Finite Group Databasebeta

← #401924

Subgroups — #401924

Order 512.

1528 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
1C2×#2367C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic, complemented
5#11202C211yesmaximal, characteristic, complemented
6#11121C212yesmaximal
7#11077C211yesmaximal, characteristic, complemented
8#11009C212yesmaximal
9#10908C212yesmaximal, complemented
10#10789C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C2×#400—21no—
22#2367—24no—
23#2367—24no—
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#2096—24no—
34#1802—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124Q814yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102—42no—
102#400—44no—
103#391—22no—
104#391D414yes—
105#390—22no—
106#390—22no—
107#388—28no—
108#388—28no—
109#385—28no—
110#384—22no—
111#384—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C22⋊C4—28no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—42no—
131#336—22no—
132#336—24no—
133#336—42no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q812yes—
144C22×Q8#4112yes—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C2×D412yes—
153C22×D4C2×Q812yes—
154C22×D4#4112yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8#4114yes—
186#124—28no—
187#124—28no—
188#124—28no—
189#124—28no—
190C4×Q8—48no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—48no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—82no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—82no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260#106—48no—
261C4⋊C8—44no—
262#103—48no—
263#103—48no—
264#102—48no—
265#102—48no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—42no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—82no—
276#95—82no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—42no—
281C24—42no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8C22⋊Q814yes—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4C22⋊Q814yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4C22⋊Q812yes—
387C22×C4C22⋊Q812yes—
388C22×C4#13512yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4#13712yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4#12314yes—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—84no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—24no—
446C42—48no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—84no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23C2×#12011yescharacteristic
475C23C2×C22⋊Q811yescharacteristic
476C23#52511yescharacteristic
477C23#54612yes—
478C23#55012yes—
479C23C2×#13511yescharacteristic
480C23C2×AGL(1,ℤ/8)11yescharacteristic
481C23C2×#13711yescharacteristic
482C23#57511yescharacteristic
483C23#57712yes—
484C23#57712yes—
485Q8—28no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—44no—
490Q8—42no—
491D4—24no—
492D4—24no—
493D4—42no—
494D4—42no—
495D4—44no—
496D4—44no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—816no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#44914yes—
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52912yes—
565C2×C4#52912yes—
566C2×C4#53211yescharacteristic
567C2×C4#53411yescharacteristic
568C2×C4#53511yescharacteristic
569C2×C4#53812yes—
570C2×C4#54012yes—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4#55112yes—
574C2×C4#55312yes—
575C2×C4#55412yes—
576C2×C4#55412yes—
577C8—44no—
578C8—82no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—82no—
589C22—82no—
590C22#240511yescharacteristic
591C22#243111yescharacteristic
592C22#245412yes—
593C22#246712yes—
594C22#284811yescharacteristic
595C22C2×#44911yescharacteristic
596C22#300711yescharacteristic
597C22#300911yescharacteristic
598C22#301011yescharacteristic
599C22#307611yescharacteristic
600C22#307911yescharacteristic
601C22#308412yes—
602C22#310711yescharacteristic
603C22#311511yescharacteristic
604C22#312512yes—
605C22#329612yes—
606C22#330612yes—
607C22#331612yes—
608C22#332612yes—
609C22#339412yes—
610C22#339712yes—
611C22#340012yes—
612C22#340112yes—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—81no—
622C4—162no—
623C4—162no—
624C4#239511yescharacteristic
625C4#245012yes—
626C4#245712yes—
627C4#248711yescharacteristic
628C4#256411yescharacteristic
629C4#263511yescharacteristic
630C2—84no—
631C2#1482511yescharacteristic
632C2#2403811yescharacteristic
633C2#2405211yescharacteristic
634C2#2562811yescharacteristic
635C2#2564111yescharacteristic
636C2#2653712yes—
637C2#2654412yes—
638C2#2707712yes—
639C2#2709012yes—
640C2#3295511yescharacteristic
641C2#3296511yescharacteristic
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
2. C2×#2096C2semidirect
4. #15152C2semidirect
5. #11202C2semidirect
7. #11077C2semidirect
9. #10908C2semidirect
10. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
C2×#20962
C2×#18022
#151522
#112022
#111212
#110772
#110092
#109082
#107892
#107882