Finite Group Databasebeta

← #628175

Subgroups — #628175

Order 512.

2712 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic, complemented
3#15608C214yesmaximal, complemented
4#15600C214yesmaximal, complemented
5#15459C214yesmaximal, complemented
6#15448C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#14945C212yesmaximal, complemented
9#14702C211yesmaximal, characteristic, complemented
10#14698C211yesmaximal, characteristic
11#14281C211yesmaximal, characteristic, complemented
12#14271C211yesmaximal, characteristic
13#14020C212yesmaximal, complemented
14#14019C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2639—28no—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#394C2212yes—
25C2×#393C2212yes—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#388C2214yes—
29C2×#387—28no—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42—24no—
118C22×C42—44no—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2314yes—
122#400—28no—
123#397—28no—
124#394—28no—
125#393—28no—
126#391—44no—
127#390—44no—
128#388—48no—
129#387—216no—
130#387—216no—
131#386—28no—
132#386—28no—
133#386—28no—
134#386—28no—
135#386—44no—
136#386—44no—
137#380—24no—
138#380—24no—
139C4×C22⋊C4—44no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95—24no—
143C2×#95D418yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—42no—
189C22×D4—42no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4#4114yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C22⋊Q8—416no—
239#121—416no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—44no—
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⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—416no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C2×D418yes—
331#95C2×D418yes—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C22×D412yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24H(5,𝔽2)12yes—
353C242−1+412yes—
354C24C2511yescharacteristic
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
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—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4#124116yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)14yes—
482C4⋊C4—232no—
483C4⋊C4—416no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C42—48no—
502C42—48no—
503C42—88no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23#52514yes—
544C23C2×#12414yes—
545C23#53714yes—
546C23#55614yes—
547C23#55614yes—
548C23#55914yes—
549C23#55914yes—
550C23C23×D411yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+411yescharacteristic
558C23#58512yes—
559C23#58512yes—
560C23#58512yes—
561Q8—88no—
562D4—88no—
563D4—816no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
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×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4C2×#12414yes—
645C2×C4C2×#12414yes—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—28no—
654C22—28no—
655C22—24no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—84no—
666C22—82no—
667C22#259218yes—
668C22C2×#52511yescharacteristic
669C22C22×#12411yescharacteristic
670C22C2×#53711yescharacteristic
671C22#345512yes—
672C22#345912yes—
673C22C2×#55611yescharacteristic
674C22C2×#55611yescharacteristic
675C22C2×#55911yescharacteristic
676C22C2×#55911yescharacteristic
677C22#348411yescharacteristic
678C22#348511yescharacteristic
679C22#348511yescharacteristic
680C22#348711yescharacteristic
681C22#349312yes—
682C22#350512yes—
683C22#350512yes—
684C22#350512yes—
685C22#351212yes—
686C22#351212yes—
687C22#353412yes—
688C22#353512yes—
689C22#356012yes—
690C22#356412yes—
691C4—24no—
692C4—24no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—82no—
703C4—84no—
704C2—28no—
705C2—162no—
706C2C2×#259211yescharacteristic
707C2#3498711yescharacteristic
708C2#3509311yescharacteristic
709C2#3537711yescharacteristic
710C2#3564911yescharacteristic
711C2#3670911yescharacteristic
712C2#3697111yescharacteristic
713C2#3975512yes—
714C2#3976412yes—
715C2#3979912yes—
716C2#3980712yes—
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
2. C22×#387C2semidirect
3. #15608C2semidirect
4. #15600C2semidirect
5. #15459C2semidirect
6. #15448C2semidirect
7. #15091C2semidirect
8. #14945C2semidirect
9. #14702C2semidirect
11. #14281C2semidirect
13. #14020C2semidirect
14. #14019C2semidirect
15. #13778C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C22×#3872
#156082
#156002
#154592
#154482
#150912
#149452
#147022
#146982
#142812
#142712
#140202
#140192
#137782