Finite Group Databasebeta

← #206476

Subgroups — #206476

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17504C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic, complemented
5#17489C211yesmaximal, characteristic
6#17486C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9863C211yesmaximal, characteristic, complemented
9#9861C211yesmaximal, characteristic, complemented
10#9859C211yesmaximal, characteristic, complemented
11#9859C211yesmaximal, characteristic, complemented
12#9858C211yesmaximal, characteristic, complemented
13#9858C211yesmaximal, characteristic, complemented
14#9857C211yesmaximal, characteristic, complemented
15#9855C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3233—22no—
19#3232—22no—
20#3231C411yescharacteristic
21#3231C411yescharacteristic
22#3230C411yescharacteristic
23#3229C411yescharacteristic
24#3228C411yescharacteristic
25#3228C411yescharacteristic
26#3227C411yescharacteristic
27#3226C411yescharacteristic
28#3160C2211yescharacteristic
29#3160C2211yescharacteristic
30#3131C2211yescharacteristic
31#3131C2211yescharacteristic
32#3091—22no—
33#3076—22no—
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#423C2211yescharacteristic
38C2×C4×M16C2211yescharacteristic
39#1950—22no—
40#1855—22no—
41#1848C2211yescharacteristic
42#1848C2211yescharacteristic
43#1837C2211yescharacteristic
44#1837C2211yescharacteristic
45#1493C2211yescharacteristic
46#1493C2211yescharacteristic
47#1493C2211yescharacteristic
48#1493C2211yescharacteristic
49#1493C2211yescharacteristic
50#1493C2211yescharacteristic
51#1493C2211yescharacteristic
52#1493C2211yescharacteristic
53#1493C2211yescharacteristic
54#1493C2211yescharacteristic
55#1493C2211yescharacteristic
56#1493C2211yescharacteristic
57#1492C2211yescharacteristic
58#1492C2211yescharacteristic
59#1492C2211yescharacteristic
60#1492C2211yescharacteristic
61#1491—22no—
62#1489—22no—
63#1488—22no—
64#1487—22no—
65#1485—22no—
66#1483—22no—
67#1482—22no—
68#1481—22no—
69#1453—22no—
70#1453—22no—
71C2×#322C2211yescharacteristic
72C2×#322C2211yescharacteristic
73C2×#322C2211yescharacteristic
74C2×C82C2211yescharacteristic
75C22×M16—21no—
76C22×M16—21no—
77#533—21no—
78#533—21no—
79#533—21no—
80#533—21no—
81#533—21no—
82#533—21no—
83#533C2×C411yescharacteristic
84#533C2×C411yescharacteristic
85#533C2×C411yescharacteristic
86#533C2×C411yescharacteristic
87#533C2311yescharacteristic
88#533C2311yescharacteristic
89#533C2311yescharacteristic
90#532C2×C411yescharacteristic
91#532C2×C411yescharacteristic
92#532C2311yescharacteristic
93C2×C4⋊Q8C2×C411yescharacteristic
94C2×C4⋊Q8C2×C411yescharacteristic
95#529—21no—
96#529—21no—
97#529—21no—
98#529—21no—
99#529—21no—
100#529—21no—
101C2×#125—21no—
102#525—21no—
103#525—21no—
104#525—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C4×#41—21no—
108C4×#41—21no—
109C4×#41—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×#117—21no—
113C22×C42C2311yescharacteristic
114#500C2×C411yescharacteristic
115#500C2×C411yescharacteristic
116#500C2×C411yescharacteristic
117#500C2×C411yescharacteristic
118#499—22no—
119#499—22no—
120#498C2×C411yescharacteristic
121#498C2×C411yescharacteristic
122#498C2×C411yescharacteristic
123#498C2×C411yescharacteristic
124#488—22no—
125#488—22no—
126#487C2×C411yescharacteristic
127#487C2×C411yescharacteristic
128#487C2×C411yescharacteristic
129#487C2×C411yescharacteristic
130#486C2×C411yescharacteristic
131#486C2×C411yescharacteristic
132#486C2×C411yescharacteristic
133#486C2×C411yescharacteristic
134#485—22no—
135#485—22no—
136#484—22no—
137#483—22no—
138#482—22no—
139#482—22no—
