Finite Group Databasebeta

← #483244

Subgroups — #483244

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62419C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#17834C211yesmaximal, characteristic, complemented
4#11980C211yesmaximal, characteristic, complemented
5#11979C211yesmaximal, characteristic, complemented
6#11978C211yesmaximal, characteristic, complemented
7#11960C211yesmaximal, characteristic, complemented
8#11959C211yesmaximal, characteristic, complemented
9#11957C211yesmaximal, characteristic, complemented
10#11946C211yesmaximal, characteristic, complemented
11#11945C211yesmaximal, characteristic, complemented
12#11944C211yesmaximal, characteristic, complemented
13#11926C211yesmaximal, characteristic, complemented
14#11925C211yesmaximal, characteristic, complemented
15C2×#1360C211yesmaximal, characteristic, complemented
16#3514—21no—
17#3396C411yescharacteristic
18#3394—21no—
19#3393C411yescharacteristic
20D4×Q16C411yescharacteristic
21#3285C411yescharacteristic
22#3285C411yescharacteristic
23#3282—21no—
24#3282—21no—
25D4×SD16—21no—
26#3280C411yescharacteristic
27#3280C411yescharacteristic
28D4×D8C411yescharacteristic
29#3212—21no—
30#3211—21no—
31#3211—21no—
32#3168C2211yescharacteristic
33#3168C2211yescharacteristic
34#3168C2211yescharacteristic
35#3147C2211yescharacteristic
36C2×#495C2211yescharacteristic
37C2×#495C2211yescharacteristic
38#2990—21no—
39C8×#41C2211yescharacteristic
40#2183—21no—
41#2169—21no—
42#2168—21no—
43#2168—21no—
44#2163C2211yescharacteristic, complemented
45#2163C2211yescharacteristic, complemented
46#2162C2211yescharacteristic, complemented
47#2150C2211yescharacteristic, complemented
48#2150C2211yescharacteristic, complemented
49C2×C4⋊C16C2211yescharacteristic, complemented
50#1574—21no—
51#1573—21no—
52#1573—21no—
53#1568C2211yescharacteristic, complemented
54#1568C2211yescharacteristic, complemented
55#1566C2211yescharacteristic, complemented
56#1564C2211yescharacteristic, complemented
57C2×#334C2211yescharacteristic, complemented
58C2×#334C2211yescharacteristic, complemented
59#1363C2211yescharacteristic, complemented
60#1363C2211yescharacteristic, complemented
61#1363C2211yescharacteristic, complemented
62#1363C2211yescharacteristic, complemented
63#1360C2211yescharacteristic
64#1360C2211yescharacteristic
65#1360C2211yescharacteristic
66#1360C2211yescharacteristic
67#1360C2211yescharacteristic, complemented
68#1360C2211yescharacteristic, complemented
69#1360C2211yescharacteristic, complemented
70#1360C2211yescharacteristic, complemented
71#1360C2211yescharacteristic, complemented
72#1360C2211yescharacteristic, complemented
73#1360C2211yescharacteristic, complemented
74#1360C2211yescharacteristic, complemented
75C2×#135—21no—
76C2×#135C2×C411yescharacteristic
77C2×#135C2×C411yescharacteristic
78C22×Q16C2×C411yescharacteristic
79C22×Q16C2×C411yescharacteristic
80C22×SD16—21no—
81C22×SD16—21no—
82C22×D8C2×C411yescharacteristic
83C22×D8C2×C411yescharacteristic
84#559—21no—
85#556—21no—
86#555—41no—
87#551—41no—
88#551—41no—
89D4×Q8—21no—
90#548—21no—
91#548—21no—
92#546—21no—
93#546—21no—
94#546—41no—
95D42—21no—
96#542—41no—
97#538—41no—
98#538—41no—
99#532—21no—
100#529—21no—
101#529—21no—
102#529—21no—
103C2×#124—21no—
104C2×#124—21no—
105#518—41no—
106C4×#41D411yescharacteristic
107C2×M32—21no—
108C2×M32—21no—
109C22×C16—21no—
110#499C2311yescharacteristic
111#495—21no—
112#495—21no—
113#495—21no—
114#495—21no—
115#495—21no—
116#495C2×C411yescharacteristic
117#495C2×C411yescharacteristic
118#495C2×C411yescharacteristic
119#495C2×C411yescharacteristic
120#495C2311yescharacteristic
121#495C2311yescharacteristic
