Finite Group Databasebeta

← #1495563

Subgroups — #1495563

Order 512.

3922 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2731C212yesmaximal, complemented
3C2×#2436C212yesmaximal, complemented
4C2×#2434C211yesmaximal, characteristic, complemented
5#15833C212yesmaximal, complemented
6#15825C214yesmaximal, complemented
7#15453C212yesmaximal, complemented
8#15315C214yesmaximal, complemented
9#15144C212yesmaximal, complemented
10#14928C214yesmaximal, complemented
11#14228C212yesmaximal, complemented
12#14084C214yesmaximal, complemented
13#13826C211yesmaximal, characteristic, complemented
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×#124C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2212yes—
21#2733—28no—
22#2731—216no—
23#2436—216no—
24#2434—28no—
25C2×#396C2214yes—
26C2×#395C2214yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2214yes—
30C2×#393C2214yes—
31C2×#393C2214yes—
32C2×#392C2214yes—
33C2×#392C2212yes—
34C2×#392C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2212yes—
39C2×#390C2214yes—
40C2×#390C2212yes—
41C2×#390C2214yes—
42C2×#390C2212yes—
43C2×#388C2214yes—
44C2×#388C2214yes—
45C2×#388C2214yes—
46C2×#387C2214yes—
47C2×#386C2214yes—
48C2×#386C2214yes—
49C2×#385C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yes—
53#1445C2212yes—
54#1444C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1437C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4×#95C2214yes—
70C4×#95C2212yes—
71C23×Q8—24no—
72C23×Q8—28no—
73C23×Q8—42no—
74C23×Q8—44no—
75C23×Q8—42no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2314yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×#124—28no—
91C2×#124—28no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—44no—
95C2×#124—44no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C22⋊Q8—42no—
99C2×#121—28no—
100C2×#121—44no—
101C2×#121—44no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#396—216no—
130#395—216no—
131#394—28no—
132#394—216no—
133#393—216no—
134#393—216no—
135#393—216no—
136#392—28no—
137#392—216no—
138#392—44no—
139#391—24no—
140#391—28no—
141#391D418yescomplemented
142#391D4116yescomplemented
143#390—216no—
144#390—28no—
145#390—48no—
146#390—44no—
147#388—216no—
148#388—216no—
149#388—216no—
150#387—216no—
151#386—216no—
152#386—216no—
153#385—216no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—28no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2414yes—
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8—42no—
194C22×Q8—432no—
195C22×Q8—416no—
196C22×Q8—432no—
197C22×Q8—416no—
198C22×Q8—416no—
199C22×Q8C2×D418yes—
200C22×Q8C2×D414yes—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—24no—
205C22×D4—28no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4—42no—
223C22×D4—44no—
224C22×D4—42no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2412yes—
248C4⋊Q8—416no—
249C4⋊Q8—416no—
250#124—432no—
251#124—432no—
252C22⋊Q8—416no—
253#121—432no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4#4118yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42C2×D418yes—
330C2×C42C2×D414yes—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—44no—
343#95—42no—
344#95C2×D4116yes—
345#95C2×D4116yes—
346#95C2×D4116yes—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24C2×#4114yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371C2×Q8—216no—
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—4128no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×Q8#127116yes—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—88no—
411C2×D4—84no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—84no—
417C2×D4—88no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4C22×D418yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+414yes—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—84no—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—48no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—88no—
521C42—84no—
522C23—24no—
523C23—28no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—22no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23#55514yes—
573C23C23×D411yescharacteristic
574C23C23×D412yes—
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+412yes—
580C23#58514yes—
581Q8—416no—
582Q8—432no—
583Q8—416no—
584Q8—88no—
585D4—816no—
586D4—816no—
587D4—832no—
588D4—816no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—432no—
619C2×C4—416no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4C2×#12714yes—
654C2×C4C2×#12718yes—
655C2×C4#54818yes—
656C22—28no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22C22×#12711yescharacteristic
680C22C2×#54812yes—
681C22D4×#4114yes—
682C22C2×#55511yescharacteristic
683C22#348412yes—
684C22#348411yescharacteristic
685C22#348411yescharacteristic
686C22#348412yes—
687C22#349714yes—
688C22#350012yes—
689C22#351112yes—
690C22#351414yes—
691C22#354012yes—
692C22#354114yes—
693C22#354411yescharacteristic
694C22#354512yes—
695C4—28no—
696C4—24no—
697C4—28no—
698C4—48no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—48no—
704C4—82no—
705C4—84no—
706C2—44no—
707C2—164no—
708C2#3539011yescharacteristic
709C2#3646512yes—
710C2#4389512yes—
711C2#4390314yes—
712C2#5174911yescharacteristic
713C2#5207612yes—
714C2#5913511yescharacteristic
715C2#5916012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2733C2semidirect
2. C2×#2731C2semidirect
3. C2×#2436C2semidirect
4. C2×#2434C2semidirect
5. #15833C2semidirect
6. #15825C2semidirect
7. #15453C2semidirect
8. #15315C2semidirect
9. #15144C2semidirect
10. #14928C2semidirect
11. #14228C2semidirect
12. #14084C2semidirect
13. #13826C2semidirect
141. #391D4semidirect
142. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#27312
C2×#24362
C2×#24342
#158332
#158252
#154532
#153152
#151442
#149282
#142282
#140842
#138262