Finite Group Databasebeta

← #206475

Subgroups — #206475

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17489C211yesmaximal, characteristic, complemented
4#17485C211yesmaximal, characteristic, complemented
5#17481C211yesmaximal, characteristic, complemented
6#17478C211yesmaximal, characteristic, complemented
7C2×#1454C211yesmaximal, characteristic, complemented
8#9863C211yesmaximal, characteristic, complemented
9#9862C211yesmaximal, characteristic, complemented
10#9860C211yesmaximal, characteristic, complemented
11#9860C211yesmaximal, characteristic, complemented
12#9858C211yesmaximal, characteristic, complemented
13#9858C211yesmaximal, characteristic, complemented
14#9857C211yesmaximal, characteristic, complemented
15#9856C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3212—22no—
19#3211—22no—
20#3210C411yescharacteristic
21#3210C411yescharacteristic
22#3209C411yescharacteristic
23#3208C411yescharacteristic
24#3207C411yescharacteristic
25#3207C411yescharacteristic
26#3206C411yescharacteristic
27#3205C411yescharacteristic
28#3147C2211yescharacteristic
29#3147C2211yescharacteristic
30#3131C2211yescharacteristic
31#3131C2211yescharacteristic
32#3035—22no—
33#3002—22no—
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#431C2211yescharacteristic
38C2×#423C2211yescharacteristic
39#1950—22no—
40#1855—22no—
41#1848C2211yescharacteristic, complemented
42#1848C2211yescharacteristic, complemented
43#1837C2211yescharacteristic, complemented
44#1837C2211yescharacteristic, complemented
45#1496—22no—
46#1495—22no—
47#1493C2211yescharacteristic, complemented
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1493C2211yescharacteristic, complemented
51#1492C2211yescharacteristic
52#1492C2211yescharacteristic
53#1492C2211yescharacteristic
54#1492C2211yescharacteristic
55#1492C2211yescharacteristic
56#1492C2211yescharacteristic
57#1492C2211yescharacteristic
58#1492C2211yescharacteristic
59#1492C2211yescharacteristic, complemented
60#1492C2211yescharacteristic, complemented
61#1492C2211yescharacteristic, complemented
62#1492C2211yescharacteristic, complemented
63#1488—22no—
64#1486—22no—
65#1482—22no—
66#1480—22no—
67#1478—22no—
68#1477—22no—
69#1454—22no—
70#1454—22no—
71C2×#322C2211yescharacteristic, complemented
72C2×#322C2211yescharacteristic, complemented
73C2×#322C2211yescharacteristic, complemented
74C2×C82C2211yescharacteristic, complemented
75C22×#41—21no—
76C2×#135—22no—
77C2×#135—22no—
78C22×M16—21no—
79#533C2×C411yescharacteristic
80#533C2×C411yescharacteristic
81#533C2311yescharacteristic
82#532—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532C2×C411yescharacteristic
89#532C2×C411yescharacteristic
90#532C2×C411yescharacteristic
91#532C2×C411yescharacteristic
92#532C2311yescharacteristic
93#532C2311yescharacteristic
94#532C2311yescharacteristic
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×#127C2×C411yescharacteristic
97#529—21no—
98#529—21no—
99#529—21no—
100#529—21no—
101#529—21no—
102#529—21no—
103C2×#124—21no—
104#525—21no—
105#525—21no—
106#525—21no—
107C2×#121—21no—
108C2×#121—21no—
109C4×#41—21no—
110C4×#41—21no—
111C4×#41—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C22×C42C2311yescharacteristic
115#495—22no—
116#495—22no—
117#494C2×C411yescharacteristic
118#494C2×C411yescharacteristic
119#493C2×C411yescharacteristic
120#493C2×C411yescharacteristic
121#492C2×C411yescharacteristic
122#492C2×C411yescharacteristic
123#492C2×C411yescharacteristic
124#492C2×C411yescharacteristic
125#488—22no—
126#488—22no—
127#487C2×C411yescharacteristic
128#487C2×C411yescharacteristic
129#487C2×C411yescharacteristic
130#487C2×C411yescharacteristic
131#486C2×C411yescharacteristic
132#486C2×C411yescharacteristic
133#486C2×C411yescharacteristic
134#486C2×C411yescharacteristic
135#464—22no—
136#464—22no—
137#463—22no—
138#463—22no—
139#462—22no—
140#461—22no—
141#460—22no—
142#459—22no—
143#452—22no—
144#452—22no—
145#451—22no—
146#450—22no—
147#449—22no—
148#449—22no—
149#448—22no—
150#447—22no—