122#495C2311yescharacteristic
123#494C2×C411yescharacteristic
124#493C2×C411yescharacteristic
125#492—21no—
126#488—21no—
127#488—21no—
128#488—21no—
129#467C2×C411yescharacteristic
130#467C2×C411yescharacteristic
131#467C2×C411yescharacteristic
132#466C2×C411yescharacteristic
133#466C2×C411yescharacteristic
134#466C2×C411yescharacteristic
135#465—21no—
136#465—21no—
137#465—21no—
138#464—21no—
139#464—21no—
140#464—21no—
141#464—21no—
142#463—21no—
143#463—21no—
144#463—21no—
145#463—21no—
146#462—21no—
147#461—21no—
148#460—21no—
149#459—21no—
150#452—21no—
151#452—21no—
152#451—21no—
153#451—21no—
154#450—21no—
155#450—21no—
156#449—21no—
157#449—21no—
158#448—21no—
159#448—21no—
160#447—21no—
161#447—21no—
162#446—41no—
163C8×Q8C2×C411yescharacteristic
164C8×Q8D411yescharacteristic
165C4×Q16C2×C411yescharacteristic
166C4×Q16C2×C411yescharacteristic
167C4×Q16C2×C411yescharacteristic
168C4×SD16—21no—
169C4×SD16—21no—
170C4×SD16—21no—
171C4×D8C2×C411yescharacteristic
172C4×D8C2×C411yescharacteristic
173C4×D8C2×C411yescharacteristic
174#435—41no—
175#435—41no—
176C8×D4—41no—
177C8×D4C2×C411yescharacteristic
178C8×D4C2×C411yescharacteristic
179C8×D4C2×C411yescharacteristic
180C8×D4D411yescharacteristic
181C8×D4D411yescharacteristic
182C8×D4D411yescharacteristic
183#432—21no—
184#432—21no—
185#431—21no—
186#431D411yescharacteristic
187#431D411yescharacteristic
188#431D411yescharacteristic
189#423—21no—
190C2×C4⋊C8—21no—
191C2×C4⋊C8—21no—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C4⋊C16C2311yescharacteristic
201C4⋊C16C2311yescharacteristic
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205#346—21no—
206#346—21no—
207C4×C16—21no—
208#335—21no—
209#335—21no—
210#335—21no—
211#334—21no—
212#334—21no—
213#334—21no—
214#334C2311yescharacteristic
215#334C2311yescharacteristic
216#334C2311yescharacteristic
217#334C2311yescharacteristic
218C2×#41—21no—
219C2×#41—21no—
220C2×#41SD1611yescharacteristic
221C22×Q8—21no—
222C22×Q8—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—41no—
226#135—81no—
227#135—81no—
228#135—81no—
229#135—81no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—81no—
234C2×Q16—81no—
235C2×Q16C22×C411yescharacteristic
236C2×Q16C22×C411yescharacteristic
237C2×Q16C22×C411yescharacteristic
238C2×SD16—21no—
239C2×SD16—21no—
240C2×SD16—21no—
241C2×SD16—21no—
242C2×SD16—21no—
243C2×SD16—21no—
244C2×SD16—21no—
245C2×SD16—21no—
246C2×SD16—21no—
247C2×SD16—81no—
248C2×SD16—81no—
249C2×SD16—81no—
250C2×SD16—81no—
251C2×D8—21no—
252C2×D8—21no—
253C2×D8—21no—
254C2×D8—81no—
255C2×D8—81no—
256C2×D8C22×C411yescharacteristic
257C2×D8C22×C411yescharacteristic
258C2×D8C22×C411yescharacteristic
259C2×M16—41no—
260C22×C8—41no—
261C22×C8—41no—
262C22×C8C22×C411yescharacteristic
263C22×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C4⋊Q8—21no—
269C4⋊Q8—41no—
270#127—21no—
271#126—41no—
272#126—41no—
273#126—41no—
274#126—41no—
275#126—41no—
276#125—21no—
277#124—21no—
278#124—21no—
279#124—21no—
280#124—21no—
281#124—21no—
282#124—21no—
283#124—21no—
284#124—41no—
285#124—41no—
286#124—41no—
287#124—41no—
288#124—41no—
289#124—41no—
290#124—41no—
291#123—41no—
292#123—41no—
293#123—41no—
294#123—41no—
295#123—41no—
296#123—41no—
297#123—41no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—41no—
302C22⋊Q8—41no—
303C22⋊Q8—41no—
304C22⋊Q8—41no—
305C22⋊Q8—41no—
306#121—21no—
307#121—21no—
308#121—21no—
309#121—41no—
310#121—41no—
311#121—41no—
312#121—41no—
313#121—41no—
314#121—41no—
315#121—41no—
316#120—41no—
317#120—41no—
318#120—41no—
319#120—41no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊C411yescharacteristic