140#481—22no—
141#480—22no—
142#479—22no—
143#479—22no—
144#478—22no—
145#478—22no—
146#477—22no—
147#476—22no—
148#475—22no—
149#474—22no—
150#432—21no—
151#432—21no—
152#431—21no—
153#431—21no—
154#431—21no—
155#431—21no—
156#431C2×C411yescharacteristic
157#431C2×C411yescharacteristic
158#431C2×C411yescharacteristic
159#431C2×C411yescharacteristic
160#431D412yes—
161#431D412yes—
162#431D412yes—
163#431D412yes—
164C2×#108—22no—
165C2×#108—22no—
166#427—22no—
167#427—22no—
168#423—21no—
169#423—21no—
170#423—21no—
171#423C2311yescharacteristic
172#423C2311yescharacteristic
173#423C2311yescharacteristic
174#423C2311yescharacteristic
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8—21no—
178C2×C4⋊C8D412yes—
179#416—22no—
180#416—22no—
181#407—21no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8D412yes—
184C4×M16—21no—
185C4×M16—21no—
186C4×M16—21no—
187C4×M16D412yes—
188C4×M16D412yes—
189C2×#97—21no—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#341—22no—
197#341—22no—
198Q8⋊C8—22no—
199Q8⋊C8—22no—
200Q8⋊C8—22no—
201Q8⋊C8—22no—
202Q8⋊C8—22no—
203Q8⋊C8—22no—
204Q8⋊C8—22no—
205Q8⋊C8—22no—
206Q8⋊C8—22no—
207Q8⋊C8—22no—
208Q8⋊C8—22no—
209Q8⋊C8—22no—
210D4⋊C8—22no—
211D4⋊C8—22no—
212D4⋊C8—22no—
213D4⋊C8—22no—
214#322—22no—
215#322—22no—
216#322—22no—
217C82—22no—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C23×C4C2×D411yescharacteristic
224C23×C4#4111yescharacteristic
225C23×C4#4111yescharacteristic
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—44no—
233C2×M16—41no—
234C2×M16—82no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—41no—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8C22×C411yescharacteristic
249C4⋊Q8C22×C411yescharacteristic
250C4⋊Q8C22×C411yescharacteristic
251C4⋊Q8C22×C411yescharacteristic
252C4⋊Q8C22×C411yescharacteristic
253C4⋊Q8C22×C411yescharacteristic
254C4⋊Q8C22×C411yescharacteristic
255#127C22×C411yescharacteristic
256#126—41no—
257#126—41no—
258#125—21no—
259#125—21no—
260#125—21no—
261#125—21no—
262#125—21no—
263#125—21no—
264#125—21no—
265#125—41no—
266#125—41no—
267#124—21no—
268#124—21no—
269#124—21no—
270#123—41no—
271#123—41no—
272#123—41no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8—21no—
287C22⋊Q8—41no—
288C22⋊Q8—41no—
289C22⋊Q8—41no—
290C22⋊Q8—41no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296#121—21no—
297#120—41no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—41no—
313C4×Q8—41no—
314C4×Q8—41no—
315C4×Q8—41no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4—41no—
326C4×D4—41no—
327C4×D4—41no—
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—21no—
335#117—41no—
336#117—41no—
337#117—41no—
338#117—41no—
339#117—41no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—41no—
345C2×C4⋊C4—41no—
346C2×C22⋊C4—41no—
347C2×C42—41no—
348C2×C42—41no—
349C2×C42C22×C411yescharacteristic
350C2×C42C22×C411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4111yescharacteristic
359C2×C42#4111yescharacteristic
360C2×C42C2411yescharacteristic
361#108—82no—
362#108—82no—
363#107—22no—
364#107—22no—
365#107—22no—
366#107—22no—
367#106—22no—
368#106—22no—
369#106—22no—
370#106—22no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—41no—
382C4⋊C8—41no—
383C4⋊C8C22⋊C412yes—
384C4⋊C8C22⋊C412yes—
385C4⋊C8C22⋊C412yes—
386C4⋊C8C22⋊C412yes—
387#103—22no—
388#103—22no—
389#103—22no—
390#103—22no—
391#102—22no—
392#102—22no—
393#102—22no—
394#102—22no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8—41no—
398C22⋊C8—41no—
399C22⋊C8C22⋊C412yes—
400C22⋊C8C22⋊C412yes—
401C22⋊C8C22⋊C412yes—
402C22⋊C8C22⋊C412yes—
403#97—22no—