151#432—21no—
152#432—21no—
153#431—21no—
154#431—21no—
155#431—21no—
156#431—21no—
157#431—21no—
158#431—21no—
159#431C2×C411yescharacteristic
160#431C2×C411yescharacteristic
161#431C2×C411yescharacteristic
162#431C2×C411yescharacteristic
163#431D412yes—
164#431D412yes—
165#431D412yes—
166#431D412yes—
167#431D412yes—
168#431D412yes—
169C2×#108—22no—
170C2×#108—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—
202D4⋊C8—22no—
203D4⋊C8—22no—
204D4⋊C8—22no—
205D4⋊C8—22no—
206D4⋊C8—22no—
207D4⋊C8—22no—
208D4⋊C8—22no—
209D4⋊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—
221C2×#41—21no—
222C2×#41—21no—
223C2×#41—21no—
224C2×#41—21no—
225C2×#41—41no—
226C2×#41—41no—
227C2×#41—82no—
228C2×#41—82no—
229C22×Q8—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—41no—
234C23×C4—41no—
235C23×C4C2×D411yescharacteristic
236C23×C4#4111yescharacteristic
237C23×C4#4111yescharacteristic
238#135—82no—
239#135—82no—
240#135—82no—
241#135—82no—
242C2×Q16—22no—
243C2×Q16—22no—
244C2×SD16—22no—
245C2×SD16—22no—
246C2×SD16—22no—
247C2×SD16—22no—
248C2×D8—22no—
249C2×D8—22no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—41no—
255C2×M16—82no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—41no—
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22×C411yescharacteristic
267C4⋊Q8C22×C411yescharacteristic
268C4⋊Q8C22×C411yescharacteristic
269C4⋊Q8C22×C411yescharacteristic
270C4⋊Q8C22×C411yescharacteristic
271#127—21no—
272#127—21no—
273#127—21no—
274#127C22×C411yescharacteristic
275#127C22×C411yescharacteristic
276#127C22×C411yescharacteristic
277#126—41no—
278#126—41no—
279#125—21no—
280#125—21no—
281#125—21no—
282#124—21no—
283#124—21no—
284#124—21no—
285#124—21no—
286#124—21no—
287#124—21no—
288#124—21no—
289#124—41no—
290#124—41no—
291#123—41no—
292#123—41no—
293#123—41no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300#121—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121—21no—
306#121—21no—
307#121—21no—
308#121—21no—
309#121—21no—
310#121—21no—
311#121—21no—
312#121—21no—
313#121—21no—
314#121—41no—
315#121—41no—
316#121—41no—
317#121—41no—
318#120—41no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—41no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—41no—
345C4×D4—41no—
346C4×D4—41no—
347C4×D4—41no—
348C4×D4—41no—
349#117—21no—
350#117—21no—
351#117—21no—
352#117—41no—
353#117—41no—
354#117—41no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—41no—
359C2×C42—41no—
360C2×C42C22×C411yescharacteristic
361C2×C42C22×C411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4111yescharacteristic
370C2×C42#4111yescharacteristic
371C2×C42C2411yescharacteristic
372#108—82no—
373#108—82no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8—22no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—41no—
387C4⋊C8—41no—
388C4⋊C8C22⋊C412yes—
389C4⋊C8C22⋊C412yes—
390C4⋊C8C22⋊C412yes—
391C4⋊C8C22⋊C412yes—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396#102—22no—
397#102—22no—
398#102—22no—
399#102—22no—
400C22⋊C8—22no—
401C22⋊C8—22no—
402C22⋊C8—22no—
403C22⋊C8—22no—
404C22⋊C8—41no—
405C22⋊C8—41no—
406C22⋊C8C22⋊C412yes—
407C22⋊C8C22⋊C412yes—
408C22⋊C8C22⋊C412yes—
409C22⋊C8C22⋊C412yes—
410#97—41no—
411C4×C8—21no—
412C4×C8—21no—
413C4×C8—21no—
414C4×C8—21no—
415C4×C8—41no—
416C4×C8C22×C411yescharacteristic
417C4×C8C22×C411yescharacteristic
418C4×C8C22×C411yescharacteristic
419C4×C8C22×C411yescharacteristic
420C4×C8C2×D412yes—
421C4×C8C2×D412yes—
422C4×C8C2×D412yes—
423C4×C8C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2×D412yes—
426C24—41no—
427C24#12411yescharacteristic
428#41—44no—
429#41—42no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—82no—
436#41—82no—
437#41—82no—
438#41—84no—
439#41—82no—
440#41—82no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—41no—
447C2×Q8—82no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—44no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—82no—
469C2×D4—82no—
470C2×D4—84no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4C2×C22⋊C411yescharacteristic
