Finite Group Databasebeta

← #206429

Subgroups — #206429

Order 512.

931 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic, complemented
4#20462C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#17489C211yesmaximal, characteristic
7#17486C211yesmaximal, characteristic, complemented
8#9854C211yesmaximal, characteristic, complemented
9#9853C211yesmaximal, characteristic, complemented
10#9851C211yesmaximal, characteristic, complemented
11#9850C211yesmaximal, characteristic, complemented
12#9848C211yesmaximal, characteristic, complemented
13#9846C211yesmaximal, characteristic, complemented
14#9845C211yesmaximal, characteristic, complemented
15#9843C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3248—22no—
19#3231C411yescharacteristic, complemented
20#3231C411yescharacteristic, complemented
21#3229C411yescharacteristic, complemented
22#3226C411yescharacteristic, complemented
23#3210C411yescharacteristic, complemented
24#3210C411yescharacteristic, complemented
25#3209C411yescharacteristic, complemented
26#3206C411yescharacteristic, complemented
27#3190—22no—
28#3170C2211yescharacteristic
29#3170C2211yescharacteristic
30#3167C2211yescharacteristic
31#3167C2211yescharacteristic
32#3108—22no—
33#3107—22no—
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#423C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1994C2211yescharacteristic
40#1994C2211yescharacteristic
41#1994C2211yescharacteristic
42#1994C2211yescharacteristic
43#1994C2211yescharacteristic
44#1994C2211yescharacteristic
45#1986—22no—
46#1962C2211yescharacteristic
47#1962C2211yescharacteristic
48#1871—22no—
49#1826—22no—
50#1806—22no—
51#1768—22no—
52#1750—22no—
53#1744C2211yescharacteristic
54#1744C2211yescharacteristic
55C4×#97C2211yescharacteristic
56C42×C8C2211yescharacteristic
57#1574—22no—
58#1573—22no—
59#1567C2211yescharacteristic
60#1567C2211yescharacteristic
61#1566C2211yescharacteristic
62#1566C2211yescharacteristic
63#1493C2211yescharacteristic
64#1493C2211yescharacteristic
65#1493C2211yescharacteristic
66#1493C2211yescharacteristic
67#1493C2211yescharacteristic
68#1493C2211yescharacteristic
69#1492C2211yescharacteristic
70#1492C2211yescharacteristic
71#1488—22no—
72#1487—22no—
73#1482—22no—
74#1481—22no—
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—
112C2×#117—21no—
113C22×C42C2311yescharacteristic
114#500C2×C411yescharacteristic
115#500C2×C411yescharacteristic
116#498C2×C411yescharacteristic
117#498C2×C411yescharacteristic
118#494C2×C411yescharacteristic
119#494C2×C411yescharacteristic
120#492C2×C411yescharacteristic
121#492C2×C411yescharacteristic
122#487C2×C411yescharacteristic
123#487C2×C411yescharacteristic
124#487C2×C411yescharacteristic
125#487C2×C411yescharacteristic
126#487C2×C411yescharacteristic
127#487C2×C411yescharacteristic
128#486C2×C411yescharacteristic
129#486C2×C411yescharacteristic
130#485—22no—
131#485—22no—
132#483—22no—
133#480—22no—
134#479—22no—
135#479—22no—
136#475—22no—
137#474—22no—
138#467—22no—
139#467—22no—
140#465—22no—
141#465—22no—
142#464—22no—
143#464—22no—
144#462—22no—
145#461—22no—
146#452—22no—
147#452—22no—
148#451—22no—
149#448—22no—
150C4×Q16—22no—
151C4×Q16—22no—
152C4×SD16—22no—
153C4×SD16—22no—
154#432—21no—
155#432—21no—
156#431—21no—
157#431—21no—
158#431—21no—
159#431—21no—
160#431C2×C411yescharacteristic
161#431C2×C411yescharacteristic
162#431C2×C411yescharacteristic
163#431C2×C411yescharacteristic
164#431D412yes—
165#431D412yes—
166#431D412yes—
167#431D412yes—
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—