404#97—22no—
405#97—41no—
406C4×C8—21no—
407C4×C8—21no—
408C4×C8—21no—
409C4×C8—21no—
410C4×C8—41no—
411C4×C8C22×C411yescharacteristic
412C4×C8C22×C411yescharacteristic
413C4×C8C22×C411yescharacteristic
414C4×C8C22×C411yescharacteristic
415C4×C8C2×D412yes—
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C4×C8C2×D412yes—
421C24#12411yescharacteristic
422#41—42no—
423#41—42no—
424#41—82no—
425#41—82no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C4×D411yescharacteristic
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D411yescharacteristic
458C22×C4#12111yescharacteristic
459C22×C4#12111yescharacteristic
460C22×C4#12111yescharacteristic
461C22×C4#12111yescharacteristic
462C22×C4#12411yescharacteristic
463C22×C4#12411yescharacteristic
464C22×C4#12411yescharacteristic
465C22×C4#12711yescharacteristic
466C22×C4#12711yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470M16—44no—
471M16—44no—
472M16—42no—
473M16—42no—
474M16—42no—
475M16—42no—
476M16—82no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—82no—
494C2×C8C2×C22⋊C412yes—
495C2×C8C2×C22⋊C412yes—
496C2×C8C2×C22⋊C412yes—
497C2×C8C2×C22⋊C412yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
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—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42C2×C22⋊C411yescharacteristic
551C42C2×C22⋊C411yescharacteristic
552C42C4×D411yescharacteristic
553C42C4×D411yescharacteristic
554C42C4×D411yescharacteristic
555C42C4×D411yescharacteristic
556C42#12111yescharacteristic
557C42#12111yescharacteristic
558C42#12111yescharacteristic
559C42#12111yescharacteristic
560C42#12711yescharacteristic
561C42#12711yescharacteristic
562C42C23×C411yescharacteristic
563C42C22×D411yescharacteristic
564C42C22×D411yescharacteristic
565C42C22×D411yescharacteristic
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#39011yescharacteristic
571C23#39011yescharacteristic
572C23C2×#12411yescharacteristic
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—82no—
581Q8—82no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—82no—
586D4—82no—
587D4—82no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
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—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4#39011yescharacteristic
621C2×C4#39011yescharacteristic
622C2×C4#39011yescharacteristic
623C2×C4#39011yescharacteristic
624C2×C4#39011yescharacteristic
625C2×C4#39011yescharacteristic
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×#12711yescharacteristic
632C2×C4#57712yes—
633C2×C4#57812yes—
634C8—24no—
635C8—22no—
636C8—22no—
637C8—22no—
638C8—22no—
639C8—42no—
640C8—42no—
641C8—42no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—82no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22C2×#39011yescharacteristic
659C22#313712yes—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C4#289812yes—
676C4#295612yes—
677C4#295712yes—
678C4#315411yescharacteristic
679C4#315611yescharacteristic
680C2—22no—
681C2—41no—
682C2—41no—
683C2—161no—
684C2#2120511yescharacteristic
685C2#2124011yescharacteristic
686C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21993C2semidirect
4. #17501C2semidirect
6. #17486C2semidirect
7. C2×#1453C2semidirect
8. #9863C2semidirect
9. #9861C2semidirect
10. #9859C2semidirect
11. #9859C2semidirect
12. #9858C2semidirect
13. #9858C2semidirect
14. #9857C2semidirect
15. #9855C2semidirect

Maximal subgroups

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

subgroupindex
#608462
#219932
#175042
#175012
#174892
#174862
C2×#14532
#98632
#98612
#98592
#98582
#98572
#98552