Finite Group Databasebeta

← #4307978

Subgroups — #4307978

Order 512.

3324 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic, complemented
3C2×#2618C211yesmaximal, characteristic
4C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#14854C214yesmaximal, complemented
6#14846C214yesmaximal, complemented
7#14789C214yesmaximal, complemented
8#14780C214yesmaximal, complemented
9#14706C212yesmaximal, complemented
10#14702C211yesmaximal, characteristic, complemented
11#14698C211yesmaximal, characteristic
12#14696C212yesmaximal, complemented
13#14347C211yesmaximal, characteristic, complemented
14#14259C212yesmaximal, complemented
15#14256C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2639—28no—
25#2627—28no—
26#2618—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#398C2214yes—
30C2×#398C2212yes—
31C2×#397C2212yes—
32C2×#396C2214yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#394C2212yes—
36C2×#393C2212yes—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4—28no—
50C2×C4×C4⋊C4C2214yes—
51C2×C4×C4⋊C4C2214yes—
52C22×#95C2211yescharacteristic
53C22×#95C2211yescharacteristic
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42—24no—
127C22×C42—24no—
128C22×C42—24no—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42—44no—
132C22×C42—44no—
133C22×C42C2314yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—28no—
138#400—28no—
139#398—28no—
140#398—216no—
141#397—28no—
142#396—216no—
143#395—28no—
144#394—28no—
145#394—28no—
146#393—28no—
147#391—44no—
148#390—44no—
149#388—48no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—28no—
155#385—28no—
156#381—28no—
157#380—24no—
158#380—24no—
159C4×C4⋊C4—216no—
160C4×C4⋊C4—216no—
161C2×#95—24no—
162C2×#95D418yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—42no—
204C22×D4—42no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4#4114yes—
241C23×C4#4114yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2412yes—
255#123—432no—
256C22⋊Q8—416no—
257#121—416no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
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⋊C4—44no—
322C2×C42—232no—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—44no—
335C2×C42—416no—
336C2×C42—416no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95C2×D418yes—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C2×#4112yes—
378C24C2×#4112yes—
379C24C2×#4112yes—
380C24C2×#4112yes—
381C24C2×#4112yes—
382C24C2511yescharacteristic
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—84no—
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—82no—
398C2×D4—82no—
399C2×D4—84no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4#121116yes—
497C22×C4#124116yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C2×#4114yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4114yes—
511C22×C4C2×#4114yes—
512C22×C4H(5,𝔽2)14yes—
513C22×C42−1+414yes—
514C4⋊C4—232no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—416no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—88no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C42—232no—
537C42—48no—
538C42—48no—
539C42—416no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23C2×#12114yes—
580C23#52514yes—
581C23#52514yes—
582C23C2×#12414yes—
583C23#52914yes—
584C23#52914yes—
585C23#53214yes—
586C23C23×D411yescharacteristic
587C23C23×D411yescharacteristic
588C23C22×#4111yescharacteristic
589C23C22×#4111yescharacteristic
590C23C22×#4111yescharacteristic
591C23C22×#4111yescharacteristic
592C23C22×#4111yescharacteristic
593C23C2×H(5,𝔽2)12yes—
594C23C2×2−1+412yes—
595C23#58512yes—
596C23#58512yes—
597Q8—88no—
598D4—88no—
599D4—816no—
600C2×C4—232no—
601C2×C4—232no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4C2×#12114yes—
681C2×C4C2×#12114yes—
682C2×C4C2×#12418yes—
683C2×C4#53918yes—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—28no—
688C22—24no—
689C22—28no—
690C22—24no—
691C22—24no—
692C22—28no—
693C22—28no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—84no—
699C22—82no—
700C22—84no—
701C22—82no—
702C22—82no—
703C22—84no—
704C22—84no—
705C22#285018yes—
706C22C22×#12111yescharacteristic
707C22C2×#52511yescharacteristic
708C22C2×#52511yescharacteristic
709C22C22×#12411yescharacteristic
710C22C2×#52911yescharacteristic
711C22C2×#52911yescharacteristic
712C22C2×#53211yescharacteristic
713C22C2×#53912yes—
714C22#345512yes—
715C22#345912yes—
716C22#346012yes—
717C22#346112yes—
718C22#346112yes—
719C22#348411yescharacteristic
720C22#348511yescharacteristic
721C22#348511yescharacteristic
722C22#348611yescharacteristic
723C22#349212yes—
724C22#349312yes—
725C22#349612yes—
726C22#349712yes—
727C22#350512yes—
728C22#350712yes—
729C4—28no—
730C4—24no—
731C4—24no—
732C4—28no—
733C4—44no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—82no—
738C4—82no—
739C4—84no—
740C4—82no—
741C4—84no—
742C2—28no—
743C2—162no—
744C2#3458412yes—
745C2#3498411yescharacteristic
746C2#3498911yescharacteristic
747C2#3500612yes—
748C2#3502112yes—
749C2#3509111yescharacteristic
750C2#3509211yescharacteristic
751C2#3509311yescharacteristic
752C2#3515311yescharacteristic
753C2#3515812yes—
754C2C2×#285011yescharacteristic
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. C2×#2627C2semidirect
4. C22×C4×C4⋊C4C2semidirect
5. #14854C2semidirect
6. #14846C2semidirect
7. #14789C2semidirect
8. #14780C2semidirect
9. #14706C2semidirect
10. #14702C2semidirect
12. #14696C2semidirect
13. #14347C2semidirect
14. #14259C2semidirect
15. #14256C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C2×#26272
C2×#26182
C22×C4×C4⋊C42
#148542
#148462
#147892
#147802
#147062
#147022
#146982
#146962
#143472
#142592
#142562