Finite Group Databasebeta

← #206474

Subgroups — #206474

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17489C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic, complemented
5#17481C211yesmaximal, characteristic
6#17479C211yesmaximal, characteristic, complemented
7C2×#1454C211yesmaximal, 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×#431C2211yescharacteristic
38C2×#423C2211yescharacteristic
39#1950—22no—
40#1855—22no—
41#1848C2211yescharacteristic
42#1848C2211yescharacteristic
43#1837C2211yescharacteristic
44#1837C2211yescharacteristic
45#1497—22no—
46#1495—22no—
47#1493C2211yescharacteristic
48#1493C2211yescharacteristic
49#1493C2211yescharacteristic
50#1493C2211yescharacteristic
51#1493C2211yescharacteristic
52#1493C2211yescharacteristic
53#1493C2211yescharacteristic
54#1493C2211yescharacteristic
55#1493C2211yescharacteristic
56#1493C2211yescharacteristic
57#1493C2211yescharacteristic
58#1493C2211yescharacteristic
59#1492C2211yescharacteristic
60#1492C2211yescharacteristic
61#1492C2211yescharacteristic
62#1492C2211yescharacteristic
63#1488—22no—
64#1487—22no—
65#1482—22no—
66#1481—22no—
67#1479—22no—
68#1477—22no—
69#1454—22no—
70#1454—22no—
71C2×#322C2211yescharacteristic
72C2×#322C2211yescharacteristic
73C2×#322C2211yescharacteristic
74C2×C82C2211yescharacteristic
75C22×M16—21no—
76#533—21no—
77#533—21no—
78#533—21no—
79#533—21no—
80#533—21no—
81#533—21no—
82#533C2×C411yescharacteristic
83#533C2×C411yescharacteristic
84#533C2×C411yescharacteristic
85#533C2×C411yescharacteristic
86#533C2311yescharacteristic
87#533C2311yescharacteristic
88#533C2311yescharacteristic
89#532C2×C411yescharacteristic
90#532C2×C411yescharacteristic
91#532C2311yescharacteristic
92C2×C4⋊Q8C2×C411yescharacteristic
93C2×C4⋊Q8C2×C411yescharacteristic
94#529—21no—
95#529—21no—
96#529—21no—
97#529—21no—
98#529—21no—
99#529—21no—
100C2×#125—21no—
101#525—21no—
102#525—21no—
103#525—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C4×#41—21no—
107C4×#41—21no—
108C4×#41—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×#117—21no—
112C22×C42C2311yescharacteristic
113#500C2×C411yescharacteristic
114#500C2×C411yescharacteristic
115#500C2×C411yescharacteristic
116#500C2×C411yescharacteristic
117#499—22no—
118#499—22no—
119#498C2×C411yescharacteristic
120#498C2×C411yescharacteristic
121#498C2×C411yescharacteristic
122#498C2×C411yescharacteristic
123#488—22no—
124#488—22no—
125#487C2×C411yescharacteristic
126#487C2×C411yescharacteristic
127#487C2×C411yescharacteristic
128#487C2×C411yescharacteristic
129#486C2×C411yescharacteristic
130#486C2×C411yescharacteristic
131#486C2×C411yescharacteristic
132#486C2×C411yescharacteristic
133#485—22no—
134#485—22no—
135#484—22no—
136#483—22no—
137#482—22no—
138#482—22no—
139#481—22no—
140#480—22no—
141#479—22no—
142#479—22no—
143#478—22no—
144#478—22no—
145#477—22no—
146#476—22no—
147#475—22no—
148#474—22no—
149#432—21no—
150#432—21no—
151#431—21no—
152#431—21no—
153#431—21no—
154#431—21no—
155#431—21no—
156#431—21no—
157#431C2×C411yescharacteristic
158#431C2×C411yescharacteristic
159#431C2×C411yescharacteristic
160#431C2×C411yescharacteristic
161#431D412yes—
162#431D412yes—
163#431D412yes—
164#431D412yes—
165#431D412yes—
166#431D412yes—
167C2×#108—22no—
168C2×#108—22no—
169#427—22no—
170#427—22no—
171#423—21no—
172#423—21no—
173#423—21no—
174#423C2311yescharacteristic
175#423C2311yescharacteristic
176#423C2311yescharacteristic
177#423C2311yescharacteristic
178C2×C4⋊C8—21no—
179C2×C4⋊C8—21no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8—21no—
182C2×C4⋊C8D412yes—
183#416—22no—
184#416—22no—
185#407—21no—
186C2×C22⋊C8—21no—
187C2×C22⋊C8—21no—
188C2×C22⋊C8D412yes—
189C4×M16—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—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—41no—
231C2×M16—82no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—41no—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C22×C411yescharacteristic
246C4⋊Q8C22×C411yescharacteristic
247C4⋊Q8C22×C411yescharacteristic
248C4⋊Q8C22×C411yescharacteristic
249C4⋊Q8C22×C411yescharacteristic
250C4⋊Q8C22×C411yescharacteristic
251C4⋊Q8C22×C411yescharacteristic
252#127C22×C411yescharacteristic
253#126—41no—
254#126—41no—
255#125—21no—
256#125—21no—
257#125—21no—
258#125—21no—
259#125—21no—
260#125—21no—
261#125—21no—
262#125—41no—
263#125—41no—
264#124—21no—
265#124—21no—
266#124—21no—
267#123—41no—
268#123—41no—
269#123—41no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
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—41no—
285C22⋊Q8—41no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288#121—21no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#120—41no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
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—41no—
310C4×Q8—41no—
311C4×Q8—41no—