486C22×C4C2×C22⋊C411yescharacteristic
487C22×C4C4×D411yescharacteristic
488C22×C4C4×D411yescharacteristic
489C22×C4C4×D411yescharacteristic
490C22×C4C4×D411yescharacteristic
491C22×C4#12111yescharacteristic
492C22×C4#12111yescharacteristic
493C22×C4#12111yescharacteristic
494C22×C4#12111yescharacteristic
495C22×C4#12411yescharacteristic
496C22×C4#12411yescharacteristic
497C22×C4#12411yescharacteristic
498C22×C4#12711yescharacteristic
499C22×C4#12711yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4111yescharacteristic
502C22×C4C2×#4111yescharacteristic
503Q16—82no—
504Q16—82no—
505SD16—82no—
506SD16—82no—
507SD16—82no—
508SD16—82no—
509D8—82no—
510D8—82no—
511M16—42no—
512M16—42no—
513M16—42no—
514M16—42no—
515M16—82no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—42no—
525C2×C8—42no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—82no—
533C2×C8C2×C22⋊C412yes—
534C2×C8C2×C22⋊C412yes—
535C2×C8C2×C22⋊C412yes—
536C2×C8C2×C22⋊C412yes—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42C2×C22⋊C411yescharacteristic
576C42C2×C22⋊C411yescharacteristic
577C42C4×D411yescharacteristic
578C42C4×D411yescharacteristic
579C42C4×D411yescharacteristic
580C42C4×D411yescharacteristic
581C42#12111yescharacteristic
582C42#12111yescharacteristic
583C42#12111yescharacteristic
584C42#12111yescharacteristic
585C42#12711yescharacteristic
586C42#12711yescharacteristic
587C42C23×C411yescharacteristic
588C42C22×D411yescharacteristic
589C42C22×D411yescharacteristic
590C42C22×D411yescharacteristic
591C23—22no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23#39011yescharacteristic
601C23#39011yescharacteristic
602C23C2×#12411yescharacteristic
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—82no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—82no—
619D4—82no—
620D4—82no—
621D4—82no—
622D4—82no—
623D4—82no—
624D4—82no—
625D4—82no—
626D4—82no—
627D4—82no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4#39011yescharacteristic
664C2×C4#39011yescharacteristic
665C2×C4#39011yescharacteristic
666C2×C4#39011yescharacteristic
667C2×C4#39011yescharacteristic
668C2×C4#39011yescharacteristic
669C2×C4C22×C22⋊C411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4C2×#12111yescharacteristic
674C2×C4C2×#12711yescharacteristic
675C2×C4#57612yes—
676C2×C4#57712yes—
677C8—22no—
678C8—22no—
679C8—22no—
680C8—22no—
681C8—42no—
682C8—42no—
683C8—42no—
684C8—42no—
685C8—81no—
686C8—81no—
687C8—81no—
688C8—81no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—82no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—164no—
708C22C2×#39011yescharacteristic
709C22#313612yes—
710C4—21no—
711C4—21no—
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—21no—
717C4—21no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4#289712yes—
724C4#295512yes—
725C4#295712yes—
726C4#315411yescharacteristic
727C4#315511yescharacteristic
728C2—22no—
729C2—41no—
730C2—41no—
731C2—161no—
732C2—161no—
733C2—161no—
734C2#2120611yescharacteristic
735C2#2124011yescharacteristic
736C2#2124211yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60782C2
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
3. #17489C2semidirect
4. #17485C2semidirect
5. #17481C2semidirect
6. #17478C2semidirect
7. C2×#1454C2semidirect
8. #9863C2semidirect
9. #9862C2semidirect
10. #9860C2semidirect
11. #9860C2semidirect
12. #9858C2semidirect
13. #9858C2semidirect
14. #9857C2semidirect
15. #9856C2semidirect
41. #1848C22semidirect
42. #1848C22semidirect
43. #1837C22semidirect
44. #1837C22semidirect
47. #1493C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
50. #1493C22semidirect
59. #1492C22semidirect
60. #1492C22semidirect
61. #1492C22semidirect
62. #1492C22semidirect
71. C2×#322C22semidirect
72. C2×#322C22semidirect
73. C2×#322C22semidirect
74. C2×C82C22semidirect

Maximal subgroups

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

subgroupindex
#607822
#219932
#174892
#174852
#174812
#174782
C2×#14542
#98632
#98622
#98602
#98582
#98572
#98562