Finite Group Databasebeta

← #367511

Subgroups — #367511

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21184C211yesmaximal, characteristic, complemented
2#20867C211yesmaximal, characteristic
3C2×#1861C211yesmaximal, characteristic
4C2×#1861C211yesmaximal, characteristic
5#20675C211yesmaximal, characteristic
6#20405C211yesmaximal, characteristic
7#20360C211yesmaximal, characteristic
8#11341C211yesmaximal, characteristic, complemented
9#11273C211yesmaximal, characteristic, complemented
10#11251C211yesmaximal, characteristic, complemented
11#11244C211yesmaximal, characteristic, complemented
12#10917C211yesmaximal, characteristic, complemented
13#10895C211yesmaximal, characteristic, complemented
14#10856C211yesmaximal, characteristic, complemented
15#10783C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#568C2211yescharacteristic
18#3428C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C2×#430C2211yescharacteristic
21C2×#430C2211yescharacteristic
22C2×#430C2211yescharacteristic
23#2911C2211yescharacteristic
24#2905C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C2×#421C2211yescharacteristic
29C2×#421C2211yescharacteristic
30C22×#104C2211yescharacteristic
31#2873C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1953—24no—
34#1861—24no—
35#1861—22no—
36#1861—22no—
37#1861C414yescomplemented
38#1861C414yescomplemented
39#1814—24no—
40#1759—22no—
41C4×#104—22no—
42C2×#344—24no—
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47C2×#344C2211yescharacteristic
48C2×#344C2211yescharacteristic
49C2×#344C2211yescharacteristic
50C2×#344C2211yescharacteristic
51C2×#344C2211yescharacteristic
52C2×#344C2211yescharacteristic
53#1740C2211yescharacteristic
54#1733C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×H(3,ℤ/4)C2211yescharacteristic
61C22×#41C2311yescharacteristic
62#568—22no—
63#568—22no—
64#568—22no—
65#568—22no—
66#568—41no—
67#568—41no—
68#568—41no—
69#568C2×C414yes—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—22no—
73C2×#131—41no—
74C22×M16—22no—
75C22×M16—22no—
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C22×M16C2311yescharacteristic
84#520—22no—
85#520—41no—
86#519Q812yes—
87#518—22no—
88#518Q812yes—
89C4×#41—22no—
90C4×#41—41no—
91C2×#117—22no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#430—22no—
99#430—22no—
100#430—22no—
101#430—22no—
102#430—22no—
103#430—41no—
104#430—41no—
105#430C2×C414yescomplemented
106C2×#108—22no—
107C2×#108—22no—
108C2×#108—22no—
109C2×#108—41no—
110#428—22no—
111#428—41no—
112#427—22no—
113#424—22no—
114#424—41no—
115#423—22no—
116#423—21no—
117#423—21no—
118#423—41no—
119#423—41no—
120#423C2×C412yescomplemented
121#423C2×C412yescomplemented
122C2×C4⋊C8—22no—
123C2×C4⋊C8—41no—
124#421—22no—
125#421—22no—
126#421—22no—
127#421—22no—
128#421C2×C414yes—
129#421D414yes—
130C2×#104—22no—
131C2×#104—22no—
132C2×#104—22no—
133C2×#104C2×C412yes—
134C2×#104C2×C412yes—
135C2×#104D412yes—
136C2×#104D412yes—
137C2×#104Q812yes—
138C2×#104Q812yes—
139#405—22no—
140#405—41no—
141C4×M16—22no—
142C4×C4⋊C4—41no—
143C4×C22⋊C4—22no—
144#376—21no—
145C43—21no—
146#344—22no—
147#344—22no—
148#344—22no—
149#344—22no—
150#344—22no—
151#344—22no—
152#344—22no—
153#344—22no—
154#344C2×C414yescomplemented
155#344D414yes—
156#342—22no—
157#339—22no—
158#339—22no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—41no—
164#339C2×C412yescomplemented
165#339C2×C412yescomplemented
166H(3,ℤ/4)—41no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41C22×C412yes—
174C2×#41C2×D412yes—
175C2×#41C2×D412yes—
176C2×#41C2×Q812yes—
177C2×#41#4112yes—
178C2×#41#4112yes—
179C22×Q8C2×Q811yescharacteristic
180C22×D4—21no—
181C22×D4C2×Q811yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2411yescharacteristic
185#131—28no—
186#131—44no—
187#131—44no—
188#131—44no—
189#131—44no—
190#131—44no—
191#131—44no—
192#131—44no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—24no—
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—28no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—42no—
234C2×M16—44no—
235C2×M16—44no—
236C2×M16—44no—
237C2×M16C22×C412yes—
238C2×M16C22×C412yes—
239C2×M16C22×C412yes—
240C2×M16C22×C412yes—
241C2×M16C2×D412yes—
242C2×M16C2×D412yes—
243C2×M16C2×D412yes—
244C2×M16C2×D412yes—
245C2×M16C2×Q812yes—
246C2×M16#4112yes—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—24no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—42no—
256C22×C8#4112yes—
257C4×Q8—24no—
258C4×Q8—42no—
259C4×D4—24no—
260C4×D4—42no—
261C4×D4—44no—