324C4×Q8D811yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—41no—
329C4×D4—41no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4—41no—
333C4×D4—41no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4—41no—
337C4×D4C22⋊C411yescharacteristic
338C4×D4C22⋊C411yescharacteristic
339C4×D4C22⋊C411yescharacteristic
340C4×D4D811yescharacteristic
341C4×D4D811yescharacteristic
342C4×D4D811yescharacteristic
343#117—41no—
344#117—41no—
345#117—41no—
346#117—41no—
347#117SD1611yescharacteristic
348#117SD1611yescharacteristic
349#117SD1611yescharacteristic
350C2×C22⋊C4—41no—
351C2×C22⋊C4—41no—
352C2×C22⋊C4—41no—
353C2×C22⋊C4—41no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—41no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359M32—81no—
360M32—81no—
361C2×C16—21no—
362C2×C16—21no—
363C2×C16—21no—
364C2×C16—21no—
365C2×C16—81no—
366#107C22×C411yescharacteristic
367#107C22×C411yescharacteristic
368#106—21no—
369C4⋊C8—21no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—41no—
374C4⋊C8—41no—
375C4⋊C8—41no—
376C4⋊C8C22⋊C411yescharacteristic
377C4⋊C8C22⋊C411yescharacteristic
378C4⋊C8C22⋊C411yescharacteristic
379C4⋊C8C22⋊C411yescharacteristic
380C4⋊C8C22⋊C411yescharacteristic
381C4⋊C8C22⋊C411yescharacteristic
382#103—21no—
383#103—21no—
384#103—21no—
385#103—21no—
386#103—21no—
387#103—21no—
388#102—21no—
389#102—21no—
390#102—21no—
391#102—21no—
392#102—21no—
393#102—21no—
394C22⋊C8—41no—
395C22⋊C8—41no—
396C22⋊C8—41no—
397C22⋊C8C22⋊C411yescharacteristic
398C22⋊C8C22⋊C411yescharacteristic
399C22⋊C8C22⋊C411yescharacteristic
400C22⋊C8C22⋊C411yescharacteristic
401C22⋊C8C22⋊C411yescharacteristic
402C22⋊C8C22⋊C411yescharacteristic
403#97—41no—
404#97—41no—
405C4×C8—41no—
406C4×C8C22×C411yescharacteristic
407C4×C8C22×C411yescharacteristic
408C4×C8C22×C411yescharacteristic
409C4×C8C2×D411yescharacteristic
410C4×C8C2×D411yescharacteristic
411C4×C8C2×D411yescharacteristic
412C4×C8C2×D411yescharacteristic
413C4×C8C2×D411yescharacteristic
414C4×C8C2×D411yescharacteristic
415C4×C8C2411yescharacteristic
416C24—41no—
417C24—41no—
418#41—42no—
419#41—81no—
420#41—81no—
421#41—81no—
422#41—81no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×Q8#10211yescharacteristic
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—81no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—84no—
450C2×D4#10211yescharacteristic
451C2×D4#10211yescharacteristic
452C2×D4#10211yescharacteristic
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×D811yescharacteristic
464C22×C4C2×D811yescharacteristic
465C22×C4C2×D811yescharacteristic
466C22×C4C2×SD1611yescharacteristic
467C22×C4C2×SD1611yescharacteristic
468C22×C4C2×SD1611yescharacteristic
469Q16—41no—
470Q16—41no—
471Q16—41no—
472Q16—81no—
473Q16—81no—
474Q16—81no—
475SD16—41no—
476SD16—41no—
477SD16—41no—
478SD16—41no—
479SD16—41no—
480SD16—41no—
481SD16—81no—
482SD16—81no—
483SD16—81no—
484SD16—81no—
485SD16—81no—
486SD16—81no—
487D8—41no—
488D8—41no—
489D8—41no—
490D8—81no—
491D8—81no—
492D8—81no—
493M16—161no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—161no—
508C2×C8—161no—
509C2×C8C2×C22⋊C411yescharacteristic
510C2×C8C2×C22⋊C411yescharacteristic
511C2×C8C2×C22⋊C411yescharacteristic
512C2×C8C2×C22⋊C411yescharacteristic
513C2×C8C2×C22⋊C411yescharacteristic
514C2×C8C2×C22⋊C411yescharacteristic
515C2×C8C23×C411yescharacteristic
516C2×C8C22×D411yescharacteristic
517C4⋊C4—21no—
518C4⋊C4—21no—
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#10211yescharacteristic
530C4⋊C4#10211yescharacteristic
531C4⋊C4#10211yescharacteristic
532C4⋊C4#10211yescharacteristic
533C4⋊C4#10211yescharacteristic
