊C8C22⋊C411yescharacteristic
392C22⋊C8C22⋊C411yescharacteristic
393C22⋊C8C22⋊C411yescharacteristic
394C22⋊C8C22⋊C411yescharacteristic
395C22⋊C8C22⋊C411yescharacteristic
396C22⋊C8C22⋊C411yescharacteristic
397#97—41no—
398#97—41no—
399C4×C8—41no—
400C4×C8C22×C411yescharacteristic
401C4×C8C22×C411yescharacteristic
402C4×C8C22×C411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410#41—42no—
411#41—42no—
412#41—81no—
413#41—84no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×Q8#10211yescharacteristic
425C2×D4—21no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4#10211yescharacteristic
430C2×D4#10211yescharacteristic
431C2×D4#10211yescharacteristic
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×D811yescharacteristic
443C22×C4C2×D811yescharacteristic
444C22×C4C2×D811yescharacteristic
445C22×C4C2×SD1611yescharacteristic
446C22×C4C2×SD1611yescharacteristic
447C22×C4C2×SD1611yescharacteristic
448Q16—21no—
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—42no—
454Q16—42no—
455Q16—81no—
456Q16#13112yes—
457SD16—42no—
458SD16—81no—
459SD16—81no—
460D8—21no—
461D8—21no—
462M16—161no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—161no—
477C2×C8—161no—
478C2×C8C2×C22⋊C411yescharacteristic
479C2×C8C2×C22⋊C411yescharacteristic
480C2×C8C2×C22⋊C411yescharacteristic
481C2×C8C2×C22⋊C411yescharacteristic
482C2×C8C2×C22⋊C411yescharacteristic
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C23×C411yescharacteristic
485C2×C8C22×D411yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
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—41no—
510C4⋊C4—41no—
511C4⋊C4#10211yescharacteristic
512C4⋊C4#10211yescharacteristic
513C4⋊C4#10211yescharacteristic
514C4⋊C4#10211yescharacteristic
515C4⋊C4#10211yescharacteristic
516C4⋊C4#10211yescharacteristic
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#10211yescharacteristic
531C22⋊C4#10211yescharacteristic
532C22⋊C4#10211yescharacteristic
533C22⋊C4#10211yescharacteristic
534C22⋊C4#10211yescharacteristic
535C22⋊C4#10211yescharacteristic
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42C2×C22⋊C411yescharacteristic
546C42C2×C22⋊C411yescharacteristic
547C42C2×C22⋊C411yescharacteristic
548C42C2×D811yescharacteristic
549C42C2×D811yescharacteristic
550C42C2×D811yescharacteristic
551C42C2×SD1611yescharacteristic
552C42C2×SD1611yescharacteristic
553C42C2×SD1611yescharacteristic
554C42C22×D411yescharacteristic
555C16—22no—
556C16—41no—
557C16—41no—
558C16—41no—
559C23—41no—
560C23C2×#10211yescharacteristic
561C23C2×#10211yescharacteristic
562C23C2×#10211yescharacteristic
563Q8—22no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—82no—
571Q8—82no—
572Q8—82no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—82no—
579D4—82no—
580D4—82no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4C2×#10211yescharacteristic
612C2×C4C2×#10211yescharacteristic
613C2×C4C2×#10211yescharacteristic
614C2×C4C2×#10211yescharacteristic
615C2×C4C2×#10211yescharacteristic
616C2×C4C2×#10211yescharacteristic
617C2×C4C2×#10211yescharacteristic
618C2×C4C2×#10211yescharacteristic
619C2×C4C2×#10211yescharacteristic
620C2×C4C22×C22⋊C411yescharacteristic
621C2×C4C22×D811yescharacteristic
622C2×C4C22×SD1611yescharacteristic
623C8—21no—
624C8—21no—
625C8—21no—
626C8—21no—
627C8—21no—
628C8—21no—
629C8—21no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8C2×#13111yescharacteristic
635C8#56811yescharacteristic
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#214312yes—
648C22C22×#10211yescharacteristic
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4#288011yescharacteristic
667C4#290111yescharacteristic
668C4#341611yescharacteristic
669C4#341711yescharacteristic
670C2—22no—
671C2—41no—
672C2—41no—
673C2—161no—
674C2#1750711yescharacteristic
675C2#3338211yescharacteristic
676C2C2×#214311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#61433C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33479C2semidirect
3. #17836C2semidirect
4. #11676C2semidirect
5. #11674C2semidirect
6. #11673C2semidirect
7. #11662C2semidirect
8. #11659C2semidirect
9. #11658C2semidirect
10. #11648C2semidirect
11. #11647C2semidirect
12. #11646C2semidirect
13. #11594C2semidirect
14. #11592C2semidirect
15. C2×#1332C2semidirect

Maximal subgroups

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

subgroupindex
#614332
#334792
#178362
#116762
#116742
#116732
#116622
#116592
#116582
#116482
#116472
#116462
#115942
#115922
C2×#13322