Finite Group Databasebeta

← #367826

Subgroups — #367826

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1945C211yesmaximal, characteristic, complemented
2#21022C211yesmaximal, characteristic, complemented
3#20872C211yesmaximal, characteristic
4#20630C211yesmaximal, characteristic
5C2×#1759C211yesmaximal, characteristic
6#11291C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11125C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#10930C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10911C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10879C211yesmaximal, characteristic, complemented
15#10694C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1945—24no—
34#1917—22no—
35#1916—22no—
36#1865C412yescomplemented
37#1864C412yescomplemented
38#1863C412yescomplemented
39#1862C412yescomplemented
40#1798—22no—
41#1797—22no—
42#1759—24no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—41no—
73C4×#41—41no—
74C2×C4×Q8C2×C412yes—
75C2×C4×D4—22no—
76C2×C4×D4C2×C412yes—
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—41no—
88#427—22no—
89#427—22no—
90C2×#107C2×C412yes—
91C2×#106C2×C412yes—
92#424—22no—
93#424—22no—
94#424—22no—
95#424—41no—
96#424—41no—
97#424—41no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104Q814yes—
107#416—22no—
108#416—22no—
109#416—41no—
110#416—41no—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C412yes—
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C412yes—
117#408—22no—
118#408—22no—
119#408—41no—
120#408—41no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C4×M16—22no—
124C4×M16—41no—
125C2×#97—21no—
126C2×#97—22no—
127C2×#97—41no—
128C2×#97—41no—
129C2×C4×C8—22no—
130C2×C4×C8—22no—
131C2×C4×C8—21no—
132C2×C4×C8—41no—
133#389C2×C412yes—
134#389C2×C412yes—
135#385—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138#376—22no—
139#376—41no—
140C2×#95—22no—
141#341—22no—
142#341—41no—
143#341—41no—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—41no—
151#339D414yes—
152#336—22no—
153#336—22no—
154#336—22no—
155#336—22no—
156#336—21no—
157#336—21no—
158#336—22no—
159#336C2×C412yes—
160#336C2×C412yes—
161#336D412yes—
162#336D412yes—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41C2×Q812yes—
174C2×#41#4112yes—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
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×M16C2×D412yes—
207C2×M16C2×Q812yes—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8C22×C412yes—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8#4112yes—
240C22×C8#4112yes—
241C4×Q8—24no—
242C4×Q8—42no—
243C4×D4—24no—
244C4×D4—42no—
245C4×D4—44no—
246C4×D4—44no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—42no—
258#117—42no—
259#117C2×D412yes—
260#117#4112yes—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C22×C412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×Q812yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297#108—44no—
298#107—24no—
299#106—24no—
300C4⋊C8—44no—
301C4⋊C8—44no—
302C4⋊C8—44no—
303#104—28no—
304#104—28no—
305#103—22no—
306#103—22no—
307#103—42no—
308#102—22no—
309#102—22no—
310#102—42no—
311C22⋊C8—44no—
312C22⋊C8—44no—
313#97—42no—
314#97—44no—
315C4×C8—44no—
316C4×C8—42no—
317#95—22no—
318#95—22no—
319#95—44no—
320C24—21no—
321C24C2×#4111yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C4×Q812yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4C4×Q812yes—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—82no—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C42−1+411yescharacteristic
400M16—22no—
401M16—22no—
402M16—22no—
403M16—22no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8#11712yes—
458C2×C8#11712yes—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—82no—
469C4⋊C4C4×D414yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—44no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—44no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42C22⋊Q814yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23C2×C4×D411yescharacteristic
506C23#52911yescharacteristic
507C23#52911yescharacteristic
508C23#54611yescharacteristic
509C23#55111yescharacteristic
510C23#55211yescharacteristic
511C23#55411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—41no—
523D4—41no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—88no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4C2×#11711yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×C4×Q811yescharacteristic
563C2×C4C4×#4111yescharacteristic
564C2×C4#51811yescharacteristic
565C2×C4#52011yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4#52511yescharacteristic
568C2×C4#52911yescharacteristic
569C2×C4#53211yescharacteristic
570C2×C4#53211yescharacteristic
571C2×C4#53311yescharacteristic
572C2×C4#53811yescharacteristic
573C2×C4#53911yescharacteristic
574C2×C4#54011yescharacteristic
575C2×C4#54211yescharacteristic
576C2×C4D4211yescharacteristic
577C2×C4#54611yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4D4×Q811yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55311yescharacteristic
584C2×C4#55311yescharacteristic
585C2×C4#55311yescharacteristic
586C2×C4#55411yescharacteristic
587C2×C4#55411yescharacteristic
588C8—22no—
589C8—22no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C8—81no—
596C8—81no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—44no—
607C22—81no—
608C22—81no—
609C22—82no—
610C22#229311yescharacteristic
611C22#235811yescharacteristic
612C22#236011yescharacteristic
613C22#245011yescharacteristic
614C22#246011yescharacteristic
615C22#252211yescharacteristic
616C22#253011yescharacteristic
617C22#295512yes—
618C22#295712yes—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—82no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4C4×C22⋊Q811yescharacteristic
629C4#233511yescharacteristic
630C4#234511yescharacteristic
631C4#235211yescharacteristic
632C4#245211yescharacteristic
633C4#246411yescharacteristic
634C4#255611yescharacteristic
635C4#255911yescharacteristic
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—82no—
641C2#1401111yescharacteristic
642C2#2129411yescharacteristic
643C2#2129611yescharacteristic
644C2#2288912yes—
645C2#2298412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1945C2semidirect
2. #21022C2semidirect
6. #11291C2semidirect
7. #11221C2semidirect
12. #10911C2semidirect
13. #10883C2semidirect
14. #10879C2semidirect
15. #10694C2semidirect
36. #1865C4semidirect
37. #1864C4semidirect
38. #1863C4semidirect
39. #1862C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19452
#210222
#208722
#206302
C2×#17592
#112912
#112212
#111252
#110682
#109302
#109292
#109112
#108832
#108792
#106942