Finite Group Databasebeta

← #366761

Subgroups — #366761

Order 512.

1545 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×#3091C211yesmaximal, characteristic
2C2×#1896C211yesmaximal, characteristic
3#20897C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic, complemented
5#20685C211yesmaximal, characteristic, complemented
6C2×#1749C211yesmaximal, characteristic, complemented
7C22×#339C211yesmaximal, characteristic, complemented
8#11376C212yesmaximal, complemented
9#11125C212yesmaximal, complemented
10#10930C212yesmaximal, complemented
11#10860C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C2×#525C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16#3091—24no—
17#3091—24no—
18C2×#484C411yescharacteristic, complemented
19C2×#481C411yescharacteristic, complemented
20C2×#485C412yescomplemented
21C2×#482C412yescomplemented
22C2×#483C411yescharacteristic, complemented
23C2×#480C411yescharacteristic, complemented
24C22×#108C2212yes—
25C2×#427C2212yes—
26C22×#104C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2212yes—
29C2×#408C2212yes—
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2212yes—
34#1896—24no—
35#1870—22no—
36#1869—22no—
37#1855—24no—
38#1818—22no—
39#1817—22no—
40#1749—24no—
41C2×#341C2212yes—
42C2×#339—24no—
43C2×#339C2211yescharacteristic
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49#1729C2212yes—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—22no—
58C22×M16—42no—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C23×C8—21no—
63C23×C8C2312yes—
64C23×C8C2312yes—
65#525—22no—
66#525—22no—
67#525—22no—
68#525—22no—
69C2×#123—21no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#120—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117C2×C412yes—
84C2×#117C2312yes—
85C2×#117C2311yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C42—24no—
90C22×C42—21no—
91C22×C42—22no—
92C22×C42—21no—
93C22×C42—42no—
94C22×C42D414yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#485—28no—
98#484—24no—
99#483—24no—
100#482—28no—
101#481—24no—
102#480—24no—
103C2×#108—24no—
104C2×#108—24no—
105C2×#108—42no—
106#427—24no—
107#427—24no—
108C2×#107C2×C412yescomplemented
109C2×#107C2×C412yescomplemented
110C2×#106C2×C412yescomplemented
111C2×#106C2×C412yescomplemented
112C2×#104—22no—
113C2×#104—22no—
114C2×#104—24no—
115C2×#104—42no—
116#416—24no—
117#416—24no—
118C2×#103C2×C412yescomplemented
119C2×#103C2×C412yescomplemented
120C2×#102C2×C412yescomplemented
121C2×#102C2×C412yescomplemented
122#408—24no—
123#408—42no—
124#407—22no—
125#407—22no—
126C2×C22⋊C8—22no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—24no—
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C412yes—
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D412yes—
136C2×C22⋊C8D414yes—
137C2×C22⋊C8D414yes—
138C2×#97—24no—
139C2×#97—24no—
140C2×#97—42no—
141#341—24no—
142#339—22no—
143#339—22no—
144#339—24no—
145#339—24no—
146#336—24no—
147#336—24no—
148#336—24no—
149#336—24no—
150C25#4111yescharacteristic
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C22×Q8C22×C411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—24no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C2×M16—22no—
188C2×M16—28no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—44no—
202C2×M16—48no—
203C2×M16—44no—
204C2×M16—42no—
205C2×M16#4114yes—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—44no—
220C22×C8—42no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D414yes—
224C22×C8C2×D414yes—
225C22×C8C2×D414yes—
226#123—42no—
227#123—42no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#120—42no—
237#120—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—42no—
242#117—24no—
243#117—24no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—22no—
251#117—21no—
252#117—42no—
253#117—42no—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—44no—
288C2×C42—48no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42#4114yes—
295#108—48no—
296#107—24no—
297#107—24no—
298#106—24no—
299#106—24no—
300#104—44no—
301#104—44no—
302#103—24no—
303#103—24no—
304#102—24no—
305#102—24no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—48no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8C22⋊C414yes—
312C22⋊C8C22⋊C414yes—
313C22⋊C8C22⋊C418yes—
314#97—48no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—21no—
319C24—42no—
320C24C4×D411yescharacteristic
321C24C4×D411yescharacteristic
322C24#12112yes—
323C24#12412yes—
324C24C2×#4111yescharacteristic
325#41—41no—
326#41—41no—
327#41—42no—
328#41—41no—
329#41—41no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—42no—
334#41—42no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—82no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—432no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—82no—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4#12412yes—
406C22×C4#12712yes—
407C22×C4#12712yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418M16—24no—
419M16—24no—
420M16—48no—
421M16—42no—
422M16—42no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—42no—
428M16—42no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—48no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8C2×C22⋊C414yes—
453C2×C8C2×C22⋊C414yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—42no—
469C22⋊C4—41no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—22no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—48no—
480C42—42no—
481C42—41no—
482C42—42no—
483C42—41no—
484C42—42no—
485C42—44no—
486C42#12314yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—48no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23#39012yes—
508C23#39012yes—
509C23#44414yes—
510C23C2×C4×D411yescharacteristic
511C23C2×#12111yescharacteristic
512C23C2×#12411yescharacteristic
513C23#55112yes—
514C23#55212yes—
515Q8—41no—
516Q8—42no—
517Q8—41no—
518D4—41no—
519D4—42no—
520D4—41no—
521D4—84no—
522D4—81no—
523D4—81no—
524D4—84no—
525D4—82no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—432no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4#39012yes—
568C2×C4#39012yes—
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4112yes—
572C2×C4#52012yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52912yes—
577C2×C4C2×#12711yescharacteristic
578C2×C4#53411yescharacteristic
579C2×C4#53611yescharacteristic
580C2×C4#53912yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#54612yes—
584C2×C4#54712yes—
585C2×C4#54712yes—
586C2×C4#54712yes—
587C8—24no—
588C8—24no—
589C8—44no—
590C8—44no—
591C8—81no—
592C8—81no—
593C22—24no—
594C22—24no—
595C22—28no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—22no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—84no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#197514yes—
614C22C2×#39011yescharacteristic
615C22#235912yes—
616C22#241211yescharacteristic
617C22#241311yescharacteristic
618C22#248812yes—
619C22C2×#44412yes—
620C22#295612yes—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—44no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4C4×#12311yescharacteristic
631C4#232211yescharacteristic
632C4#233912yes—
633C4#242512yes—
634C4#250111yescharacteristic
635C4#253911yescharacteristic
636C2—21no—
637C2—21no—
638C2—24no—
639C2—22no—
640C2—42no—
641C2—161no—
642C2#1399611yescharacteristic
643C2C2×#197511yescharacteristic
644C2#2124111yescharacteristic
645C2#2295014yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1855C2semidirect
5. #20685C2semidirect
6. C2×#1749C2semidirect
7. C22×#339C2semidirect
8. #11376C2semidirect
9. #11125C2semidirect
10. #10930C2semidirect
11. #10860C2semidirect
18. C2×#484C4semidirect
19. C2×#481C4semidirect
20. C2×#485C4semidirect
21. C2×#482C4semidirect
22. C2×#483C4semidirect
23. C2×#480C4semidirect
108. C2×#107C2×C4semidirect
109. C2×#107C2×C4semidirect
110. C2×#106C2×C4semidirect
111. C2×#106C2×C4semidirect
118. C2×#103C2×C4semidirect
119. C2×#103C2×C4semidirect
120. C2×#102C2×C4semidirect
121. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30912
C2×#18962
#208972
C2×#18552
#206852
C2×#17492
C22×#3392
#113762
#111252
#109302
#108602