179C2×#104—22no—
180C2×#104—22no—
181#407—21no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8D412yes—
184C4×M16—21no—
185C2×#97—22no—
186C2×C4×C8—22no—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C4×C4⋊C4—21no—
192C4×C4⋊C4D412yes—
193C4×C22⋊C4—21no—
194C4×C22⋊C4D412yes—
195#376—21no—
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198#339—22no—
199#339—22no—
200#339—22no—
201#339—22no—
202#339—22no—
203#339—22no—
204#336—22no—
205#336—22no—
206#335—22no—
207#335—22no—
208#334—22no—
209#334—22no—
210Q8⋊C8—22no—
211Q8⋊C8—22no—
212Q8⋊C8—22no—
213Q8⋊C8—22no—
214Q8⋊C8—22no—
215Q8⋊C8—22no—
216D4⋊C8—22no—
217D4⋊C8—22no—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4#4111yescharacteristic
226C2×Q16—22no—
227C2×Q16—22no—
228C2×SD16—22no—
229C2×SD16—22no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—41no—
235C2×M16—82no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—41no—
239C22×C8C2×D412yes—
240C22×C8C2×D412yes—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C22×C411yescharacteristic
248C4⋊Q8C22×C411yescharacteristic
249C4⋊Q8C22×C411yescharacteristic
250C4⋊Q8C22×C411yescharacteristic
251C4⋊Q8C22×C411yescharacteristic
252C4⋊Q8C22×C411yescharacteristic
253C4⋊Q8C22×C411yescharacteristic
254#127C22×C411yescharacteristic
255#126—41no—
256#126—41no—
257#125—21no—
258#125—21no—
259#125—21no—
260#125—21no—
261#125—21no—
262#125—21no—
263#125—21no—
264#125—41no—
265#125—41no—
266#124—21no—
267#124—21no—
268#124—21no—
269#123—41no—
270#123—41no—
271#123—41no—
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—21no—
285C22⋊Q8—21no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288C22⋊Q8—41no—
289C22⋊Q8—41no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296#120—41no—
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—21no—
310C4×Q8—21no—
311C4×Q8—41no—
312C4×Q8—41no—
313C4×Q8—41no—
314C4×Q8—41no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—41no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4—41no—
326C4×D4—41no—
327#117—21no—
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—44no—
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—22no—
345C2×C4⋊C4—41no—
346C2×C4⋊C4—41no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—41no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—41no—
354C2×C42—41no—
355C2×C42C22×C411yescharacteristic
356C2×C42C22×C411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42#4111yescharacteristic
366C2×C42#4111yescharacteristic
367C2×C42#4111yescharacteristic
368C2×C42C2411yescharacteristic
369#107—22no—
370#107—22no—
371#106—22no—
372#106—22no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—41no—
384C4⋊C8—41no—
385C4⋊C8C22⋊C412yes—
386C4⋊C8C22⋊C412yes—
387C4⋊C8C22⋊C412yes—
388C4⋊C8C22⋊C412yes—
389#104—82no—
390#104—82no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396#103—22no—
397#102—22no—
398#102—22no—
399C22⋊C8—22no—
400C22⋊C8—22no—
401C22⋊C8—41no—
402C22⋊C8—41no—
403C22⋊C8C22⋊C412yes—
404C22⋊C8C22⋊C412yes—
405C22⋊C8C22⋊C412yes—
406C22⋊C8C22⋊C412yes—
407#97—44no—
408#97—41no—
409C4×C8—21no—
410C4×C8—21no—
411C4×C8—44no—
412C4×C8—41no—
413C4×C8C22×C411yescharacteristic
414C4×C8C22×C411yescharacteristic
415C4×C8C22×C411yescharacteristic
416C4×C8C22×C411yescharacteristic
417C4×C8C2×D412yes—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C4×C8C2×D412yes—
421#95—22no—
422#95—22no—
423C24#12111yescharacteristic
424#41—42no—
425#41—42no—
426#41—82no—
427#41—82no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—41no—
