Finite Group Databasebeta

← #483202

Subgroups — #483202

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60713C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic, complemented
3#17798C211yesmaximal, characteristic, complemented
4#11980C211yesmaximal, characteristic, complemented
5#11979C211yesmaximal, characteristic, complemented
6#11978C211yesmaximal, characteristic, complemented
7#11977C211yesmaximal, characteristic, complemented
8#11970C211yesmaximal, characteristic, complemented
9#11968C211yesmaximal, characteristic, complemented
10#11959C211yesmaximal, characteristic, complemented
11#11957C211yesmaximal, characteristic, complemented
12#11946C211yesmaximal, characteristic, complemented
13#11945C211yesmaximal, characteristic, complemented
14#11944C211yesmaximal, characteristic, complemented
15#11943C211yesmaximal, characteristic, complemented
16#3510—21no—
17#3183—21no—
18#3183—21no—
19#3175C411yescharacteristic
20#3175C411yescharacteristic
21#3175C411yescharacteristic
22#3175C411yescharacteristic
23#3173—21no—
24#3173—21no—
25#3168C2211yescharacteristic
26#3168C2211yescharacteristic
27#3168C2211yescharacteristic
28#3168C2211yescharacteristic
29#3160C2211yescharacteristic
30C2×#495C2211yescharacteristic
31#3140—21no—
32#3059C411yescharacteristic
33#3058C411yescharacteristic
34#3057—21no—
35#2989—21no—
36#2950C411yescharacteristic
37#2949—21no—
38#2948C411yescharacteristic
39C4×#131C2211yescharacteristic
40#2164—21no—
41#2164—21no—
42#2163C2211yescharacteristic
43#2163C2211yescharacteristic, complemented
44#2162C2211yescharacteristic
45#2162C2211yescharacteristic, complemented
46#2150C2211yescharacteristic, complemented
47C2×C4⋊C16C2211yescharacteristic, complemented
48#2113—21no—
49#2108—21no—
50#1569—21no—
51#1569—21no—
52#1568C2211yescharacteristic
53#1568C2211yescharacteristic, complemented
54#1566C2211yescharacteristic
55#1566C2211yescharacteristic, complemented
56#1565—21no—
57#1564C2211yescharacteristic, complemented
58C2×#334C2211yescharacteristic, complemented
59#1363C2211yescharacteristic, complemented
60#1363C2211yescharacteristic, complemented
61#1363C2211yescharacteristic, complemented
62#1363C2211yescharacteristic, complemented
63#1363C2211yescharacteristic, complemented
64#1363C2211yescharacteristic, complemented
65#1363C2211yescharacteristic, complemented
66#1363C2211yescharacteristic, complemented
67#1360C2211yescharacteristic
68#1360C2211yescharacteristic
69#1360C2211yescharacteristic
70#1360C2211yescharacteristic
71#1360C2211yescharacteristic
72#1360C2211yescharacteristic
73#1360C2211yescharacteristic
74#1360C2211yescharacteristic
75C22×Q16C2×C411yescharacteristic
76C22×SD16—21no—
77C22×D8C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131D411yescharacteristic
80#559—41no—
81#559—41no—
82#556—41no—
83#556—41no—
84#551—41no—
85#551—41no—
86#551—41no—
87#551—41no—
88#542—21no—
89#542—21no—
90#542—21no—
91#542—21no—
92#536—21no—
93#535—21no—
94#533—21no—
95#529—21no—
96#529—21no—
97#529—21no—
98#529—21no—
99C2×#124—21no—
100#519—21no—
101#518—21no—
102C4×#41D411yescharacteristic
103C2×M32—21no—
104C22×C16—21no—
105#500C2×C411yescharacteristic
106#500C2×C411yescharacteristic
107#499—21no—
108#499C2311yescharacteristic
109#499C2311yescharacteristic
110#498—21no—
111#495—21no—
112#495—21no—
113#495C2×C411yescharacteristic
114#495C2×C411yescharacteristic
115#495C2311yescharacteristic
116#495C2311yescharacteristic
117#489—21no—
118#489—21no—
119#489—21no—
120#489—21no—
121#488—21no—
122#488—21no—
123#488—21no—
124#488—21no—
125#470—21no—
126#470—21no—
127#469—21no—
128#469—21no—
129#468—21no—
130#468—21no—
131#468—21no—
132#468—21no—
133#467C2×C411yescharacteristic
134#467C2×C411yescharacteristic
135#467C2×C411yescharacteristic
136#467C2×C411yescharacteristic
137#466C2×C411yescharacteristic
138#466C2×C411yescharacteristic
139#466C2×C411yescharacteristic
140#466C2×C411yescharacteristic
141#465—21no—
142#465—21no—
143#465—21no—
144#465—21no—
145#446—41no—
146#442—21no—
147#442—21no—
148#441—21no—
149#441—21no—
150#440—21no—
151#440—21no—
152#440—21no—
153#440—21no—
154C4×Q16C2×C411yescharacteristic
155C4×Q16C2×C411yescharacteristic
156C4×Q16C2×C411yescharacteristic
157C4×Q16C2×C411yescharacteristic
158C4×SD16—21no—
159C4×SD16—21no—
160C4×SD16—21no—
161C4×SD16—21no—
162C4×D8C2×C411yescharacteristic
