Finite Group Databasebeta

← #403021

Subgroups — #403021

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11292C212yesmaximal, complemented
6#11077C211yesmaximal, characteristic, complemented
7#11042C211yesmaximal, characteristic
8#10970C212yesmaximal
9#10806C211yesmaximal, characteristic, complemented
10#10789C212yesmaximal, complemented
11#10731C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#393—21no—
21#2362—24no—
22#2362—24no—
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#387—21no—
28C2×#380C2212yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2024—24no—
33#1987—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8Q814yescomplemented
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100#393—44no—
101#391—22no—
102#391D414yes—
103#389—22no—
104#389—22no—
105#388—28no—
106#388—28no—
107#387—28no—
108#380—24no—
109#380—42no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#336—22no—
133#336—42no—
134C25C2×D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—44no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8C2×D412yes—
148C22×Q8C2×Q812yes—
149C22×Q8#4112yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
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—22no—
186C4⋊Q8—28no—
187C4⋊Q8—48no—
188C4⋊Q8—44no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C4×Q8—48no—
194C4×Q8—44no—
195C4×Q8—44no—
196#117—48no—
197#117—44no—
198#117—44no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
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—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C4⋊C4#4114yes—
233C2×C4⋊C4#4112yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C4⋊C8—44no—
265#103—48no—
266#103—48no—
267C22⋊C8—48no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—44no—
278#95—82no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24C22⋊Q812yes—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—48no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×Q8C22⋊Q814yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4#12012yes—
388C22×C4#12012yes—
389C22×C4C22⋊Q812yes—
390C22×C4#13512yes—
391C22×C4AGL(1,ℤ/8)12yes—
392C22×C4#13712yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)12yes—
403C2×C8—42no—
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—
415C2×C8—42no—
416C2×C8—42no—
417C4⋊C4—216no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4C22⋊Q814yes—
438C4⋊C4#12414yes—
439C22⋊C4—216no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—88no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—48no—
448C42—48no—
449C42—41no—
450C42—41no—
451C42—42no—
452C42—48no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C2×#12011yescharacteristic
477C23C2×C22⋊Q811yescharacteristic
478C23#52511yescharacteristic
479C23#54612yes—
480C23#54612yes—
481C23C2×#13511yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×#13711yescharacteristic
484C23#57511yescharacteristic
485C23#57712yes—
486C23#57812yes—
487Q8—28no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—44no—
493Q8—816no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
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—82no—
559C2×C4—82no—
560C2×C4#45214yes—
561C2×C4C2×C22⋊Q811yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4C2×#12411yescharacteristic
567C2×C4#52912yes—
568C2×C4#52912yes—
569C2×C4#53511yescharacteristic
570C2×C4#53511yescharacteristic
571C2×C4#53812yes—
572C2×C4#53812yes—
573C2×C4#54612yes—
574C2×C4#54612yes—
575C2×C4#55412yes—
576C2×C4#55412yes—
577C2×C4#55512yes—
578C2×C4#55912yes—
579C8—82no—
580C8—82no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22#240411yescharacteristic
593C22#243511yescharacteristic
594C22#245012yes—
595C22#245912yes—
596C22#284811yescharacteristic
597C22C2×#45211yescharacteristic
598C22#300611yescharacteristic
599C22#301011yescharacteristic
600C22#301111yescharacteristic
601C22#307611yescharacteristic
602C22#307911yescharacteristic
603C22#308212yes—
604C22#310611yescharacteristic
605C22#311611yescharacteristic
606C22#312112yes—
607C22#330012yes—
608C22#330312yes—
609C22#330812yes—
610C22#331112yes—
611C22#331212yes—
612C22#331312yes—
613C22#332412yes—
614C22#333112yes—
615C4—24no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—81no—
623C4—82no—
624C4—84no—
625C4—162no—
626C4#239811yescharacteristic
627C4#245012yes—
628C4#245912yes—
629C4#256311yescharacteristic
630C4#258011yescharacteristic
631C4#263111yescharacteristic
632C2—82no—
633C2#1486511yescharacteristic
634C2#2396711yescharacteristic
635C2#2397911yescharacteristic
636C2#2588611yescharacteristic
637C2#2589611yescharacteristic
638C2#2641512yes—
639C2#2643112yes—
640C2#2673312yes—
641C2#2675012yes—
642C2#3295811yescharacteristic
643C2#3296411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2024C2semidirect
3. C2×#1987C2semidirect
5. #11292C2semidirect
6. #11077C2semidirect
9. #10806C2semidirect
10. #10789C2semidirect
61. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#20242
C2×#19872
#151252
#112922
#110772
#110422
#109702
#108062
#107892
#107312