Finite Group Databasebeta

← #674813

Subgroups — #674813

Order 512.

2806 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic
3C2×#2492C212yesmaximal, complemented
4#15631C214yesmaximal
5#15509C214yesmaximal, complemented
6#15236C212yesmaximal, complemented
7#15125C212yesmaximal, complemented
8#15004C214yesmaximal, complemented
9#14232C214yesmaximal, complemented
10#13942C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic
12#13875C212yesmaximal, complemented
13#13804C211yesmaximal, characteristic, complemented
14C4×#391C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2862—28no—
21#2536—28no—
22#2492—216no—
23C2×#400C2214yes—
24C2×#398C2212yes—
25C2×#394C2212yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2214yes—
33C2×#391C2211yescharacteristic
34C2×#390C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C43C2211yescharacteristic
49#1446C2214yes—
50#1445C2212yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1437C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76C4×#95C2212yes—
77C4×#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2314yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—44no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—44no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122#400—216no—
123#398—28no—
124#394—28no—
125#393—216no—
126#393—216no—
127#391—28no—
128#391—24no—
129#391—216no—
130#391—216no—
131#391—216no—
132#391D418yescomplemented
133#390—28no—
134#388—216no—
135#388—216no—
136#382—28no—
137#382—24no—
138#382D418yes—
139#380—28no—
140#380—28no—
141#380—44no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—28no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C43—24no—
175C25C2412yes—
176C22×Q8—22no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—28no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—28no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D414yes—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—42no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C4⋊Q8—432no—
217C22⋊Q8—432no—
218C4×Q8—416no—
219C4×D4—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4#4118yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42#4114yes—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328#95C2×D418yes—
329#95C2×D4116yes—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24C2×#4114yes—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—416no—
356C2×Q8—44no—
357C2×Q8—88no—
358C2×Q8—88no—
359C2×Q8—88no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D418yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+414yes—
459C22×C42−1+412yes—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—28no—
480C42—28no—
481C42—48no—
482C42—48no—
483C42—48no—
484C42—48no—
485C42—84no—
486C42—88no—
487C42#124116yes—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+412yes—
532C23#58512yes—
533C23#58512yes—
534Q8—416no—
535Q8—832no—
536D4—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—816no—
616C2×C4C2×#12414yes—
617C2×C4C2×#12414yes—
618C2×C4C2×#12414yes—
619C2×C4#53214yes—
620C2×C4#54214yes—
621C2×C4D4214yes—
622C2×C4#54714yes—
623C2×C4#54818yes—
624C2×C4#54814yes—
625C2×C4#55514yes—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C22×#12411yescharacteristic
642C22C2×#53211yescharacteristic
643C22#344411yescharacteristic
644C22C2×#54211yescharacteristic
645C22#345511yescharacteristic
646C22C2×D4211yescharacteristic
647C22C2×#54711yescharacteristic
648C22C2×#54811yescharacteristic
649C22C2×#54812yes—
650C22D4×#4112yes—
651C22C2×#55511yescharacteristic
652C22#347512yes—
653C22#347512yes—
654C22#348912yes—
655C22#350012yes—
656C22#350712yes—
657C22#350712yes—
658C22#350912yes—
659C22#350911yescharacteristic
660C22#351011yescharacteristic
661C22#351412yes—
662C22#356112yes—
663C22#356512yes—
664C4—24no—
665C4—28no—
666C4—24no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—48no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4#249118yes—
678C2—84no—
679C2C2×#249111yescharacteristic
680C2#3481311yescharacteristic
681C2#3562811yescharacteristic
682C2#3615111yescharacteristic
683C2#3619511yescharacteristic
684C2#3653412yes—
685C2#3937411yescharacteristic
686C2#3950111yescharacteristic
687C2#4000112yes—
688C2#5913012yes—
689C2#5913812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2862C2semidirect
3. C2×#2492C2semidirect
5. #15509C2semidirect
6. #15236C2semidirect
7. #15125C2semidirect
8. #15004C2semidirect
9. #14232C2semidirect
10. #13942C2semidirect
12. #13875C2semidirect
13. #13804C2semidirect
14. C4×#391C2semidirect
132. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#25362
C2×#24922
#156312
#155092
#152362
#151252
#150042
#142322
#139422
#139052
#138752
#138042
C4×#3912
C4×#3902