Finite Group Databasebeta

← #385293

Subgroups — #385293

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21098C211yesmaximal, characteristic
2C2×#1929C211yesmaximal, characteristic
3#20886C211yesmaximal, characteristic
4#20852C211yesmaximal, characteristic
5#20768C211yesmaximal, characteristic, complemented
6#11341C211yesmaximal, characteristic, complemented
7#11326C211yesmaximal, characteristic, complemented
8#11275C211yesmaximal, characteristic, complemented
9#11250C211yesmaximal, characteristic, complemented
10#11104C211yesmaximal, characteristic
11#11080C211yesmaximal, characteristic
12#10909C211yesmaximal, characteristic, complemented
13#10873C211yesmaximal, characteristic, complemented
14#10872C211yesmaximal, characteristic, complemented
15#10780C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2905C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30C2×#408C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33#1938—22no—
34#1937—22no—
35#1929C412yescomplemented
36#1929C412yescomplemented
37#1929C412yescomplemented
38#1929C412yescomplemented
39#1866—22no—
40#1866—22no—
41#1855—22no—
42#1855—22no—
43#1838—22no—
44#1837—22no—
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49#1738C2211yescharacteristic
50#1738C2211yescharacteristic
51#1738C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64#568—22no—
65#568—41no—
66#568—41no—
67#568—41no—
68C2×#131—22no—
69C2×#131—41no—
70C22×M16—22no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#430—41no—
86#430—41no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—21no—
91C2×#108—21no—
92C2×#108—21no—
93C2×#108—21no—
94C2×#108—41no—
95C2×#108—41no—
96C2×#108—41no—
97C2×#108—41no—
98C2×#108C2×C412yes—
99C2×#108C2×C412yes—
100#427—22no—
101#427—41no—
102C2×#107—21no—
103C2×#106—21no—
104#424—22no—
105#424—22no—
106#424—41no—
107#424—41no—
108#423—22no—
109#423—22no—
110#423—41no—
111#423—41no—
112C2×C4⋊C8—41no—
113#421—22no—
114#421—41no—
115#421—41no—
116#421—41no—
117C2×#104—22no—
118C2×#104—22no—
119C2×#104—41no—
120C2×#104—41no—
121C2×#104C2×C412yes—
122C2×#104C2×C412yes—
123#416C2×C412yes—
124#416C2×C412yes—
125#416D412yes—
126#416D412yes—
127C2×#103Q811yescharacteristic
128C2×#103Q811yescharacteristic
129C2×#102Q811yescharacteristic
130C2×#102Q811yescharacteristic
131#408—21no—
132#408—21no—
133#408—22no—
134#408—41no—
135#408C2×C412yes—
136#408C2×C412yes—
137#407—22no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140#405—21no—
141#405—21no—
142#405—22no—
143#405—41no—
144#405C2×C412yes—
145#405C2×C412yes—
146C2×#97—21no—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150#344—41no—
151#344—41no—
152#344—41no—
153#344—41no—
154#341—22no—
155#341—22no—
156#341—22no—
157#341—21no—
158#341—21no—
159#341—21no—
160#341—21no—
161#341D412yes—
162#341D412yes—
163#339—22no—
164#339—22no—
165#339—41no—
166#339—41no—
167#339C2×C412yes—
168#339C2×C412yes—
169#336—21no—
170#336—21no—
171#336—41no—
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176C2×#41—21no—
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×#41C22×C412yes—
187C2×#41C2×D412yes—
188C22×Q8C2×Q811yescharacteristic
189C22×D4—21no—
190C22×D4C2×Q811yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C2411yescharacteristic
194#131—44no—
195#131—44no—
196#131—44no—
197#131—44no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
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—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—42no—
242C2×M16—44no—
243C2×M16—44no—
244C2×M16C22×C412yes—
245C2×M16C22×C412yes—
246C2×M16C2×D412yes—
247C2×M16C2×D412yes—
248C2×M16#4112yes—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—42no—
266C22×C8C22×C411yescharacteristic
267C22×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C22×C411yescharacteristic
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×Q811yescharacteristic
275C22×C8C2×Q811yescharacteristic
276C22×C8#4111yescharacteristic
277C22×C8#4111yescharacteristic
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117C22×C412yes—
288#117C2×D412yes—
289#117#4112yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C2×Q811yescharacteristic
293C2×C4⋊C4C2×Q811yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307#108—42no—
308#108—42no—
309#108—42no—
310#108—42no—
311#108—42no—
312#108—42no—