312C4×Q8—41no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—41no—
320C4×D4—41no—
321C4×D4—41no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325#117—21no—
326#117—21no—
327#117—21no—
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—41no—
333#117—41no—
334#117—41no—
335#117—41no—
336#117—41no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342C2×C4⋊C4—41no—
343C2×C22⋊C4—41no—
344C2×C42—41no—
345C2×C42—41no—
346C2×C42C22×C411yescharacteristic
347C2×C42C22×C411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42#4111yescharacteristic
356C2×C42#4111yescharacteristic
357C2×C42C2411yescharacteristic
358#108—82no—
359#108—82no—
360#107—22no—
361#107—22no—
362#107—22no—
363#107—22no—
364#106—22no—
365#106—22no—
366#106—22no—
367#106—22no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—41no—
381C4⋊C8—41no—
382C4⋊C8C22⋊C412yes—
383C4⋊C8C22⋊C412yes—
384C4⋊C8C22⋊C412yes—
385C4⋊C8C22⋊C412yes—
386#103—22no—
387#103—22no—
388#103—22no—
389#103—22no—
390#102—22no—
391#102—22no—
392#102—22no—
393#102—22no—
394C22⋊C8—22no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—41no—
399C22⋊C8—41no—
400C22⋊C8C22⋊C412yes—
401C22⋊C8C22⋊C412yes—
402C22⋊C8C22⋊C412yes—
403C22⋊C8C22⋊C412yes—
404#97—41no—
405C4×C8—21no—
406C4×C8—21no—
407C4×C8—21no—
408C4×C8—21no—
409C4×C8—41no—
410C4×C8C22×C411yescharacteristic
411C4×C8C22×C411yescharacteristic
412C4×C8C22×C411yescharacteristic
413C4×C8C22×C411yescharacteristic
414C4×C8C2×D412yes—
415C4×C8C2×D412yes—
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C24#12411yescharacteristic
421#41—44no—
422#41—42no—
423#41—42no—
424#41—82no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—44no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—82no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C4×D411yescharacteristic
454C22×C4C4×D411yescharacteristic
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4#12111yescharacteristic
458C22×C4#12111yescharacteristic
459C22×C4#12111yescharacteristic
460C22×C4#12111yescharacteristic
461C22×C4#12411yescharacteristic
462C22×C4#12411yescharacteristic
463C22×C4#12411yescharacteristic
464C22×C4#12711yescharacteristic
465C22×C4#12711yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469M16—42no—
470M16—42no—
471M16—42no—
472M16—42no—
473M16—82no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—82no—
491C2×C8C2×C22⋊C412yes—
492C2×C8C2×C22⋊C412yes—
493C2×C8C2×C22⋊C412yes—
494C2×C8C2×C22⋊C412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
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—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
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—
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—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C4×D411yescharacteristic
550C42C4×D411yescharacteristic
551C42C4×D411yescharacteristic
552C42C4×D411yescharacteristic
553C42#12111yescharacteristic
554C42#12111yescharacteristic
555C42#12111yescharacteristic
556C42#12111yescharacteristic
557C42#12711yescharacteristic
558C42#12711yescharacteristic
559C42C23×C411yescharacteristic
560C42C22×D411yescharacteristic
561C42C22×D411yescharacteristic
562C42C22×D411yescharacteristic
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23#39011yescharacteristic
568C23#39011yescharacteristic
569C23C2×#12411yescharacteristic
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—82no—
578Q8—82no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—82no—
583D4—82no—
584D4—82no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
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—41no—
610C2×C4—41no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4#39011yescharacteristic
618C2×C4#39011yescharacteristic
619C2×C4#39011yescharacteristic
620C2×C4#39011yescharacteristic
621C2×C4#39011yescharacteristic
622C2×C4#39011yescharacteristic
623C2×C4C22×C22⋊C411yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12711yescharacteristic
629C2×C4#57712yes—
630C2×C4#57812yes—
631C8—22no—
632C8—22no—
633C8—22no—
634C8—22no—
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—81no—
640C8—81no—
641C8—81no—
642C8—81no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—82no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×#39011yescharacteristic
656C22#313712yes—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—81no—
672C4#289812yes—
673C4#295512yes—
674C4#295712yes—
675C4#315411yescharacteristic
676C4#315611yescharacteristic
677C2—22no—
678C2—41no—
679C2—41no—
680C2—161no—
681C2#2120511yescharacteristic
682C2#2124011yescharacteristic
683C2#2124211yescharacteristic
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. #17486C2semidirect
6. #17479C2semidirect
7. C2×#1454C2semidirect
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
#174892
#174862
#174812
#174792
C2×#14542
#98632
#98612
#98592
#98582
#98572
#98552