Finite Group Databasebeta

← #399577

Subgroups — #399577

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2083C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#14697C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic, complemented
7#11357C211yesmaximal, characteristic, complemented
8#11325C212yesmaximal, complemented
9#11208C212yesmaximal, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#385—21no—
20#2285—24no—
21#2285—24no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25C2×#380—21no—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31#2084—24no—
32#2083—24no—
33#1854—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22⋊#95—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—24no—
78C22×C42—22no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8Q818yes—
92#389—22no—
93#389—22no—
94#389—24no—
95#385—44no—
96#384—22no—
97#384—22no—
98#384D414yes—
99#384D414yes—
100#382—24no—
101#382—24no—
102#380—28no—
103#380—28no—
104#380—22no—
105#380—24no—
106#380D414yes—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—24no—
128C25C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×Q814yes—
162C23×C4#4112yes—
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×Q814yes—
178C22×C8C2×Q814yes—
179#117—48no—
180#117—48no—
181#117—48no—
182#117—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C42—216no—
241C2×C42—216no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—28no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4114yes—
259#107—48no—
260#106—48no—
261C22⋊C8—216no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24#12112yes—
285C24#12312yes—
286C24C4⋊Q812yes—
287C24C22×D411yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4#12012yes—
383C22×C4#12012yes—
384C22×C4#12112yes—
385C22×C4C22⋊Q818yes—
386C22×C4#12312yes—
387C22×C4#12312yes—
388C22×C4#12312yes—
389C22×C4#12518yes—
390C22×C4C4⋊Q812yes—
391C22×C4#13512yes—
392C22×C4#13512yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4#13712yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q812yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C42−1+412yes—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—84no—
446C42—28no—
447C42—28no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—24no—
452C42—416no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23#39814yes—
488C23C2×#12011yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×#12311yescharacteristic
491C23C2×#12311yescharacteristic
492C23C2×C4⋊Q811yescharacteristic
493C23#53511yescharacteristic
494C23#53611yescharacteristic
495C23C2×#13511yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×#13711yescharacteristic
499C23#57511yescharacteristic
500C23#57511yescharacteristic
501C23#57511yescharacteristic
502C23#57511yescharacteristic
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4#39814yes—
584C2×C4#44914yes—
585C2×C4#45214yes—
586C2×C4C2×C22⋊Q812yes—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4C2×#12512yes—
590C2×C4#53912yes—
591C2×C4#53912yes—
592C2×C4D4211yescharacteristic
593C2×C4D4211yescharacteristic
594C2×C4#54611yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54712yes—
597C2×C4D4×Q812yes—
598C2×C4D4×Q812yes—
599C2×C4#55212yes—
600C2×C4#55412yes—
601C2×C4#55412yes—
602C8—44no—
603C8—44no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22C2×#39811yescharacteristic
620C22#240811yescharacteristic
621C22#243411yescharacteristic
622C22#248611yescharacteristic
623C22#250211yescharacteristic
624C22#262311yescharacteristic
625C22#284911yescharacteristic
626C22C2×#44911yescharacteristic
627C22C2×#45211yescharacteristic
628C22#300511yescharacteristic
629C22#300511yescharacteristic
630C22#303511yescharacteristic
631C22#303511yescharacteristic
632C22#303611yescharacteristic
633C22#303711yescharacteristic
634C22#309111yescharacteristic
635C22#309111yescharacteristic
636C22#309111yescharacteristic
637C22#309111yescharacteristic
638C22#309411yescharacteristic
639C22#309411yescharacteristic
640C22#309511yescharacteristic
641C22#309511yescharacteristic
642C22#316011yescharacteristic
643C22#316211yescharacteristic
644C22#316511yescharacteristic
645C22#316611yescharacteristic
646C22#321211yescharacteristic
647C22#321211yescharacteristic
648C22#321411yescharacteristic
649C22#321611yescharacteristic
650C22#322811yescharacteristic
651C22#323111yescharacteristic
652C22#324011yescharacteristic
653C22#324011yescharacteristic
654C4—24no—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C4—84no—
663C4—161no—
664C4—161no—
665C4#244912yes—
666C4#244912yes—
667C4#247312yes—
668C4#247312yes—
669C2—44no—
670C2#1478611yescharacteristic
671C2#2365611yescharacteristic
672C2#2365711yescharacteristic
673C2#2421011yescharacteristic
674C2#2421311yescharacteristic
675C2#2581211yescharacteristic
676C2#2581711yescharacteristic
677C2#2765011yescharacteristic
678C2#2765211yescharacteristic
679C2#2795111yescharacteristic
680C2#2795511yescharacteristic
681C2#2991011yescharacteristic
682C2#2991111yescharacteristic
683C2#3298011yescharacteristic
684C2#3298411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#2083C2semidirect
6. #11357C2semidirect
7. #11357C2semidirect
8. #11325C2semidirect
9. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#20842
C2×#20832
C2×#18542
#146972
#113572
#113252
#112082
#107312
#107122