Finite Group Databasebeta

← #377000

Subgroups — #377000

Order 512.

1089 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2961C211yesmaximal, characteristic, complemented
2#23451C211yesmaximal, characteristic
3C2×#1938C211yesmaximal, characteristic
4C2×#1937C211yesmaximal, characteristic
5#21047C211yesmaximal, characteristic
6#11386C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11053C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10887C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10882C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17#2961—24no—
18#2961—24no—
19C2×#446—22no—
20C2×#446—22no—
21C2×#424C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C22×#104C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#408C2211yescharacteristic
30C2×#408C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2297C2211yescharacteristic
37#2026C412yescomplemented
38#2025C412yescomplemented
39#2024C412yescomplemented
40#2023C412yescomplemented
41#1938—24no—
42#1937—24no—
43#1924—22no—
44#1924—22no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72#533—22no—
73#533—22no—
74#533—22no—
75#533—22no—
76C2×C4⋊Q8C2×C412yes—
77C2×#125C2×C412yes—
78C2×C22⋊Q8—22no—
79C2×C4×Q8—22no—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#446—216no—
87#446—216no—
88#424—22no—
89#424—22no—
90#424—22no—
91#424—22no—
92#424D414yes—
93#424D414yes—
94#423—22no—
95#423—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—41no—
106C2×#104—41no—
107#416—22no—
108#416—22no—
109#416—22no—
110#416—22no—
111C2×#103C2×C412yes—
112C2×#103C2×C412yes—
113C2×#102C2×C412yes—
114C2×#102C2×C412yes—
115#408—22no—
116#408—22no—
117#408—22no—
118#408—22no—
119C2×C22⋊C8—22no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122#405—22no—
123#405—22no—
124#405—22no—
125#405—22no—
126C2×#97—22no—
127C2×#97—22no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—22no—
131C2×#97—22no—
132C2×#97—41no—
133C2×#97D412yes—
134C2×#97D412yes—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—22no—
138C2×C4×C8—22no—
139C2×C4×C8—41no—
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#386—22no—
145C4×C22⋊C4—22no—
146C2×#95—22no—
147#341—22no—
148#341—22no—
149#341—41no—
150#341—41no—
151#340C2×C412yes—
152#340C2×C412yes—
153#340C2×C412yes—
154#340C2×C412yes—
155#339—22no—
156#339—22no—
157#339—22no—
158#339—22no—
159#339—22no—
160#339—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336C2×C412yes—
166#336C2×C412yes—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C22×Q8—22no—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
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×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16C2×D412yes—
211C2×M16C2×D412yes—
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×C8—21no—
235C22×C8—21no—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C22×C8C2×D412yes—
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8#4112yes—
244C22×C8#4112yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247#125—24no—
248#125—24no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117C2×D412yes—
262#117C2×D412yes—
263#117#4112yes—
264#117#4112yes—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C412yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
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—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C4⋊C8—28no—
299C4⋊C8—28no—
300C4⋊C8—28no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303#104—44no—
304#104—44no—
305#103—24no—
306#103—24no—
307#102—24no—
308#102—24no—
309C22⋊C8—44no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312#97—24no—
313#97—24no—
314#97—24no—
315#97—24no—
316#97—44no—
317C4×C8—24no—
318C4×C8—24no—
319C4×C8—44no—
320#95—22no—
321#95—22no—
322#95—44no—
323C24—21no—
324C24C2×#4111yescharacteristic
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341C2×Q8—28no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—82no—
349C2×Q8—82no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C42−1+411yescharacteristic
401C22×C42−1+411yescharacteristic
402C22×C42−1+411yescharacteristic
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
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—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
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—41no—
458C2×C8—41no—
459C2×C8#12014yes—
460C2×C8#12412yes—
461C2×C8#12412yes—
462C4⋊C4—216no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C4×D414yes—
471C4⋊C4C4×D414yes—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23C2×C4×D411yescharacteristic
508C23#52511yescharacteristic
509C23#52511yescharacteristic
510C23#52911yescharacteristic
511C23#52911yescharacteristic
512C23#54111yescharacteristic
513C23#54211yescharacteristic
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—88no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—81no—
524D4—81no—
525D4—81no—
526D4—81no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C4×#4111yescharacteristic
565C2×C4C4×#4111yescharacteristic
566C2×C4#51911yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4C2×#12011yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4C2×#12411yescharacteristic
571C2×C4#53211yescharacteristic
572C2×C4#53611yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54711yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#54811yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55611yescharacteristic
585C2×C4#55611yescharacteristic
586C2×C4#55611yescharacteristic
587C2×C4#55611yescharacteristic
588C2×C4#55911yescharacteristic
589C2×C4#55911yescharacteristic
590C8—24no—
591C8—22no—
592C8—22no—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—42no—
597C8—81no—
598C8—81no—
599C22—22no—
600C22—22no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—81no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22C4×#12011yescharacteristic
613C22C4×#12411yescharacteristic
614C22#232311yescharacteristic
615C22#245811yescharacteristic
616C22#245811yescharacteristic
617C22#246911yescharacteristic
618C22#246911yescharacteristic
619C22#295612yes—
620C22#295712yes—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4#236211yescharacteristic
631C4#236211yescharacteristic
632C4#236711yescharacteristic
633C4#236711yescharacteristic
634C4#243011yescharacteristic
635C4#243611yescharacteristic
636C4#259211yescharacteristic
637C4#260411yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—161no—
643C2#1418811yescharacteristic
644C2#2146812yes—
645C2#2165911yescharacteristic
646C2#2166211yescharacteristic
647C2#2213212yes—
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 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2961C2semidirect
6. #11386C2semidirect
7. #11269C2semidirect
11. #10887C2semidirect
12. #10887C2semidirect
13. #10883C2semidirect
14. #10882C2semidirect
15. #10696C2semidirect
37. #2026C4semidirect
38. #2025C4semidirect
39. #2024C4semidirect
40. #2023C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
#234512
C2×#19382
C2×#19372
#210472
#113862
#112692
#110532
#109342
#108872
#108832
#108822
#106962