313#108—44no—
314#107—42no—
315#106—42no—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C4⋊C8—44no—
319#104—42no—
320#104—42no—
321#104—44no—
322#104—44no—
323#103C4⋊C412yes—
324#103C4⋊C412yes—
325#103C4⋊C412yes—
326#103C4⋊C412yes—
327#102C4⋊C412yes—
328#102C4⋊C412yes—
329#102C4⋊C412yes—
330#102C4⋊C412yes—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8—22no—
335C22⋊C8—44no—
336#97—22no—
337#97—22no—
338#97—42no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8—42no—
342C24—21no—
343C242−1+411yescharacteristic
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C4⋊C412yes—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4C2×C4⋊C412yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—81no—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C42−1+411yescharacteristic
408C22×C42−1+411yescharacteristic
409M16—22no—
410M16—22no—
411M16—22no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—48no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
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—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—48no—
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×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8C2×C4⋊C411yescharacteristic
492C2×C8C2×C4⋊C411yescharacteristic
493C2×C8C2×C4⋊C411yescharacteristic
494C2×C8C2×C4⋊C411yescharacteristic
495C2×C8C4×D411yescharacteristic
496C2×C8C4×D411yescharacteristic
497C2×C8C4×D411yescharacteristic
498C2×C8C4×D411yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4C2×C4⋊C412yes—
506C4⋊C4C2×C4⋊C412yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—81no—
531C23—81no—
532C23#52011yescharacteristic
533C23#53911yescharacteristic
534C23#54111yescharacteristic
535C23#54711yescharacteristic
536C23D4×Q811yescharacteristic
537C23#56211yescharacteristic
538C23#56311yescharacteristic
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—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4C22×C4⋊C411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C4×#4111yescharacteristic
579C2×C4#51811yescharacteristic
580C2×C4#52011yescharacteristic
581C2×C4#52011yescharacteristic
582C2×C4#53511yescharacteristic
583C2×C4#53511yescharacteristic
584C2×C4#53611yescharacteristic
585C2×C4#53711yescharacteristic
586C2×C4#53811yescharacteristic
587C2×C4#54011yescharacteristic
588C2×C4#54211yescharacteristic
589C2×C4#54211yescharacteristic
590C2×C4#54411yescharacteristic
591C2×C4#54411yescharacteristic
592C2×C4D4211yescharacteristic
593C2×C4#54611yescharacteristic
594C2×C4#54611yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54711yescharacteristic
597C2×C4#55111yescharacteristic
598C2×C4#55211yescharacteristic
599C2×C4#55411yescharacteristic
600C2×C4#55411yescharacteristic
601C2×C4#55411yescharacteristic
602C2×C4H(3,𝔽4)11yescharacteristic
603C2×C4#56211yescharacteristic
604C8—21no—
605C8—21no—
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—21no—
611C8—21no—
612C8—42no—
613C8—42no—
614C8—42no—
615C8—42no—
616C8—42no—
617C8—81no—
618C8—81no—
619C8—81no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—161no—
634C22—161no—
635C22#233411yescharacteristic
636C22D4×C4⋊C411yescharacteristic
637C22#237911yescharacteristic
638C22#269811yescharacteristic
639C22#270411yescharacteristic
640C22#270611yescharacteristic
641C22#272011yescharacteristic
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—82no—
648C4—161no—
649C4#231711yescharacteristic
650C4#232411yescharacteristic
651C4#234911yescharacteristic
652C4#235911yescharacteristic
653C4#267011yescharacteristic
654C4#267211yescharacteristic
655C4#276311yescharacteristic
656C4#276811yescharacteristic
657C2—21no—
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—82no—
664C2#1440311yescharacteristic
665C2#2269411yescharacteristic
666C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #20768C2semidirect
6. #11341C2semidirect
7. #11326C2semidirect
8. #11275C2semidirect
9. #11250C2semidirect
12. #10909C2semidirect
13. #10873C2semidirect
14. #10872C2semidirect
35. #1929C4semidirect
36. #1929C4semidirect
37. #1929C4semidirect
38. #1929C4semidirect

Maximal subgroups

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

subgroupindex
#210982
C2×#19292
#208862
#208522
#207682
#113412
#113262
#112752
#112502
#111042
#110802
#109092
#108732
#108722
#107802