Finite Group Databasebeta

← #4397514

Subgroups — #4397514

Order 512.

3192 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2436C211yesmaximal, characteristic
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2332C211yesmaximal, characteristic, complemented
5#16806C212yesmaximal, complemented
6#16505C212yesmaximal, complemented
7#15855C214yesmaximal, complemented
8#15411C214yesmaximal, complemented
9#15322C214yesmaximal, complemented
10#15301C214yesmaximal, complemented
11#15097C212yesmaximal, complemented
12#15070C212yesmaximal, complemented
13#14336C211yesmaximal, characteristic, complemented
14#14290C211yesmaximal, characteristic
15#13953C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22#2816—28no—
23#2436—28no—
24#2434—28no—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#393C2212yes—
33C2×#393C2214yes—
34#2332—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#386C2214yes—
45C2×#385C2214yes—
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#381C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1446C2214yes—
56#1445C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78#1434C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—44no—
83C23×Q8—44no—
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×#124—24no—
96C2×#124—24no—
97C2×#124—44no—
98C2×#124—44no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8D418yes—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—28no—
133#398—28no—
134#396—48no—
135#395—28no—
136#395—28no—
137#394—28no—
138#394—216no—
139#393—28no—
140#393—216no—
141#391—28no—
142#391—24no—
143#391—24no—
144#391—28no—
145#388—28no—
146#388—28no—
147#388—48no—
148#387—28no—
149#387—28no—
150#386—216no—
151#385—216no—
152#381—24no—
153#381—24no—
154#381—44no—
155#381—44no—
156C4×C4⋊C4—28no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4D418yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—432no—
207C22×Q8—416no—
208C22×Q8—416no—
209C22×Q8—432no—
210C22×Q8—416no—
211C22×Q8C2×D414yes—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—44no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4#4114yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C4⋊Q8—432no—
248#124—416no—
249#124—416no—
250C22⋊Q8—232no—
251#121—416no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D418yes—
313C2×C22⋊C4C2×D418yes—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C2×D418yes—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C2×#4112yes—
367C24C2511yescharacteristic
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—4128no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—84no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—88no—
399C2×D4—84no—
400C2×D4—84no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
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—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
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—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4#121116yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+418yes—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C22⋊C4—82no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—48no—
514C42—84no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×#12114yes—
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+414yes—
558C23#58512yes—
559C23#58512yes—
560Q8—416no—
561Q8—416no—
562Q8—432no—
563Q8—88no—
564D4—88no—
565D4—816no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—432no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
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×C4C2×#12114yes—
642C2×C4C2×#12114yes—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—44no—
647C22—44no—
648C22—48no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22C22×#12111yescharacteristic
661C22#344912yes—
662C22#348411yescharacteristic
663C22#348411yescharacteristic
664C22#348412yes—
665C22#348612yes—
666C22#349012yes—
667C22#349912yes—
668C22#350012yes—
669C22#350912yes—
670C22#351011yescharacteristic
671C22#351011yescharacteristic
672C22#354114yes—
673C22#354714yes—
674C22#355414yes—
675C22#356114yes—
676C4—24no—
677C4—24no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—84no—
688C4—82no—
689C2—44no—
690C2—162no—
691C2#3532711yescharacteristic
692C2#3534811yescharacteristic
693C2#3566711yescharacteristic
694C2#4050312yes—
695C2#4231112yes—
696C2#4645512yes—
697C2#4703212yes—
698C2#4794914yes—
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×#2434C2semidirect
4. C2×#2332C2semidirect
5. #16806C2semidirect
6. #16505C2semidirect
7. #15855C2semidirect
8. #15411C2semidirect
9. #15322C2semidirect
10. #15301C2semidirect
11. #15097C2semidirect
12. #15070C2semidirect
13. #14336C2semidirect
15. #13953C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#24362
C2×#24342
C2×#23322
#168062
#165052
#158552
#154112
#153222
#153012
#150972
#150702
#143362
#142902
#139532