Finite Group Databasebeta

← #415796

Subgroups — #415796

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#11095C214yesmaximal, complemented
8#10969C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#121—21no—
12C22×#121C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#476—24no—
16C2×#476—24no—
17C2×#476—24no—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C22⋊C8C2214yes—
23C22×C4×C8C2211yescharacteristic
24#2563—24no—
25#2563—24no—
26#2537—24no—
27#2537—24no—
28C2×#394—22no—
29C2×#393—22no—
30C2×#390—21no—
31C2×#384C2211yescharacteristic
32C2×#382—21no—
33C2×#380—21no—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990—24no—
38#1954—24no—
39C4×#102—24no—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8—21no—
44C23×D4—22no—
45C23×D4D411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8D414yescomplemented
57C2×#121—28no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121—28no—
61C2×#121—44no—
62C2×#121—44no—
63C2×#121—44no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4D414yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#476—232no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8D414yescomplemented
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×C22⋊C8—28no—
105C2×C22⋊C8—28no—
106C2×C22⋊C8—44no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110#394—216no—
111#393—216no—
112#390—28no—
113#384—22no—
114#384D414yes—
115#382—28no—
116#380—28no—
117#380—28no—
118#380—28no—
119#380—24no—
120#380—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yescomplemented
123C2×#95—21no—
124C2×#95—24no—
125C2×#95—22no—
126#340—28no—
127#340—24no—
128#340—44no—
129C25—21no—
130C25C2×D412yes—
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—82no—
135C22×Q8—82no—
136C22×D4—28no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—48no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—84no—
155C22×D4SD1614yes—
156C22×D4C2×D412yes—
157C22×D4#4114yes—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C2×D414yes—
194C22×C8#4114yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199#121—416no—
200#121—48no—
201#121—432no—
202#121—48no—
203C4×D4—28no—
204C4×D4—28no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—84no—
257C2×C22⋊C4—84no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267#106—216no—
268C4⋊C8—28no—
269#102—216no—
270#102—216no—
271C22⋊C8—416no—
272C4×C8—44no—
273#95—28no—
274#95—216no—
275#95—24no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—41no—
289C24—41no—
290C24—42no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24C2×SD1614yes—
298C24C22×D411yescharacteristic
299C2×Q8—48no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—416no—
318C2×D4—416no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—88no—
331C2×D4—816no—
332C2×D4—816no—
333C2×D4#12414yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C2×SD1612yes—
397C22×C4#13512yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4#13712yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—28no—
414C2×C8—24no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—44no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—28no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—84no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—816no—
446C42—48no—
447C42#12014yes—
448C42#12114yes—
449C42#12114yes—
450C42#12714yes—
451C23—28no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23—82no—
494C23—81no—
495C23—81no—
496C23—82no—
497C23C2×#12011yescharacteristic
498C23C2×#12011yescharacteristic
499C23C2×#12111yescharacteristic
500C23#54612yes—
501C23#54712yes—
502C23C22×SD1611yescharacteristic
503C23C2×#13511yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23#57612yes—
507C23#57712yes—
508Q8—816no—
509D4—24no—
510D4—24no—
511D4—42no—
512D4—44no—
513D4—42no—
514D4—416no—
515D4—44no—
516D4—832no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—816no—
556C2×C4—84no—
557C2×C4#39214yes—
558C2×C4#39214yes—
559C2×C4#44814yes—
560C2×C4#44914yes—
561C2×C4#45014yes—
562C2×C4#45214yes—
563C2×C4#46514yes—
564C2×C4#46814yes—
565C2×C4C2×#12011yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4#52512yes—
569C2×C4C2×#12411yescharacteristic
570C2×C4#52912yes—
571C2×C4C2×#12711yescharacteristic
572C2×C4#53211yescharacteristic
573C2×C4#53511yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#53712yes—
576C2×C4#54212yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#54812yes—
580C2×C4#54812yes—
581C2×C4#54812yes—
582C2×C4#55012yes—
583C8—44no—
584C8—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—84no—
594C22—84no—
595C22—82no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22C2×#39211yescharacteristic
608C22#240712yes—
609C22#242712yes—
610C22#243211yescharacteristic
611C22#243511yescharacteristic
612C22C2×#45011yescharacteristic
613C22C2×#44811yescharacteristic
614C22C2×#44911yescharacteristic
615C22C2×#45211yescharacteristic
616C22#300811yescharacteristic
617C22#301011yescharacteristic
618C22#301212yes—
619C22C2×#46511yescharacteristic
620C22C2×#46811yescharacteristic
621C22#306212yes—
622C22#329512yes—
623C22#330312yes—
624C22#332512yes—
625C22#333112yes—
626C22#333512yes—
627C22#334112yes—
628C22#335412yes—
629C22#335712yes—
630C4—24no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4—84no—
635C4—82no—
636C4#200018yes—
637C4#242612yes—
638C4#243111yescharacteristic
639C4#243611yescharacteristic
640C4#254012yes—
641C4#262011yescharacteristic
642C4#285011yescharacteristic
643C2—84no—
644C2—84no—
645C2#1521511yescharacteristic
646C2C2×#200011yescharacteristic
647C2#2339311yescharacteristic
648C2#2411712yes—
649C2#2411812yes—
650C2#2533912yes—
651C2#2534612yes—
652C2#2569911yescharacteristic
653C2#2570411yescharacteristic
654C2#2588111yescharacteristic
655C2#2588611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2537C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#476C2semidirect
2. C2×#2563C2semidirect
4. C2×#1990C2semidirect
5. C2×#1954C2semidirect
6. C2×C4×#102C2semidirect
7. #11095C2semidirect
8. #10969C2semidirect
9. #10849C2semidirect
56. C2×C4⋊Q8D4semidirect
97. C2×C4⋊C8D4semidirect
122. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#25632
C2×#25372
C2×#19902
C2×#19542
C2×C4×#1022
#110952
#109692
#108492