Finite Group Databasebeta

← #366762

Subgroups — #366762

Order 512.

1489 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3091C211yesmaximal, characteristic
2C2×#1871C211yesmaximal, characteristic
3#20895C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic, complemented
5C2×#1826C211yesmaximal, characteristic, complemented
6C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
7C2×#1729C211yesmaximal, characteristic, complemented
8#11376C212yesmaximal, complemented
9#11125C212yesmaximal, complemented
10#10929C212yesmaximal, complemented
11#10859C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C22×#117C2211yescharacteristic
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×#104C2212yes—
27C2×#416C2212yes—
28C2×#408C2211yescharacteristic
29C2×#408C2211yescharacteristic
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C4×C8C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1871—24no—
35#1869—22no—
36#1869—22no—
37#1855—24no—
38#1826—24no—
39C8×C22⋊C4—24no—
40C2×#341C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44#1729—22no—
45#1729C2211yescharacteristic
46#1729C2212yes—
47#1729C2212yes—
48#1729C2211yescharacteristic
49C2×#336—22no—
50C2×#336C2212yes—
51C22×#41C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C22×M16—24no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—42no—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2312yes—
61C23×C8—21no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8D412yes—
67C23×C8D412yes—
68C23×C8C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71#525—22no—
72#525—22no—
73#525—22no—
74#525—22no—
75C2×#123—21no—
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×#121C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×#120—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117C2×C412yes—
90C2×#117C2312yes—
91C2×#117C2311yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C42—21no—
97C22×C42C2312yes—
98#485—28no—
99#484—24no—
100#483—24no—
101#482—28no—
102#481—24no—
103#480—24no—
104C2×#108—24no—
105C2×#108—24no—
106C2×#108—42no—
107#427—24no—
108#427—24no—
109C2×#107C2×C412yes—
110C2×#107C2×C412yes—
111C2×#106C2×C412yes—
112C2×#106C2×C412yes—
113C2×#104—24no—
114C2×#104—24no—
115C2×#104—42no—
116#416—24no—
117#416—24no—
118C2×#103C2×C412yes—
119C2×#103C2×C412yes—
120C2×#102C2×C412yes—
121C2×#102C2×C412yes—
122#408—24no—
123#408—42no—
124#407—22no—
125#407—22no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D414yes—
134C2×C22⋊C8D414yes—
135C2×C4×C8—24no—
136C2×C4×C8—24no—
137C2×C4×C8—42no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4D414yes—
140C2×#95—21no—
141#341—24no—
142#339—24no—
143#339—24no—
144#339—24no—
145#336—22no—
146#336—22no—
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—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—48no—
197C2×M16—48no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16#4114yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8—48no—
223C22×C8—48no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8C22×C412yes—
227C22×C8C22×C412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D414yes—
230C22×C8C2×D414yes—
231C22×C8C2×D412yes—
232C22×C8#4114yes—
233#123—42no—
234#123—42no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#120—42no—
244#120—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249#117—24no—
250#117—24no—
251#117—22no—
252#117—22no—
253#117—22no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—22no—
258#117—21no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117—42no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C2×D414yes—
297#108—48no—
298#107—24no—
299#107—24no—
300#106—24no—
301#106—24no—
302#104—48no—
303#103—24no—
304#103—24no—
305#102—24no—
306#102—24no—
307C22⋊C8—44no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—44no—
311C22⋊C8C22⋊C414yes—
312C22⋊C8C22⋊C414yes—
313C22⋊C8C22⋊C418yes—
314C4×C8—48no—
315#95—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—21no—
320C24—42no—
321C24C4×D411yescharacteristic
322C24C4×D411yescharacteristic
323C24#12112yes—
324C24#12412yes—
325C24C2×#4111yescharacteristic
326#41—42no—
327#41—42no—
328#41—42no—
329#41—42no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—42no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12412yes—
402C22×C4#12712yes—
403C22×C4#12712yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414M16—24no—
415M16—24no—
416M16—48no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421M16—42no—
422M16—42no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—48no—
440C2×C8—48no—
441C2×C8—48no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8C2×C22⋊C414yes—
453C2×C8C2×C22⋊C414yes—
454C2×C8#12314yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C22⋊C4—216no—
468C22⋊C4—42no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—41no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C42—24no—
478C42—24no—
479C42—42no—
480C42—41no—
481C42—42no—
482C42—41no—
483C42—42no—
484C42—42no—
485C42—44no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—42no—
500C23—42no—
501C23—48no—
502C23—48no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23#39012yes—
509C23#39012yes—
510C23#44314yes—
511C23C2×C4×D411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12411yescharacteristic
514C23#52912yes—
515C23#53912yes—
516Q8—42no—
517Q8—42no—
518D4—42no—
519D4—42no—
520D4—84no—
521D4—82no—
522D4—84no—
523D4—82no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4#39012yes—
565C2×C4#39012yes—
566C2×C4C22×C22⋊C411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C4×#4112yes—
569C2×C4#52012yes—
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4C2×#12711yescharacteristic
574C2×C4#53411yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#54612yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#54712yes—
580C2×C4#54712yes—
581C2×C4#54712yes—
582C2×C4#55112yes—
583C2×C4#55212yes—
584C8—24no—
585C8—24no—
586C8—24no—
587C8—42no—
588C8—44no—
589C8—81no—
590C8—81no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—48no—
605C22—48no—
606C22—42no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#197414yes—
614C22C2×#39011yescharacteristic
615C22C4×#12311yescharacteristic
616C22#232211yescharacteristic
617C22#242512yes—
618C22#248812yes—
619C22C2×#44312yes—
620C22#295512yes—
621C4—22no—
622C4—22no—
623C4—48no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4#233912yes—
630C4#235912yes—
631C4#241211yescharacteristic
632C4#241311yescharacteristic
633C4#250111yescharacteristic
634C4#253911yescharacteristic
635C2—22no—
636C2—24no—
637C2—22no—
638C2—42no—
639C2—161no—
640C2#1399611yescharacteristic
641C2C2×#197411yescharacteristic
642C2#2124211yescharacteristic
643C2#2159212yes—
644C2#2209512yes—
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 ◁ C24×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. C2×#1826C2semidirect
6. C2×C8×C22⋊C4C2semidirect
7. C2×#1729C2semidirect
8. #11376C2semidirect
9. #11125C2semidirect
10. #10929C2semidirect
11. #10859C2semidirect
18. C2×#484C4semidirect
19. C2×#481C4semidirect
20. C2×#485C4semidirect
21. C2×#482C4semidirect
22. C2×#483C4semidirect
23. C2×#480C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30912
C2×#18712
#208952
C2×#18552
C2×#18262
C2×C8×C22⋊C42
C2×#17292
#113762
#111252
#109292
#108592