262C4×D4—44no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—24no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—42no—
280#117—42no—
281#117C22×C412yes—
282#117C2×D412yes—
283#117C2×D412yes—
284#117C2×Q812yes—
285#117C2×Q812yes—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—42no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4#4112yes—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×Q811yescharacteristic
313C2×C42C2×Q811yescharacteristic
314#108—28no—
315#108—44no—
316#108—44no—
317#108—44no—
318#107—42no—
319#106—42no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8—44no—
325#104—28no—
326#104—28no—
327#104—44no—
328#104C4⋊C414yes—
329#104C4⋊C414yes—
330#104C4⋊C414yes—
331#104C4⋊C414yes—
332#97—42no—
333C4×C8—42no—
334#95—42no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41C2×C4⋊C414yes—
350#41C4×D414yes—
351#41#12114yes—
352#41#12514yes—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C4×Q812yes—
357C2×Q8C22⋊Q812yes—
358C2×Q8C4⋊Q812yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C4×Q812yes—
369C2×D4C22⋊Q812yes—
370C2×D4C4⋊Q812yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—82no—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C42−1+411yescharacteristic
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—48no—
420M16—48no—
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—
437M16C2×C4⋊C414yes—
438M16C4×D414yes—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—48no—
453C2×C8—48no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—44no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4C22⋊Q814yes—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—44no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—84no—
505C42C2×C4⋊C411yescharacteristic
506C42C2×C4⋊C411yescharacteristic
507C42C2×C4⋊C411yescharacteristic
508C42C2×C4⋊C411yescharacteristic
509C23—22no—
510C23—22no—
511C23—22no—
512C23—81no—
513C23—161no—
514C23C4×#4111yescharacteristic
515C23C2×C22⋊Q811yescharacteristic
516C23#52911yescharacteristic
517C23#54611yescharacteristic
518C23#54711yescharacteristic
519C23#55111yescharacteristic
520C23#55411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8#38914yes—
525D4—24no—
526D4—24no—
527D4—24no—
528D4—24no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4#38914yes—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—164no—
558C2×C4—161no—
559C2×C4—161no—
560C2×C4—161no—
561C2×C4C22×C4⋊C411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×Q811yescharacteristic
565C2×C4#51811yescharacteristic
566C2×C4#52011yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4C2×#12511yescharacteristic
571C2×C4C2×C4⋊Q811yescharacteristic
572C2×C4#53311yescharacteristic
573C2×C4#53811yescharacteristic
574C2×C4#53911yescharacteristic
575C2×C4#54311yescharacteristic
576C2×C4#54411yescharacteristic
577C2×C4D4211yescharacteristic
578C2×C4D4211yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54611yescharacteristic
582C2×C4#54611yescharacteristic
583C2×C4#54711yescharacteristic
584C2×C4D4×Q811yescharacteristic
585C2×C4#55111yescharacteristic
586C2×C4#55211yescharacteristic
587C2×C4#55411yescharacteristic
588C2×C4#55411yescharacteristic
589C8—24no—
590C8—24no—
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—81no—
597C22—24no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C4×C22⋊Q811yescharacteristic
609C22#234911yescharacteristic
610C22#235911yescharacteristic
611C22#242511yescharacteristic
612C22#243711yescharacteristic
613C22#244011yescharacteristic
614C22#250511yescharacteristic
615C4—22no—
616C4—24no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—161no—
625C4C2×#38911yescharacteristic
626C4#232911yescharacteristic
627C4D4×C4⋊C411yescharacteristic
628C4#235811yescharacteristic
629C4#242411yescharacteristic
630C4#243811yescharacteristic
631C4#247311yescharacteristic
632C4#254911yescharacteristic
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—44no—
641C2#1400411yescharacteristic
subgroupG/H
centerC4#2424
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14004
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21184C2semidirect
8. #11341C2semidirect
9. #11273C2semidirect
10. #11251C2semidirect
11. #11244C2semidirect
12. #10917C2semidirect
13. #10895C2semidirect
14. #10856C2semidirect
37. #1861C4semidirect
38. #1861C4semidirect
105. #430C2×C4semidirect
120. #423C2×C4semidirect
121. #423C2×C4semidirect
154. #344C2×C4semidirect
164. #339C2×C4semidirect
165. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#211842
#208672
C2×#18612
#206752
#204052
#203602
#113412
#112732
#112512
#112442
#109172
#108952
#108562
#107832