Finite Group Databasebeta

← #355747

Subgroups — #355747

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2#21058C211yesmaximal, characteristic
3C2×#1866C211yesmaximal, characteristic
4#20427C211yesmaximal, characteristic, complemented
5#20384C211yesmaximal, characteristic
6#10933C211yesmaximal, characteristic
7#10929C211yesmaximal, characteristic
8#10911C211yesmaximal, characteristic, complemented
9#10909C211yesmaximal, characteristic, complemented
10#10887C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10860C211yesmaximal, characteristic, complemented
13#10808C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2297C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1938C418yescomplemented
34#1928C412yescomplemented
35#1927C412yescomplemented
36#1926C412yescomplemented
37#1925C412yescomplemented
38#1866C418yescomplemented
39#1768—21no—
40#1768—21no—
41#1767—21no—
42#1767—21no—
43#1753—21no—
44#1753—21no—
45#1752—21no—
46#1752—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C4×#95C2211yescharacteristic
65C22×#41C2311yescharacteristic
66C2×#131—22no—
67C2×#131—41no—
68C2×#131—41no—
69C2×#131—41no—
70C2×#131—41no—
71C2×#131C2×C414yes—
72C22×M16—21no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C23×C8—21no—
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#108—41no—
89C2×#108—41no—
90C2×#108C2×C414yes—
91#427—41no—
92#427—41no—
93C2×#107C2×C412yes—
94C2×#106C2×C412yes—
95#424—22no—
96#424—22no—
97#424C2×C414yes—
98#424C2×C414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×#104—41no—
104C2×#104—41no—
105C2×#104—41no—
106C2×#104—41no—
107C2×#104C2×C414yes—
108C2×#104C2×C414yes—
109#416—41no—
110#416—41no—
111#416C2×C414yes—
112#416C2×C414yes—
113C2×#103C2×C412yes—
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C412yes—
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119#408—22no—
120#408C2×C414yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123#405—41no—
124#405—41no—
125C4×M16—41no—
126C4×M16—41no—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—41no—
130C2×#97—41no—
131C2×#97—41no—
132C2×#97—41no—
133C2×#97C2×C414yes—
134C2×#97D411yescharacteristic
135C2×#97D411yescharacteristic
136C2×#97D411yescharacteristic
137C2×#97D411yescharacteristic
138C2×C4×C8—21no—
139C2×C4×C8—22no—
140C2×C4×C8—21no—
141C2×C4×C8—41no—
142C2×C4×C8—41no—
143#391C2×C412yes—
144#390C2×C412yes—
145#386—22no—
146C4×C4⋊C4C2×C412yes—
147C4×C4⋊C4C2×C412yes—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151#376—21no—
152C2×#95—22no—
153C43—21no—
154#341—41no—
155#341—41no—
156#341C2×C414yes—
157#340C2×C412yes—
158#340C2×C412yes—
159#339—41no—
160#339—41no—
161#339—41no—
162#339—41no—
163#339C2×C414yes—
164#339C2×C414yes—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336C2×C414yes—
175#336C2×C412yes—
176#336C2×C412yes—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41C22×C412yes—
188C2×#41C22×C412yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4—21no—
191C22×D4C22×C411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197#131—44no—
198#131—44no—
199#131—44no—
200#131—44no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—24no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—42no—
223C2×M16—42no—
224C2×M16C22×C412yes—
225C2×M16C22×C412yes—
226C2×M16C22×C412yes—
227C2×M16C22×C412yes—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8—42no—
253C22×C8C22×C412yes—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C412yes—
257C22×C8C22×C412yes—
258C22×C8C22×C412yes—
259C22×C8C22×C412yes—
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8#4112yes—
265C22×C8#4112yes—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117C22×C412yes—
277#117C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C22×C412yes—
307C2×C42C22×C412yes—
308C2×C42C22×C412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311#108—44no—
312#107—42no—
313#106—42no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317C4⋊C8—24no—
318#104—44no—
319#104—44no—
320#103—42no—
321#103C4214yes—
322#103C4214yes—
323#102—42no—
324#102C4214yes—
325#102C4214yes—
326C22⋊C8—24no—
327C22⋊C8—24no—
328#97—44no—
329#97—41no—
330#97—41no—
331#97—41no—
332#97—41no—
333#97—42no—
334#97—42no—
335C4×C8—42no—
336C4×C8—44no—
337C4×C8—42no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—44no—
343C24—21no—
344C24C2×#4111yescharacteristic
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8C2×C4212yes—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C2×C4212yes—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419C22×C42−1+411yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—48no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—48no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8C2×C4212yes—
482C2×C8C2×C4212yes—
483C2×C8C4×D412yes—
484C2×C8C4×D412yes—
485C2×C8C4×D412yes—
486C2×C8C4×D412yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4C2×C4212yes—
494C4⋊C4C2×C4212yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—82no—
524C42—82no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23C2×C4×D411yescharacteristic
533C23C4×#4111yescharacteristic
534C23C4×#4111yescharacteristic
535C23#53911yescharacteristic
536C23#53911yescharacteristic
537C23#54011yescharacteristic
538C23#54211yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543D4—22no—
544D4—22no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—41no—
550D4—41no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4C22×C4211yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4#51811yescharacteristic
591C2×C4#51811yescharacteristic
592C2×C4#51811yescharacteristic
593C2×C4#51811yescharacteristic
594C2×C4#51911yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4#52011yescharacteristic
598C2×C4#52011yescharacteristic
599C2×C4#52011yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4#52011yescharacteristic
602C2×C4#53411yescharacteristic
603C2×C4#53511yescharacteristic
604C2×C4#53611yescharacteristic
605C2×C4#53711yescharacteristic
606C2×C4#53811yescharacteristic
607C2×C4#53811yescharacteristic
608C2×C4#53811yescharacteristic
609C2×C4#53911yescharacteristic
610C2×C4#54111yescharacteristic
611C2×C4#54211yescharacteristic
612C2×C4#54211yescharacteristic
613C2×C4#54211yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—22no—
618C8—22no—
619C8—22no—
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C8—42no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—82no—
637C22—82no—
638C22C42×D411yescharacteristic
639C22#232211yescharacteristic
640C22#232411yescharacteristic
641C22#233211yescharacteristic
642C22#233411yescharacteristic
643C22#233511yescharacteristic
644C22#233511yescharacteristic
645C22#295612yes—
646C22#295712yes—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#227411yescharacteristic
657C4#227511yescharacteristic
658C4#232111yescharacteristic
659C4#232311yescharacteristic
660C4#232511yescharacteristic
661C4#232611yescharacteristic
662C4#233411yescharacteristic
663C4#233711yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—82no—
669C2#1378911yescharacteristic
670C2#2051211yescharacteristic
671C2#2051311yescharacteristic
672C2#2209712yes—
673C2#2216712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20427C2semidirect
8. #10911C2semidirect
9. #10909C2semidirect
10. #10887C2semidirect
11. #10882C2semidirect
12. #10860C2semidirect
13. #10808C2semidirect
15. #10696C2semidirect
33. #1938C4semidirect
34. #1928C4semidirect
35. #1927C4semidirect
36. #1926C4semidirect
37. #1925C4semidirect
38. #1866C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
#210582
C2×#18662
#204272
#203842
#109332
#109292
#109112
#109092
#108872
#108822
#108602
#108082
#107912
#106962