Finite Group Databasebeta

← #7625040

Subgroups — #7625040

Order 512.

1195 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62474C211yesmaximal, characteristic
2#62473C211yesmaximal, characteristic, complemented
3#33955C214yesmaximal, complemented
4#33918C214yesmaximal, complemented
5#33917C214yesmaximal, complemented
6#33915C214yesmaximal, complemented
7#33797C211yesmaximal, characteristic, complemented
8#33797C211yesmaximal, characteristic, complemented
9#33795C211yesmaximal, characteristic, complemented
10#33794C211yesmaximal, characteristic, complemented
11#33794C211yesmaximal, characteristic, complemented
12#33793C211yesmaximal, characteristic, complemented
13#33751C211yesmaximal, characteristic, complemented
14#33751C211yesmaximal, characteristic, complemented
15#33736C211yesmaximal, characteristic, complemented
16#33731C211yesmaximal, characteristic, complemented
17#33731C211yesmaximal, characteristic, complemented
18#33726C211yesmaximal, characteristic, complemented
19#33479C211yesmaximal, characteristic, complemented
20#3511—21no—
21#3509—21no—
22#3383—21no—
23#3383C2212yes—
24#3380—21no—
25#3380C2212yes—
26#3342—21no—
27#3342C2212yes—
28#3341—21no—
29#3341C2212yes—
30#3339—21no—
31#3339—21no—
32#3339C2212yes—
33#3339C2212yes—
34#3336—21no—
35#3336—21no—
36#3336C2212yes—
37#3336C2212yes—
38#3231—21no—
39#3231—21no—
40#3229—21no—
41#3228—21no—
42#3228—21no—
43#3227—21no—
44#3170C2211yescharacteristic
45#3170C2211yescharacteristic
46#3170C2211yescharacteristic
47#3169C2211yescharacteristic
48#3169C2211yescharacteristic
49#3169C2211yescharacteristic
50#3160C2211yescharacteristic
51#3160C2211yescharacteristic
52#3160C2211yescharacteristic
53#3160C2211yescharacteristic
54C2×#500C2211yescharacteristic
55C2×#500C2211yescharacteristic
56C8×#41C2211yescharacteristic
57#2255C2214yes—
58#2255C2214yes—
59#2254C2211yescharacteristic
60#2254C2211yescharacteristic
61#2254C2211yescharacteristic
62#2254C2211yescharacteristic
63#2254C2211yescharacteristic
64#2253C2214yes—
65#2252C2211yescharacteristic
66#2252C2211yescharacteristic
67#2252C2211yescharacteristic
68#2252C2211yescharacteristic
69#2248C2211yescharacteristic
70#2241C2211yescharacteristic
71#2241C2211yescharacteristic
72#2241C2211yescharacteristic
73#2240C2211yescharacteristic
74#2240C2211yescharacteristic
75#2240C2211yescharacteristic
76#2237C2214yes—
77#2237C2214yes—
78#2235C2214yes—
79#2234C2214yes—
80#2234C2214yes—
81#2233C2214yes—
82#2229C2214yes—
83#2229C2214yes—
84#2228C2214yes—
85#2228C2214yes—
86#2226C2214yes—
87#2225C2214yes—
88#2220C2214yes—
89#2220C2214yes—
90#2217C2214yes—
91#2183C2212yes—
92#2175C2214yes—
93#2175C2214yes—
94C4×SD32C2214yes—
95#2169C2212yes—
96#2168C2212yes—
97#2168C2212yes—
98#2162C2211yescharacteristic
99#2162C2211yescharacteristic
100#2162C2211yescharacteristic
101#2159C2214yes—
102#2159C2214yes—
103#2158C2214yes—
104C4×M32C2211yescharacteristic
105C4×M32C2211yescharacteristic
106C2×C4×C16C2211yescharacteristic
107#557—22no—
108#557—41no—
109#556—22no—
110#556—41no—
111#556—41no—
112#555—22no—
113#554—22no—
114#553—22no—
115#553—22no—
116#552—22no—
117#552—22no—
118#552—41no—
119#552—41no—
120#548—41no—
121#548—41no—
122#547—41no—
123#542—41no—
124#542—41no—
125#541—41no—
126#541—41no—
127#537—41no—
128#536—41no—
129#533—21no—
130#533—21no—
131#533—21no—
132#533—21no—
133#533—21no—
134#533—21no—
135#533—21no—
136#532—21no—
137#532—21no—
138#532—21no—
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8—21no—
141#520—41no—
142#519—41no—
