Finite Group Databasebeta

← #669716

Subgroups — #669716

Order 512.

3016 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2500C211yesmaximal, characteristic
3C2×#2346C211yesmaximal, characteristic, complemented
4C2×#2296C211yesmaximal, characteristic, complemented
5#15612C214yesmaximal, complemented
6#15411C214yesmaximal, complemented
7#15141C212yesmaximal, complemented
8#15137C212yesmaximal, complemented
9#15074C214yesmaximal, complemented
10#14141C212yesmaximal, complemented
11#14136C211yesmaximal, characteristic
12#14129C211yesmaximal, characteristic, complemented
13#14079C214yesmaximal, complemented
14#14006C212yesmaximal, complemented
15C4×#381C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2854—28no—
22#2500—28no—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#394C2214yes—
26#2346—28no—
27#2296—28no—
28C2×#390C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2211yescharacteristic
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2214yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376C2212yes—
54#1444C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1437C2212yes—
67#1437C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74#1434C2212yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—48no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—28no—
93C2×#117D418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2312yes—
128#398—48no—
129#396—48no—
130#394—216no—
131#390—28no—
132#390—24no—
133#390—24no—
134#390—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—28no—
140#388—216no—
141#388—216no—
142#388—216no—
143#388—216no—
144#386—28no—
145#385—28no—
146#385—28no—
147#384—24no—
148#382—24no—
149#382—28no—
150#381—28no—
151#380D418yes—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—216no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156#376—28no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—44no—
191C22×D4—44no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#123—432no—
221#121—416no—
222#117—232no—
223#117—232no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4#4118yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×D418yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42—416no—
312C2×C42C2×D418yes—
313#95—22no—
314#95—28no—
315#95—22no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335#95#4118yes—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4112yes—
354C24C2511yescharacteristic
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—88no—
364C2×D4—88no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
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—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4#121116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C42−1+414yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—44no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—216no—
485C42—216no—
486C42—416no—
487C42—48no—
488C42—48no—
489C42—44no—
490C42—44no—
491C42—48no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
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—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23C2×#12114yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4112yes—
538C23C22×#4112yes—
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23#58512yes—
542C23#58512yes—
543D4—816no—
544D4—88no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—232no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4C2×#12114yes—
624C2×C4C2×#12114yes—
625C2×C4#52514yes—
626C2×C4#53614yes—
627C2×C4#54718yes—
628C2×C4#55218yes—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22C22×#12111yescharacteristic
648C22C2×#52511yescharacteristic
649C22#344411yescharacteristic
650C22C2×#53611yescharacteristic
651C22#345511yescharacteristic
652C22C2×#54712yes—
653C22D4×#4112yes—
654C22D4×#4112yes—
655C22C2×#55212yes—
656C22#347512yes—
657C22#348312yes—
658C22#348411yescharacteristic
659C22#348711yescharacteristic
660C22#348812yes—
661C22#349412yes—
662C22#351112yes—
663C22#351412yes—
664C22#351512yes—
665C22#352412yes—
666C22#354712yes—
667C22#356512yes—
668C4—24no—
669C4—48no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—48no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#241318yes—
681C2—44no—
682C2—162no—
683C2C2×#241311yescharacteristic
684C2#3474311yescharacteristic
685C2#3556311yescharacteristic
686C2#3623112yes—
687C2#3639112yes—
688C2#3807412yes—
689C2#3986612yes—
690C2#5911512yes—
691C2#5912012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2346C2semidirect
4. C2×#2296C2semidirect
5. #15612C2semidirect
6. #15411C2semidirect
7. #15141C2semidirect
8. #15137C2semidirect
9. #15074C2semidirect
10. #14141C2semidirect
12. #14129C2semidirect
13. #14079C2semidirect
14. #14006C2semidirect
15. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#25002
C2×#23462
C2×#22962
#156122
#154112
#151412
#151372
#150742
#141412
#141362
#141292
#140792
#140062
C4×#3812