163C4×D8C2×C411yescharacteristic
164C4×D8C2×C411yescharacteristic
165C4×D8C2×C411yescharacteristic
166#436—41no—
167#435—41no—
168#435—41no—
169#432—21no—
170#432—21no—
171#431—21no—
172#427—21no—
173#427C2×C411yescharacteristic
174#427C2×C411yescharacteristic
175#423—21no—
176#423—21no—
177C2×C4⋊C8—21no—
178#419—21no—
179#419—21no—
180#418—21no—
181#418—21no—
182#405C2×C411yescharacteristic
183#405C2×C411yescharacteristic
184#405C2×C411yescharacteristic
185#405D411yescharacteristic
186#405D411yescharacteristic
187#405D411yescharacteristic
188C4×M16D411yescharacteristic
189C4×M16D411yescharacteristic
190C4×M16D411yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16C2311yescharacteristic
197C4⋊C16C2311yescharacteristic
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C22⋊C16—21no—
201C22⋊C16—21no—
202C22⋊C16—21no—
203C22⋊C16—21no—
204#346—21no—
205#346—21no—
206C4×C16—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×#41C22⋊C411yescharacteristic
219C22×Q8—21no—
220C22×D4—21no—
221C2×Q16—21no—
222C2×Q16—42no—
223C2×Q16C22×C411yescharacteristic
224C2×Q16C22×C411yescharacteristic
225C2×SD16—21no—
226C2×SD16—21no—
227C2×SD16—21no—
228C2×SD16—21no—
229C2×SD16—42no—
230C2×SD16—42no—
231C2×D8—21no—
232C2×D8—42no—
233C2×D8C22×C411yescharacteristic
234C2×D8C22×C411yescharacteristic
235#131—41no—
236#131—41no—
237#131—42no—
238C2×M16—41no—
239C2×M16—41no—
240C2×M16C22⋊C411yescharacteristic
241C2×M16C22⋊C411yescharacteristic
242C2×M16C22⋊C411yescharacteristic
243C2×M16C22⋊C411yescharacteristic
244C2×M16C22⋊C411yescharacteristic
245C2×M16C22⋊C411yescharacteristic
246C22×C8—41no—
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—41no—
256#127—41no—
257#126—41no—
258#126—41no—
259#126—41no—
260#126—41no—
261#126—41no—
262#126—41no—
263#126—41no—
264#126—41no—
265#125—21no—
266#125—21no—
267#124—21no—
268#124—21no—
269#124—21no—
270#124—21no—
271#124—41no—
272#124—41no—
273#124—41no—
274#124—41no—
275#124—41no—
276#124—41no—
277#124—41no—
278#124—41no—
279#123—41no—
280#123—41no—
281#123—41no—
282#123—41no—
283#123—41no—
284#123—41no—
285#123—41no—
286#123—41no—
287#123—41no—
288#123—41no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—41no—
294C22⋊Q8—41no—
295C22⋊Q8—41no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C22⋊Q8—41no—
299#121—21no—
300#121—21no—
301#121—21no—
302#121—21no—
303#120—41no—
304#120—41no—
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×Q8D811yescharacteristic
314C4×Q8SD1611yescharacteristic
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—
325C4×D4D811yescharacteristic
326C4×D4D811yescharacteristic
327C4×D4D811yescharacteristic
328C4×D4SD1611yescharacteristic
329C4×D4SD1611yescharacteristic
330C4×D4SD1611yescharacteristic
331#117—21no—
332#117—21no—
333#117—41no—
334#117—41no—
335#117—41no—
336#117—41no—
337#117—41no—
338#117—41no—
339#117C22⋊C411yescharacteristic
340#117C22⋊C411yescharacteristic
341#117C22⋊C411yescharacteristic
342C2×C22⋊C4—41no—
343C2×C22⋊C4—41no—
344C2×C22⋊C4—41no—
345C2×C22⋊C4—41no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349M32—42no—
350C2×C16—21no—
351C2×C16—21no—
352C2×C16—21no—
353C2×C16—21no—
354C2×C16—42no—
355#107—21no—
356#107—21no—
357#107C22×C411yescharacteristic
358#107C22×C411yescharacteristic
359#107C22×C411yescharacteristic
360#107C22×C411yescharacteristic
361#106—21no—
362#106—21no—
363#106—21no—
364#106—21no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—41no—
370C4⋊C8—41no—
371C4⋊C8—41no—
372#103—21no—
373#103—21no—
374#103—21no—
375#103—21no—
376#103—21no—
377#103—21no—
378#103—21no—
379#103—21no—
380#102—21no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—21no—
385#102—21no—
386#102—21no—
387#102—21no—
388C22⋊C8—41no—
389C22⋊C8—41no—
390C22⋊C8—41no—
391#97—41no—
392#97—41no—
393#97C22⋊C411yescharacteristic
394#97C22⋊C411yescharacteristic
395#97C22⋊C411yescharacteristic
396#97C22⋊C411yescharacteristic
397#97C22⋊C411yescharacteristic
398#97C22⋊C411yescharacteristic
399C4×C8—41no—
400C4×C8C22×C411yescharacteristic
401C4×C8C22×C411yescharacteristic
402C4×C8C22×C411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410C24—41no—
411#41—41no—
412#41—41no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8#10211yescharacteristic
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4#10211yescharacteristic