143C4×#41D411yescharacteristic
144C2×SD32C2314yes—
145C2×M32C2312yes—
146C2×M32C2312yes—
147C22×C16C2312yes—
148#500—21no—
149#500—21no—
150#500—21no—
151#500—21no—
152#500C2312yes—
153#500C2312yes—
154#500C2311yescharacteristic
155#500C2311yescharacteristic
156#500C2311yescharacteristic
157#500C2311yescharacteristic
158#500C2311yescharacteristic
159#500C2311yescharacteristic
160#499—21no—
161#499—21no—
162#499—21no—
163#499—21no—
164#499C2312yes—
165#499C2312yes—
166#499C2312yes—
167#499C2312yes—
168#498—21no—
169#498—21no—
170#498—21no—
171#498—21no—
172#494C2311yescharacteristic
173#493C2311yescharacteristic
174#487—21no—
175#487—21no—
176#487—21no—
177#486—21no—
178#486—21no—
179#486—21no—
180#485—22no—
181#485—22no—
182#485—22no—
183#485—22no—
184#483—22no—
185#483—22no—
186#482—22no—
187#482—22no—
188#482—22no—
189#482—22no—
190#481—22no—
191#481—22no—
192#479—22no—
193#479—22no—
194#479—22no—
195#479—22no—
196#478—22no—
197#478—22no—
198#478—22no—
199#478—22no—
200#476—22no—
201#476—22no—
202#475—22no—
203#475—22no—
204#467—21no—
205#467—21no—
206#467—21no—
207#467C2312yes—
208#467C2312yes—
209#467C2312yes—
210#466—21no—
211#466—21no—
212#466—21no—
213#466C2312yes—
214#466C2312yes—
215#466C2312yes—
216C8×Q8D411yescharacteristic
217C8×Q8C2311yescharacteristic
218C4×Q16—21no—
219C4×Q16—21no—
220C4×Q16—21no—
221C4×Q16C2312yes—
222C4×Q16C2312yes—
223C4×Q16C2312yes—
224C4×D8—21no—
225C4×D8—21no—
226C4×D8—21no—
227C4×D8C2312yes—
228C4×D8C2312yes—
229C4×D8C2312yes—
230C8×D4D411yescharacteristic
231C8×D4D411yescharacteristic
232C8×D4D411yescharacteristic
233C8×D4C2311yescharacteristic
234C8×D4C2311yescharacteristic
235C8×D4C2311yescharacteristic
236#431D411yescharacteristic
237#431D411yescharacteristic
238#431D411yescharacteristic
239#427—21no—
240#427—21no—
241#427—21no—
242#427—21no—
243#427C2312yes—
244#427C2312yes—
245#427C2312yes—
246#427C2312yes—
247C2×#107—21no—
248C2×#107—21no—
249C2×#107C2312yes—
250C2×#107C2312yes—
251C2×C4×C8C2311yescharacteristic
252C2×C4×C8C2311yescharacteristic
253C2×C4×C8C2311yescharacteristic
254#367C2314yes—
255#367C2314yes—
256#367C2314yes—
257#367C2314yes—
258#367C2314yes—
259#366C2314yes—
260#366C2314yes—
261#366C2314yes—
262#366C2314yes—
263C4⋊C16C2312yes—
264C4⋊C16C2312yes—
265C4⋊C16C2312yes—
266#358C2314yes—
267#358C2314yes—
268#358C2314yes—
269#357C2314yes—
270#357C2314yes—
271#357C2314yes—
272C22⋊C16C2312yes—
273C22⋊C16C2312yes—
274C22⋊C16C2312yes—
275#346C2312yes—
276#346C2312yes—
277C4×C16C2311yescharacteristic
278C4×C16C2311yescharacteristic
279C4×C16C2312yes—
280C4×C16C2311yescharacteristic
281C4×C16C2311yescharacteristic
282C2×#41—41no—
283C2×#41—41no—
284C2×#41—41no—
285C2×#41D811yescharacteristic
286C22×Q8—41no—
287C22×Q8—41no—
288C2×Q16—21no—
289C2×Q16C2412yes—
290C2×D8—21no—
291C2×D8C2412yes—
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C22×C8C2×D411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—42no—
308C4⋊Q8—41no—
309C4⋊Q8—41no—
310C4⋊Q8—41no—
311C4⋊Q8—41no—
312#127—21no—
313#126—42no—
314#126—42no—
315#126—42no—
316#126—42no—
317#126—42no—
318#126—42no—
319#125—22no—
320#125—22no—
321#125—22no—
322#125—22no—
323#125—42no—
324#125—42no—
325#125—42no—
326#125—42no—
327#125—41no—
328#125—41no—
329#125—41no—
330#124—42no—
331#124—42no—
332#124—42no—
333#124—41no—
334#124—41no—
335#124—41no—
336#123—42no—
337#123—42no—
338#123—42no—
339#123—42no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—42no—
344C22⋊Q8—41no—
345C22⋊Q8—41no—
346C22⋊Q8—41no—
347C22⋊Q8—41no—
348C22⋊Q8—41no—
349C22⋊Q8—41no—
