Finite Group Databasebeta

← #411955

Subgroups — #411955

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2025C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4C2×#1762C211yesmaximal, characteristic
5#13941C211yesmaximal, characteristic
6#11385C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11066C212yesmaximal
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2552—24no—
20#2552—24no—
21C2×#398—22no—
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2025—28no—
31#1802—24no—
32#1762—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—21no—
43#1439—21no—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—24no—
55C2×#125—42no—
56C2×#125—42no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4Q814yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102D414yes—
92C2×#102Q814yes—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96#398—216no—
97#390—24no—
98#390—42no—
99#389—28no—
100#389—24no—
101#389—24no—
102#384—28no—
103#384—22no—
104#384D414yes—
105#382—22no—
106#382—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D414yes—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—42no—
127C25—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×D412yes—
140C22×D4C2×Q812yes—
141C22×D4C2×Q812yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D412yes—
177C22×C8C2×Q812yes—
178#125—48no—
179#125—48no—
180#125—48no—
181#125—48no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—44no—
185C4×D4—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4C2×Q812yes—
231C2×C4⋊C4#4112yes—
232C2×C4⋊C4#4112yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C4⋊C8—44no—
257#102—28no—
258#102—28no—
259#102—48no—
260#97—44no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—42no—
265#95—42no—
266#95—82no—
267#95—82no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C22×D411yescharacteristic
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4C4⋊Q814yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—48no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4#12412yes—
365C22×C4#12412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C42−1+411yescharacteristic
379C22×C42−1+411yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—416no—
415C4⋊C4—416no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4#12114yes—
427C4⋊C4C22⋊Q814yes—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C42—48no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42C22⋊Q814yes—
437C42C22⋊Q814yes—
438C42#12414yes—
439C42#12414yes—
440C23—24no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—82no—
462C23—82no—
463C23C2×#12011yescharacteristic
464C23C2×C22⋊Q811yescharacteristic
465C23C2×#12411yescharacteristic
466C23D4211yescharacteristic
467C23#54611yescharacteristic
468C23#54811yescharacteristic
469C23D4×Q811yescharacteristic
470C23#57512yes—
471C23#57512yes—
472C23#57612yes—
473C23#57712yes—
474D4—24no—
475D4—24no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—816no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4#39314yes—
535C2×C4#39314yes—
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4C2×#12411yescharacteristic
542C2×C4C2×C4⋊Q811yescharacteristic
543C2×C4#53211yescharacteristic
544C2×C4#53311yescharacteristic
545C2×C4#53311yescharacteristic
546C2×C4#53511yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#54211yescharacteristic
549C2×C4#54211yescharacteristic
550C2×C4#54411yescharacteristic
551C2×C4#54411yescharacteristic
552C2×C4D4211yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54711yescharacteristic
555C2×C4#54711yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4D4×Q811yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55411yescharacteristic
560C2×C4#55511yescharacteristic
561C2×C4#55511yescharacteristic
562C2×C4#55611yescharacteristic
563C2×C4#55611yescharacteristic
564C8—42no—
565C8—42no—
566C8—82no—
567C22—44no—
568C22—48no—
569C22—82no—
570C22—82no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—82no—
578C22C2×#39311yescharacteristic
579C22#240811yescharacteristic
580C22#240911yescharacteristic
581C22#243911yescharacteristic
582C22#244011yescharacteristic
583C22#249111yescharacteristic
584C22#249211yescharacteristic
585C22#300511yescharacteristic
586C22#300511yescharacteristic
587C22#300811yescharacteristic
588C22#301011yescharacteristic
589C22#308012yes—
590C22#308112yes—
591C22#313612yes—
592C22#314212yes—
593C22#326712yes—
594C22#326812yes—
595C22#330712yes—
596C22#330912yes—
597C22#337012yes—
598C22#337112yes—
599C22#338812yes—
600C22#338912yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4—162no—
610C4#243911yescharacteristic
611C4#244011yescharacteristic
612C4#249111yescharacteristic
613C4#249211yescharacteristic
614C4#254711yescharacteristic
615C4#256311yescharacteristic
616C4#262811yescharacteristic
617C4#285711yescharacteristic
618C2—84no—
619C2#1512511yescharacteristic
620C2#2345311yescharacteristic
621C2#2345611yescharacteristic
622C2#2424911yescharacteristic
623C2#2425011yescharacteristic
624C2#2433911yescharacteristic
625C2#2434011yescharacteristic
626C2#2603712yes—
627C2#2605912yes—
628C2#2777912yes—
629C2#2782912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2552C2semidirect
6. #11385C2semidirect
7. #11292C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20252
C2×#18022
C2×#17622
#139412
#113852
#112922
#110662
#108502
#108432
#108352