436C2×Q8—41no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D411yescharacteristic
460C22×C4C4×D411yescharacteristic
461C22×C4C4×D411yescharacteristic
462C22×C4#12111yescharacteristic
463C22×C4#12111yescharacteristic
464C22×C4#12111yescharacteristic
465C22×C4#12111yescharacteristic
466C22×C4#12111yescharacteristic
467C22×C4#12411yescharacteristic
468C22×C4#12411yescharacteristic
469C22×C4#12711yescharacteristic
470C22×C4#12711yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4111yescharacteristic
474Q16—82no—
475Q16—82no—
476SD16—82no—
477SD16—82no—
478M16—42no—
479M16—42no—
480M16—42no—
481M16—42no—
482M16—82no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—42no—
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—44no—
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—
530C4⋊C4—41no—
531C22⋊C4—44no—
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—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42C2×C22⋊C411yescharacteristic
561C42C2×C22⋊C411yescharacteristic
562C42C4×D411yescharacteristic
563C42C4×D411yescharacteristic
564C42C4×D411yescharacteristic
565C42C4×D411yescharacteristic
566C42#12111yescharacteristic
567C42#12111yescharacteristic
568C42#12411yescharacteristic
569C42#12411yescharacteristic
570C42#12711yescharacteristic
571C42#12711yescharacteristic
572C42C23×C411yescharacteristic
573C42C22×D411yescharacteristic
574C42C22×D411yescharacteristic
575C42C2×#4111yescharacteristic
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23#39011yescharacteristic
581C23#39011yescharacteristic
582C23C2×#12111yescharacteristic
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—82no—
591Q8—82no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—82no—
596D4—82no—
597D4—82no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4#39011yescharacteristic
637C2×C4#39011yescharacteristic
638C2×C4#39011yescharacteristic
639C2×C4#39011yescharacteristic
640C2×C4#39011yescharacteristic
641C2×C4#39011yescharacteristic
642C2×C4C22×C22⋊C411yescharacteristic
643C2×C4C2×C4×D411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×#12111yescharacteristic
646C2×C4C2×#12411yescharacteristic
647C2×C4C2×#12711yescharacteristic
648C2×C4#57712yes—
649C2×C4#57812yes—
650C8—22no—
651C8—22no—
652C8—22no—
653C8—22no—
654C8—42no—
655C8—42no—
656C8—81no—
657C8—81no—
658C8—81no—
659C8—81no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—82no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22C2×#39011yescharacteristic
673C22#306312yes—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—44no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4#290112yes—
692C4#294712yes—
693C4#295712yes—
694C4#315411yescharacteristic
695C4#315611yescharacteristic
696C2—22no—
697C2—41no—
698C2—41no—
699C2—161no—
700C2#2121011yescharacteristic
701C2#2123411yescharacteristic
702C2#2124011yescharacteristic
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
3. #21844C2semidirect
4. #20462C2semidirect
5. #17817C2semidirect
7. #17486C2semidirect
8. #9854C2semidirect
9. #9853C2semidirect
10. #9851C2semidirect
11. #9850C2semidirect
12. #9848C2semidirect
13. #9846C2semidirect
14. #9845C2semidirect
15. #9843C2semidirect
19. #3231C4semidirect
20. #3231C4semidirect
21. #3229C4semidirect
22. #3226C4semidirect
23. #3210C4semidirect
24. #3210C4semidirect
25. #3209C4semidirect
26. #3206C4semidirect

Maximal subgroups

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

subgroupindex
#609472
#218562
#218442
#204622
#178172
#174892
#174862
#98542
#98532
#98512
#98502
#98482
#98462
#98452
#98432