431C2×D4#10211yescharacteristic
432C2×D4#10211yescharacteristic
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×D811yescharacteristic
442C22×C4C2×D811yescharacteristic
443C22×C4C2×D811yescharacteristic
444C22×C4C2×SD1611yescharacteristic
445C22×C4C2×SD1611yescharacteristic
446C22×C4C2×SD1611yescharacteristic
447Q16—41no—
448Q16—41no—
449Q16—42no—
450SD16—42no—
451SD16—42no—
452SD16—41no—
453SD16—41no—
454SD16—41no—
455SD16—41no—
456D8—41no—
457D8—41no—
458D8—42no—
459M16—22no—
460M16—22no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—161no—
466M16—161no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—161no—
481C2×C8C2×C22⋊C411yescharacteristic
482C2×C8C2×C22⋊C411yescharacteristic
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C411yescharacteristic
486C2×C8C2×C22⋊C411yescharacteristic
487C2×C8C23×C411yescharacteristic
488C2×C8C22×D411yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4#10211yescharacteristic
508C4⋊C4#10211yescharacteristic
509C4⋊C4#10211yescharacteristic
510C4⋊C4#10211yescharacteristic
511C4⋊C4#10211yescharacteristic
512C4⋊C4#10211yescharacteristic
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊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#10211yescharacteristic
536C22⋊C4#10211yescharacteristic
537C22⋊C4#10211yescharacteristic
538C22⋊C4#10211yescharacteristic
539C22⋊C4#10211yescharacteristic
540C22⋊C4#10211yescharacteristic
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C2×C22⋊C411yescharacteristic
550C42C2×D811yescharacteristic
551C42C2×D811yescharacteristic
552C42C2×D811yescharacteristic
553C42C2×SD1611yescharacteristic
554C42C2×SD1611yescharacteristic
555C42C2×SD1611yescharacteristic
556C42C22×D411yescharacteristic
557C16—41no—
558C16—41no—
559C16—41no—
560C16—41no—
561C23—41no—
562C23—41no—
563C23—82no—
564C23—161no—
565C23—161no—
566C23C2×#10211yescharacteristic
567C23C2×#10211yescharacteristic
568C23C2×#10211yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—82no—
575Q8—161no—
576Q8—161no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—82no—
585D4—161no—
586D4—161no—
587D4—161no—
588D4—161no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
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—82no—
608C2×C4—161no—
609C2×C4—161no—
610C2×C4—161no—
611C2×C4—161no—
612C2×C4C2×#10211yescharacteristic
613C2×C4C2×#10211yescharacteristic
614C2×C4C2×#10211yescharacteristic
615C2×C4C2×#10211yescharacteristic
616C2×C4C2×#10211yescharacteristic
617C2×C4C2×#10211yescharacteristic
618C2×C4C2×#10211yescharacteristic
619C2×C4C2×#10211yescharacteristic
620C2×C4C2×#10211yescharacteristic
621C2×C4C22×C22⋊C411yescharacteristic
622C2×C4C22×D811yescharacteristic
623C2×C4C22×SD1611yescharacteristic
624C8—21no—
625C8—21no—
626C8—21no—
627C8—21no—
628C8—21no—
629C8—21no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8#41312yes—
635C8#57511yescharacteristic
636C8#57511yescharacteristic
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—322no—
653C22C22×#10211yescharacteristic
654C4—21no—
655C4—21no—
656C4—21no—
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—
671C4C2×#41311yescharacteristic
672C4#289611yescharacteristic
673C2—41no—
674C2—41no—
675C2—41no—
676C2—161no—
677C2—161no—
678C2#1770411yescharacteristic
679C2#3338511yescharacteristic
680C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#3058C4
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33423C2semidirect
3. #17798C2semidirect
4. #11980C2semidirect
5. #11979C2semidirect
6. #11978C2semidirect
7. #11977C2semidirect
8. #11970C2semidirect
9. #11968C2semidirect
10. #11959C2semidirect
11. #11957C2semidirect
12. #11946C2semidirect
13. #11945C2semidirect
14. #11944C2semidirect
15. #11943C2semidirect
43. #2163C22semidirect
45. #2162C22semidirect
46. #2150C22semidirect
47. C2×C4⋊C16C22semidirect
53. #1568C22semidirect
55. #1566C22semidirect
57. #1564C22semidirect
58. C2×#334C22semidirect
59. #1363C22semidirect
60. #1363C22semidirect
61. #1363C22semidirect
62. #1363C22semidirect
63. #1363C22semidirect
64. #1363C22semidirect
65. #1363C22semidirect
66. #1363C22semidirect

Maximal subgroups

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

subgroupindex
#607132
#334232
#177982
#119802
#119792
#119782
#119772
#119702
#119682
#119592
#119572
#119462
#119452
#119442
#119432