350C22⋊Q8—41no—
351C22⋊Q8—41no—
352C22⋊Q8—41no—
353C22⋊Q8—41no—
354C22⋊Q8—41no—
355C22⋊Q8—42no—
356C22⋊Q8—42no—
357#121—22no—
358#121—22no—
359#121—22no—
360#121—42no—
361#121—42no—
362#121—42no—
363#121—41no—
364#121—41no—
365#121—41no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—42no—
370C4×Q8—42no—
371C4×Q8—41no—
372C4×Q8—41no—
373C4×Q8—41no—
374C4×Q8—41no—
375C4×Q8—41no—
376C4×Q8—41no—
377C4×Q8—41no—
378C4×Q8—41no—
379C4×Q8—41no—
380C4×Q8—41no—
381C4×Q8—41no—
382C4×Q8D811yescharacteristic
383C4×Q8C2×D411yescharacteristic
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—42no—
388C4×D4—42no—
389C4×D4—41no—
390C4×D4—41no—
391C4×D4—41no—
392C4×D4D811yescharacteristic
393C4×D4D811yescharacteristic
394C4×D4D811yescharacteristic
395C4×D4C2×D411yescharacteristic
396C4×D4C2×D411yescharacteristic
397C4×D4C2×D411yescharacteristic
398#117—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402#117—41no—
403#117—41no—
404#117—41no—
405#117—42no—
406#117—42no—
407#117—42no—
408#117D811yescharacteristic
409#117D811yescharacteristic
410#117D811yescharacteristic
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—41no—
414C2×C4⋊C4—41no—
415C2×C4⋊C4—42no—
416C2×C42C2×D411yescharacteristic
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419SD32—28no—
420M32—24no—
421M32—24no—
422C2×C16—24no—
423C2×C16C2412yes—
424C2×C16C2412yes—
425C2×C16C2412yes—
426C2×C16C2412yes—
427#107—21no—
428#107—21no—
429#107—21no—
430#107—21no—
431#107—22no—
432#107—22no—
433#107—22no—
434#107—22no—
435#107—21no—
436#107—21no—
437#107C2412yes—
438#107C2412yes—
439#107C2412yes—
440#107C2412yes—
441#107C2412yes—
442#107C2412yes—
443#106—22no—
444#106—22no—
445#106—22no—
446#106—22no—
447#106—22no—
448#106—22no—
449#106—22no—
450#106—22no—
451C4⋊C8C2×D411yescharacteristic
452C4⋊C8C2×D411yescharacteristic
453C4⋊C8C2×D411yescharacteristic
454C4⋊C8C2×D411yescharacteristic
455C4⋊C8C2×D411yescharacteristic
456C4⋊C8C2×D411yescharacteristic
457#103—22no—
458#103—22no—
459#103—22no—
460#103—22no—
461#103—22no—
462#103—22no—
463#102—22no—
464#102—22no—
465#102—22no—
466#102—22no—
467#102—22no—
468#102—22no—
469C22⋊C8C2×D411yescharacteristic
470C22⋊C8C2×D411yescharacteristic
471C22⋊C8C2×D411yescharacteristic
472C22⋊C8C2×D411yescharacteristic
473C22⋊C8C2×D411yescharacteristic
474C22⋊C8C2×D411yescharacteristic
475C4×C8C2×D411yescharacteristic
476C4×C8C2×D411yescharacteristic
477C4×C8C2×D411yescharacteristic
478C4×C8C2×D411yescharacteristic
479C4×C8C2×D411yescharacteristic
480C4×C8C2×D411yescharacteristic
481C4×C8C2411yescharacteristic
482C4×C8C2411yescharacteristic
483C4×C8C2411yescharacteristic
484C4×C8C2411yescharacteristic
485#41—42no—
486#41—84no—
487#41—162no—
488#41—162no—
489C2×Q8—22no—
490C2×Q8—41no—
491C2×Q8—41no—
492C2×Q8—41no—
493C2×Q8—41no—
494C2×Q8—42no—
495C2×Q8—41no—
496C2×Q8—41no—
497C2×Q8—42no—
498C2×Q8—84no—
499C2×Q8—162no—
500C2×Q8C2×D811yescharacteristic
501C2×D4—22no—
502C2×D4—42no—
503C2×D4—42no—
504C2×D4—42no—
505C2×D4—41no—
506C2×D4—41no—
507C2×D4C2×D811yescharacteristic
508C2×D4C2×D811yescharacteristic
509C2×D4C2×D811yescharacteristic
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4C2×D811yescharacteristic
516C22×C4C2×D811yescharacteristic
517C22×C4C2×D811yescharacteristic
518C22×C4C2×D811yescharacteristic
519C22×C4C2×D811yescharacteristic
520C22×C4C2×D811yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524Q16—24no—
525Q16—42no—
526D8—24no—
527D8—42no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8C22×D411yescharacteristic
