Finite Group Databasebeta

← #373445

Subgroups — #373445

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21072C211yesmaximal, characteristic
2C2×#1929C211yesmaximal, characteristic
3#20852C211yesmaximal, characteristic
4#20460C211yesmaximal, characteristic, complemented
5#20422C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic, complemented
7#11250C211yesmaximal, characteristic, complemented
8#11108C211yesmaximal, characteristic
9#11080C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic, complemented
11#10901C211yesmaximal, characteristic, complemented
12#10894C211yesmaximal, characteristic, complemented
13#10873C211yesmaximal, characteristic, complemented
14#10828C211yesmaximal, characteristic, complemented
15#10780C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C2×#421C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#416C2211yescharacteristic
27#2879C2211yescharacteristic
28C2×#408C2211yescharacteristic
29#2873C2211yescharacteristic
30#2873C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33#1929—22no—
34#1929—22no—
35#1929C412yescomplemented
36#1929C412yescomplemented
37#1929C412yescomplemented
38#1929C412yescomplemented
39#1855—22no—
40#1855—22no—
41#1778—22no—
42C4×#108—22no—
43#1765—22no—
44#1764—22no—
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49#1738C2211yescharacteristic
50#1738C2211yescharacteristic
51#1738C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1735C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64#568—22no—
65#568—22no—
66#568—22no—
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—22no—
86#430—41no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108—21no—
92C2×#108—21no—
93C2×#108—22no—
94C2×#108—22no—
95C2×#108—21no—
96C2×#108—21no—
97C2×#108—41no—
98C2×#108—41no—
99C2×#108—41no—
100C2×#108C2×C412yes—
101C2×#108C2×C412yes—
102C2×#108D412yes—
103C2×#108D412yes—
104#423—22no—
105#423—41no—
106C2×C4⋊C8—41no—
107#421—22no—
108#421—22no—
109#421—22no—
110#421—41no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—41no—
114C2×#104—41no—
115C2×#104C2×C412yes—
116C2×#104C2×C412yes—
117#416C2×C412yes—
118#416C2×C412yes—
119#416D412yes—
120#416D412yes—
121C2×#103Q811yescharacteristic
122C2×#103Q811yescharacteristic
123C2×#102Q811yescharacteristic
124C2×#102Q811yescharacteristic
125#408—21no—
126#408—21no—
127#408—22no—
128#408—41no—
129#408C2×C412yes—
130#408C2×C412yes—
131#407—22no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134#405—21no—
135#405—21no—
136#405—22no—
137#405—22no—
138#405—22no—
139#405—41no—
140#405—41no—
141#405—41no—
142#405C2×C412yes—
143#405C2×C412yes—
144C4×M16—22no—
145C4×M16—22no—
146C2×#97—21no—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150#344—22no—
151#344—22no—
152#344—22no—
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#340—21no—
162#340—21no—
163#339—22no—
164#339—22no—
165#339—22no—
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—28no—
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—42no—
238C2×M16—44no—
239C2×M16—44no—
240C2×M16C22×C412yes—
241C2×M16C22×C412yes—
242C2×M16C2×D412yes—
243C2×M16C2×D412yes—
244C2×M16#4112yes—
245C2×M16#4112yes—
246C2×M16#4112yes—
247C2×M16#4112yes—
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—24no—
308#108—24no—
309#108—24no—
310#108—24no—
311#108—42no—
312#108—42no—
313#108—42no—
314#108—42no—
315#108—44no—
316C4⋊C8—44no—
317#104—28no—
318#104—42no—
319#104—42no—
320#104—44no—
321#103C4⋊C412yes—
322#103C4⋊C412yes—
323#103C4⋊C412yes—
324#103C4⋊C412yes—
325#102C4⋊C412yes—
326#102C4⋊C412yes—
327#102C4⋊C412yes—
328#102C4⋊C412yes—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—44no—
334#97—22no—
335#97—22no—
336#97—42no—
337#97—42no—
338#97—42no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8—42no—
342C4×C8—42no—
343C4×C8—42no—
344C24—21no—
345C242−1+411yescharacteristic
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8C2×C4⋊C412yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4C2×C4⋊C412yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—81no—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411M16—22no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—48no—
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—
442M16—41no—
443M16#12314yes—
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—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—48no—
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×C8—41no—
492C2×C8—41no—
493C2×C8C2×C4⋊C411yescharacteristic
494C2×C8C2×C4⋊C411yescharacteristic
495C2×C8C2×C4⋊C411yescharacteristic
496C2×C8C2×C4⋊C411yescharacteristic
497C2×C8C4×D411yescharacteristic
498C2×C8C4×D411yescharacteristic
499C2×C8C4×D411yescharacteristic
500C2×C8C4×D411yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4C2×C4⋊C412yes—
508C4⋊C4C2×C4⋊C412yes—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—81no—
533C23—81no—
534C23#52011yescharacteristic
535C23#54211yescharacteristic
536C23#54211yescharacteristic
537C23#54711yescharacteristic
538C23D4×Q811yescharacteristic
539C23#55211yescharacteristic
540C23#55611yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545D4—22no—
546D4—22no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—41no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4C22×C4⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4C4×#4111yescharacteristic
582C2×C4C4×#4111yescharacteristic
583C2×C4#51811yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4#52511yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#52911yescharacteristic
590C2×C4#53311yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53811yescharacteristic
593C2×C4#53811yescharacteristic
594C2×C4D4211yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#55011yescharacteristic
599C2×C4#55111yescharacteristic
600C2×C4#55311yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55411yescharacteristic
603C2×C4#55411yescharacteristic
604C2×C4#55411yescharacteristic
605C2×C4#55511yescharacteristic
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—24no—
611C8—21no—
612C8—21no—
613C8—21no—
614C8—21no—
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C8—81no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—161no—
636C22—161no—
637C22#233711yescharacteristic
638C22D4×C4⋊C411yescharacteristic
639C22#237311yescharacteristic
640C22#242311yescharacteristic
641C22#246511yescharacteristic
642C22#246811yescharacteristic
643C22#247411yescharacteristic
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—82no—
650C4—161no—
651C4#229611yescharacteristic
652C4C4×#12311yescharacteristic
653C4#234911yescharacteristic
654C4#236311yescharacteristic
655C4#241811yescharacteristic
656C4#245611yescharacteristic
657C4#257411yescharacteristic
658C4#258711yescharacteristic
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—82no—
666C2#1411811yescharacteristic
667C2#2269411yescharacteristic
668C2#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
4. #20460C2semidirect
6. #11326C2semidirect
7. #11250C2semidirect
10. #10910C2semidirect
11. #10901C2semidirect
12. #10894C2semidirect
13. #10873C2semidirect
14. #10828C2semidirect
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
#210722
C2×#19292
#208522
#204602
#204222
#113262
#112502
#111082
#110802
#109102
#109012
#108942
#108732
#108282
#107802