534C4⋊C4#10211yescharacteristic
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—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4#10211yescharacteristic
559C22⋊C4#10211yescharacteristic
560C22⋊C4#10211yescharacteristic
561C22⋊C4#10211yescharacteristic
562C22⋊C4#10211yescharacteristic
563C22⋊C4#10211yescharacteristic
564C42—41no—
565C42—41no—
566C42—41no—
567C42C2×C22⋊C411yescharacteristic
568C42C2×C22⋊C411yescharacteristic
569C42C2×C22⋊C411yescharacteristic
570C42C2×D811yescharacteristic
571C42C2×D811yescharacteristic
572C42C2×D811yescharacteristic
573C42C2×SD1611yescharacteristic
574C42C2×SD1611yescharacteristic
575C42C2×SD1611yescharacteristic
576C42C22×D411yescharacteristic
577C16—41no—
578C16—41no—
579C16—41no—
580C16—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—82no—
585C23—82no—
586C23—161no—
587C23—161no—
588C23—161no—
589C23—161no—
590C23C2×#10211yescharacteristic
591C23C2×#10211yescharacteristic
592C23C2×#10211yescharacteristic
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597Q8—41no—
598Q8—41no—
599Q8—41no—
600Q8—161no—
601Q8—161no—
602D4—22no—
603D4—22no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—41no—
611D4—82no—
612D4—82no—
613D4—82no—
614D4—161no—
615D4—161no—
616D4—161no—
617D4—161no—
618D4—161no—
619D4—161no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—82no—
638C2×C4—161no—
639C2×C4—161no—
640C2×C4—161no—
641C2×C4—161no—
642C2×C4—161no—
643C2×C4—161no—
644C2×C4C2×#10211yescharacteristic
645C2×C4C2×#10211yescharacteristic
646C2×C4C2×#10211yescharacteristic
647C2×C4C2×#10211yescharacteristic
648C2×C4C2×#10211yescharacteristic
649C2×C4C2×#10211yescharacteristic
650C2×C4C2×#10211yescharacteristic
651C2×C4C2×#10211yescharacteristic
652C2×C4C2×#10211yescharacteristic
653C2×C4C22×C22⋊C411yescharacteristic
654C2×C4C22×D811yescharacteristic
655C2×C4C22×SD1611yescharacteristic
656C8—21no—
657C8—21no—
658C8—21no—
659C8—21no—
660C8—21no—
661C8—21no—
662C8—21no—
663C8—81no—
664C8—81no—
665C8—81no—
666C8—81no—
667C8#57611yescharacteristic
668C8#57711yescharacteristic
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—322no—
687C22—322no—
688C22#214312yes—
689C22C22×#10211yescharacteristic
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4#288811yescharacteristic
707C4#289711yescharacteristic
708C2—22no—
709C2—41no—
710C2—41no—
711C2—161no—
712C2—161no—
713C2—161no—
714C2#1772011yescharacteristic
715C2C2×#214311yescharacteristic
716C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62419C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33503C2semidirect
3. #17834C2semidirect
4. #11980C2semidirect
5. #11979C2semidirect
6. #11978C2semidirect
7. #11960C2semidirect
8. #11959C2semidirect
9. #11957C2semidirect
10. #11946C2semidirect
11. #11945C2semidirect
12. #11944C2semidirect
13. #11926C2semidirect
14. #11925C2semidirect
15. C2×#1360C2semidirect
44. #2163C22semidirect
45. #2163C22semidirect
46. #2162C22semidirect
47. #2150C22semidirect
48. #2150C22semidirect
49. C2×C4⋊C16C22semidirect
53. #1568C22semidirect
54. #1568C22semidirect
55. #1566C22semidirect
56. #1564C22semidirect
57. C2×#334C22semidirect
58. C2×#334C22semidirect
59. #1363C22semidirect
60. #1363C22semidirect
61. #1363C22semidirect
62. #1363C22semidirect
67. #1360C22semidirect
68. #1360C22semidirect
69. #1360C22semidirect
70. #1360C22semidirect
71. #1360C22semidirect
72. #1360C22semidirect
73. #1360C22semidirect
74. #1360C22semidirect

Maximal subgroups

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

subgroupindex
#624192
#335032
#178342
#119802
#119792
#119782
#119602
#119592
#119572
#119462
#119452
#119442
#119262
#119252
C2×#13602