535C2×C8C22×D411yescharacteristic
536C2×C8C22×D411yescharacteristic
537C2×C8C22×D411yescharacteristic
538C2×C8C22×D411yescharacteristic
539C2×C8C22×D411yescharacteristic
540C2×C8C22×D411yescharacteristic
541C2×C8C2511yescharacteristic
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—41no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—41no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4C2×D811yescharacteristic
575C4⋊C4C2×D811yescharacteristic
576C4⋊C4C2×D811yescharacteristic
577C4⋊C4C2×D811yescharacteristic
578C4⋊C4C2×D811yescharacteristic
579C4⋊C4C2×D811yescharacteristic
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C22⋊C4—42no—
584C22⋊C4—42no—
585C22⋊C4—42no—
586C22⋊C4—42no—
587C22⋊C4—42no—
588C22⋊C4—42no—
589C22⋊C4—42no—
590C22⋊C4—42no—
591C22⋊C4—42no—
592C22⋊C4—42no—
593C22⋊C4C2×D811yescharacteristic
594C22⋊C4C2×D811yescharacteristic
595C22⋊C4C2×D811yescharacteristic
596C22⋊C4C2×D811yescharacteristic
597C22⋊C4C2×D811yescharacteristic
598C22⋊C4C2×D811yescharacteristic
599C42—42no—
600C42—42no—
601C42—42no—
602C42—42no—
603C42—42no—
604C42—42no—
605C42—42no—
606C42C2×D811yescharacteristic
607C42C2×D811yescharacteristic
608C42C2×D811yescharacteristic
609C42C2×D811yescharacteristic
610C42C2×D811yescharacteristic
611C42C2×D811yescharacteristic
612C42C22×D411yescharacteristic
613C42C22×D411yescharacteristic
614C42C22×D411yescharacteristic
615C42C22×D411yescharacteristic
616C16—22no—
617C16—22no—
618C16—22no—
619C162−1+414yes—
620C23—42no—
621C23C22×D811yescharacteristic
622C23C22×D811yescharacteristic
623C23C22×D811yescharacteristic
624Q8—41no—
625Q8—42no—
626Q8—42no—
627Q8—84no—
628Q8—82no—
629Q8—82no—
630Q8—82no—
631Q8—82no—
632Q8—82no—
633Q8—82no—
634Q8—162no—
635D4—41no—
636D4—41no—
637D4—41no—
638D4—42no—
639D4—42no—
640D4—84no—
641D4—82no—
642D4—82no—
643D4—162no—
644D4—162no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—162no—
667C2×C4—162no—
668C2×C4—162no—
669C2×C4C22×D811yescharacteristic
670C2×C4C22×D811yescharacteristic
671C2×C4C22×D811yescharacteristic
672C2×C4C22×D811yescharacteristic
673C2×C4C22×D811yescharacteristic
674C2×C4C22×D811yescharacteristic
675C2×C4C22×D811yescharacteristic
676C2×C4C22×D811yescharacteristic
677C2×C4C22×D811yescharacteristic
678C2×C4C22×D811yescharacteristic
679C2×C4C22×D811yescharacteristic
680C2×C4C23×D411yescharacteristic
681C8—21no—
682C8—21no—
683C8—21no—
684C8—21no—
685C8—21no—
686C8—21no—
687C8#57812yes—
688C8C2×2−1+412yes—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22C23×D811yescharacteristic
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#341412yes—
719C4#341612yes—
720C4#349911yescharacteristic
721C4C2×#57811yescharacteristic
722C2—41no—
723C2—41no—
724C2—41no—
725C2—162no—
726C2#6247111yescharacteristic
727C2C2×#341411yescharacteristic
728C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62474C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #62473C2semidirect
3. #33955C2semidirect
4. #33918C2semidirect
5. #33917C2semidirect
6. #33915C2semidirect
7. #33797C2semidirect
8. #33797C2semidirect
9. #33795C2semidirect
10. #33794C2semidirect
11. #33794C2semidirect
12. #33793C2semidirect
13. #33751C2semidirect
14. #33751C2semidirect
15. #33736C2semidirect
16. #33731C2semidirect
17. #33731C2semidirect
18. #33726C2semidirect
19. #33479C2semidirect

Maximal subgroups

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

subgroupindex
#624742
#624732
#339552
#339182
#339172
#339152
#337972
#337952
#337942
#337932
#337512
#337362
#337312
#337262
#334792