Finite Group Databasebeta

← #729356

Subgroups — #729356

Order 512.

3680 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2436C211yesmaximal, characteristic
2C2×#2618C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic, complemented
4C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#16628C214yesmaximal, complemented
6#15500C214yesmaximal, complemented
7#15321C214yesmaximal, complemented
8#14927C212yesmaximal, complemented
9#14877C214yesmaximal, complemented
10#14729C212yesmaximal, complemented
11#14152C211yesmaximal, characteristic
12#14149C211yesmaximal, characteristic, complemented
13#14019C212yesmaximal, complemented
14#13996C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2618—28no—
24#2436—28no—
25C2×#398C2212yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#395—28no—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#394C2212yes—
33C2×#393C2212yes—
34C2×#393C2212yes—
35Q8×C22⋊C4—28no—
36C2×#391C2214yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2211yescharacteristic
54C22×#95C2212yes—
55#1445C2212yes—
56#1445C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—48no—
82C23×Q8D418yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#124—28no—
94C2×#124—48no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—44no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121—24no—
101C2×C4×Q8—28no—
102C2×C4×Q8—48no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—48no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134#398—28no—
135#396—216no—
136#396—48no—
137#395—28no—
138#395—216no—
139#395—28no—
140#394—28no—
141#393—28no—
142#393—28no—
143#391—28no—
144#391—28no—
145#391—216no—
146#388—48no—
147#386—28no—
148#386—28no—
149#386—28no—
150#386—28no—
151#386—44no—
152#385—44no—
153#381—24no—
154#381—24no—
155#380—28no—
156#380D418yes—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—28no—
160C2×#95—28no—
161C2×#95—24no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—232no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8—432no—
201C22×Q8C2×D414yes—
202C22×Q8C2×D418yes—
203C22×Q8C2×D414yes—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—44no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D418yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253#124—432no—
254#123—432no—
255#121—416no—
256C4×Q8—432no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—432no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D418yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4#4114yes—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C2×#4112yes—
369C24C2×#4112yes—
370C24C2×#4112yes—
371C24H(5,𝔽2)12yes—
372C242−1+412yes—
373C24C2511yescharacteristic
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—216no—
378C2×Q8—28no—
379C2×Q8—216no—
380C2×Q8—216no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—4128no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—84no—
398C2×D4—84no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4#121116yes—
482C22×C4C22×D414yes—
483C22×C4C22×D418yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)18yes—
494C22×C42−1+418yes—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—416no—
503C4⋊C4—88no—
504C22⋊C4—216no—
505C22⋊C4—28no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—416no—
519C42—48no—
520C42—48no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C2×#12114yes—
555C23#52914yes—
556C23#54214yes—
557C23#54614yes—
558C23#54814yes—
559C23#54814yes—
560C23#55014yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)14yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×2−1+411yescharacteristic
569C23C2×2−1+414yes—
570Q8—232no—
571Q8—432no—
572Q8—416no—
573D4—88no—
574D4—816no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—416no—
600C2×C4—432no—
601C2×C4—432no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—88no—
648C2×C4C2×#12114yes—
649C2×C4C2×#12114yes—
650C2×C4#53418yes—
651C22—28no—
652C22—24no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—24no—
657C22—28no—
658C22—28no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—84no—
663C22—84no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22#242618yes—
671C22C22×#12111yescharacteristic
672C22C2×#52911yescharacteristic
673C22C2×#53412yes—
674C22C2×#54211yescharacteristic
675C22C2×#54611yescharacteristic
676C22C2×#54811yescharacteristic
677C22C2×#54811yescharacteristic
678C22C2×#55011yescharacteristic
679C22#348412yes—
680C22#348612yes—
681C22#349012yes—
682C22#349912yes—
683C22#350912yes—
684C22#351114yes—
685C22#352014yes—
686C22#353614yes—
687C22#355214yes—
688C4—24no—
689C4—24no—
690C4—28no—
691C4—48no—
692C4—44no—
693C4—44no—
694C4—48no—
695C4—44no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C2—28no—
702C2—162no—
703C2C2×#242611yescharacteristic
704C2#3498111yescharacteristic
705C2#3566711yescharacteristic
706C2#3574212yes—
707C2#3913312yes—
708C2#3972112yes—
709C2#4054512yes—
710C2#4300712yes—
711C2#4388812yes—
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. C22×#395C2semidirect
4. C2×Q8×C22⋊C4C2semidirect
5. #16628C2semidirect
6. #15500C2semidirect
7. #15321C2semidirect
8. #14927C2semidirect
9. #14877C2semidirect
10. #14729C2semidirect
12. #14149C2semidirect
13. #14019C2semidirect
14. #13996C2semidirect
15. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24362
C2×#26182
C22×#3952
C2×Q8×C22⋊C42
#166282
#155002
#153212
#149272
#148772
#147292
#141522
#141492
#140192
#139962
#137762