Finite Group Databasebeta

← #367502

Subgroups — #367502

Order 512.

1227 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21165C211yesmaximal, characteristic, complemented
2#20885C211yesmaximal, characteristic
3C2×#1861C211yesmaximal, characteristic
4#20863C211yesmaximal, characteristic
5#20655C211yesmaximal, characteristic
6C2×C4×#104C211yesmaximal, characteristic
7#20343C211yesmaximal, characteristic
8#11341C211yesmaximal, characteristic, complemented
9#11289C211yesmaximal, characteristic, complemented
10#11215C211yesmaximal, characteristic, complemented
11#11212C211yesmaximal, characteristic, complemented
12#10897C211yesmaximal, characteristic, complemented
13#10878C211yesmaximal, characteristic, complemented
14#10839C211yesmaximal, characteristic, complemented
15#10772C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17#3428C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C2×#430C2211yescharacteristic
21#2911C2211yescharacteristic
22#2905C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C22×#104C2211yescharacteristic
27C22×#104C2211yescharacteristic
28#2896C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2296C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1946—22no—
34#1945—22no—
35#1866—24no—
36#1861—24no—
37#1860C414yescomplemented
38#1859C414yescomplemented
39#1806—24no—
40C4×#104—24no—
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43#1742C2211yescharacteristic
44#1740C2211yescharacteristic
45#1733—22no—
46#1733C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339—22no—
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1728C2211yescharacteristic
61C22×#41C2311yescharacteristic
62#568—22no—
63#568—22no—
64#568—22no—
65#568—41no—
66C2×#131—22no—
67C2×#131—41no—
68C22×M16—22no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75#520—22no—
76#520C2×C414yes—
77#519—21no—
78#518—21no—
79#518—41no—
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C4×#41—41no—
84C4×#41—41no—
85C4×#41Q814yes—
86C2×C4×Q8—21no—
87C2×C4×D4—21no—
88C2×C4×D4—41no—
89C2×#117—21no—
90C2×#117—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C42—21no—
97C22×C42—41no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#430—22no—
102#430—22no—
103#430—41no—
104#430—41no—
105C2×#108—22no—
106C2×#108—22no—
107#428—22no—
108#428C2×C414yescomplemented
109#427—22no—
110#424—22no—
111#424—22no—
112#424—22no—
113#424—41no—
114#423—22no—
115#423—22no—
116#423—22no—
117#423—41no—
118C2×C4⋊C8—41no—
119#421—22no—
120#421—22no—
121C2×#104—22no—
122C2×#104—22no—
123C2×#104—22no—
124C2×#104C2×C412yes—
125C2×#104C2×C412yes—
126C2×#104D414yes—
127C2×#104D414yes—
128C2×#104Q814yes—
129#419—21no—
130#418—21no—
131#417C2×C414yes—
132#416—22no—
133C4×M16—22no—
134C4×M16—41no—
135C2×#97—21no—
136C2×C4×C8—21no—
137#389—22no—
138#385—41no—
139#384C2×C412yescomplemented
140#382C2×C412yescomplemented
141#380—41no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—41no—
144C43—22no—
145C43—22no—
146#344—22no—
147#344—41no—
148#343C2×C414yescomplemented
149#342—41no—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—22no—
154#339—22no—
155#339—22no—
156#339—22no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—22no—
162#339—22no—
163#339C2×C412yescomplemented
164#339C2×C412yescomplemented
165#339D412yes—
166#339D412yes—
167#336—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41C22×C412yes—
175C2×#41C2×D412yes—
176C2×#41C2×D412yes—
177C2×#41C2×Q812yes—
178C2×#41C2×Q812yes—
179C2×#41#4112yes—
180C22×Q8#4111yescharacteristic
181C22×D4—21no—
182C22×D4#4111yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—41no—
188C23×C4C2411yescharacteristic
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—21no—
200C2×M16—28no—
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—42no—
224C2×M16—44no—
225C2×M16C22×C412yes—
226C2×M16C22×C412yes—
227C2×M16C2×D412yes—
228C2×M16C2×D412yes—
229C2×M16C2×D412yes—
230C2×M16C2×Q812yes—
231C2×M16#4112yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—42no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—42no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—42no—
249C4×D4—44no—
250C4×D4—44no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—24no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—24no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—42no—
271#117—42no—
272#117—42no—
273#117C22×C412yes—
274#117C2×D412yes—
275#117C2×Q812yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4#4112yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—41no—
312C2×C42—44no—
313C2×C42—48no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×Q812yes—
321C2×C42C2×Q812yes—
322C2×C42#4112yes—
323#108—44no—
324#108—44no—
325#107—24no—
326#106—24no—
327C4⋊C8—44no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330#104—28no—
331#104—44no—
332#104C4⋊C418yescomplemented
333#104C4⋊C418yescomplemented
334#103—24no—
335#102—24no—
336#97—42no—
337C4×C8—42no—
338#95—42no—
339#95—42no—
340C24—21no—
341C24H(5,𝔽2)11yescharacteristic
342#41—22no—
343#41—22no—
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—22no—
353#41—22no—
354#41C2×C4⋊C414yes—
355#41C4×Q814yes—
356#41#12114yes—
357#41C4⋊Q814yes—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8C4×D412yes—
362C2×Q8C22⋊Q812yes—
363C2×Q8#12512yes—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4C4×D412yes—
374C2×D4C22⋊Q812yes—
375C2×D4#12512yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—84no—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C42−1+411yescharacteristic
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—48no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16C2×C4⋊C414yes—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—48no—
452C2×C8—41no—
453C2×C8—41no—
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—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C4×D414yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—22no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—44no—
503C42—44no—
504C42—416no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42C2×C4⋊C412yes—
515C42C2×C4⋊C412yes—
516C42C22⋊Q812yes—
517C42C22⋊Q812yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—42no—
522C23—82no—
523C23#51811yescharacteristic
524C23#53911yescharacteristic
525C23#54311yescharacteristic
526C23#54611yescharacteristic
527C23#54611yescharacteristic
528C23#54611yescharacteristic
529C23#55411yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#38914yes—
534D4—24no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4#38914yes—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—88no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—164no—
578C2×C4C22×C4⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×C4×Q811yescharacteristic
582C2×C4C4×#4111yescharacteristic
583C2×C4#52011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4C2×#12511yescharacteristic
589C2×C4#52911yescharacteristic
590C2×C4C2×C4⋊Q811yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53811yescharacteristic
593C2×C4#54411yescharacteristic
594C2×C4D4211yescharacteristic
595C2×C4D4211yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4D4×Q811yescharacteristic
601C2×C4#55111yescharacteristic
602C2×C4#55111yescharacteristic
603C2×C4#55211yescharacteristic
604C2×C4#55411yescharacteristic
605C2×C4#55411yescharacteristic
606C8—24no—
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C8—81no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22#232911yescharacteristic
624C22#234911yescharacteristic
625C22#235811yescharacteristic
626C22#242511yescharacteristic
627C22#243711yescharacteristic
628C22#247311yescharacteristic
629C22#254911yescharacteristic
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—161no—
641C4C2×#38911yescharacteristic
642C4C4×C22⋊Q811yescharacteristic
643C4D4×C4⋊C411yescharacteristic
644C4#235911yescharacteristic
645C4#242411yescharacteristic
646C4#243811yescharacteristic
647C4#244011yescharacteristic
648C4#250511yescharacteristic
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—21no—
654C2—21no—
655C2—44no—
656C2#1400411yescharacteristic
657C2#2289912yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2358
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
1. #21165C2semidirect
8. #11341C2semidirect
9. #11289C2semidirect
10. #11215C2semidirect
11. #11212C2semidirect
12. #10897C2semidirect
13. #10878C2semidirect
14. #10839C2semidirect
37. #1860C4semidirect
38. #1859C4semidirect
108. #428C2×C4semidirect
139. #384C2×C4semidirect
140. #382C2×C4semidirect
148. #343C2×C4semidirect
163. #339C2×C4semidirect
164. #339C2×C4semidirect
332. #104C4⋊C4semidirect
333. #104C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211652
#208852
C2×#18612
#208632
#206552
C2×C4×#1042
#203432
#113412
#112892
#112152
#112122
#108972
#108782
#108392
#107722