Finite Group Databasebeta

← #537740

Subgroups — #537740

Order 512.

3700 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C212yesmaximal, complemented
3C2×#2323C212yesmaximal, complemented
4#14537C212yesmaximal, complemented
5#14524C214yesmaximal, complemented
6#14410C212yesmaximal
7#14232C212yesmaximal, complemented
8#14145C212yesmaximal
9#14011C214yesmaximal, complemented
10#13978C212yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13839C211yesmaximal, characteristic
15#13817C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2409C4116yescomplemented
23C2×#395C2212yes—
24C2×#393C2212yes—
25Q8×C22⋊C4—216no—
26#2323—216no—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1445C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1440C2211yescharacteristic
56#1440C2211yescharacteristic
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2211yescharacteristic
61#1438C2211yescharacteristic
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2212yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C4×#95C2214yes—
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2×C418yes—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8C2×C4116yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4C2×C418yes—
101C2×#117—28no—
102C2×#117—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131#395C2×C4116yes—
132#393C2×C4116yes—
133#391—28no—
134#391—216no—
135#391—216no—
136#391C2×C4116yes—
137#391D4116yescomplemented
138#389—28no—
139#389—216no—
140#388—216no—
141#388—216no—
142#387—216no—
143#386C2×C4116yes—
144#385C2×C4116yes—
145#382—42no—
146#380—28no—
147#380—28no—
148#380—216no—
149#380C2×C418yes—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—42no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—216no—
154C4×C22⋊C4—42no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—264no—
183C22×Q8—24no—
184C22×Q8—28no—
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—416no—
192C22×Q8—416no—
193C22×Q8—432no—
194C22×Q8—416no—
195C22×Q8C22×C418yes—
196C22×Q8C22×C418yes—
197C22×Q8C2×D418yes—
198C22×Q8#4118yes—
199C22×D4—24no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—42no—
203C22×D4C22×C414yes—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C22×C414yes—
217C23×C4C22×C414yes—
218C23×C4C22×C418yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C22⋊Q8—264no—
233C4×Q8—432no—
234C4×Q8—432no—
235C4×D4—416no—
236#117—416no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C4116yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C2×D418yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C418yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C22×C414yes—
314C2×C42C2×D418yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95C22×C418yes—
336#95C22×C4116yes—
337#95C22×C4116yes—
338#95C2×D4116yes—
339#95C2×D4116yes—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C23×C412yes—
356C24C23×C412yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—216no—
360C2×Q8—232no—
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C2×Q8—48no—
370C2×Q8—4128no—
371C2×Q8C4×D4132yes—
372C2×D4—28no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—48no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—232no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C23×C418yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C418yes—
462C22×C4C23×C412yes—
463C22×C4C22×D418yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+414yes—
473C4⋊C4—216no—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—88no—
482C22⋊C4—216no—
483C22⋊C4—232no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—88no—
494C42—28no—
495C42—216no—
496C42—416no—
497C42—416no—
498C42—48no—
499C42—88no—
500C42—84no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23#51914yes—
543C23C24×C411yescharacteristic
544C23C23×D412yes—
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+412yes—
549C23C2×2−1+412yes—
550C23#58512yes—
551C23#58512yes—
552Q8—264no—
553Q8—432no—
554D4—416no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—432no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4C2×C4×D4116yes—
634C2×C4C2×C4×D418yes—
635C2×C4#51814yes—
636C2×C4D4214yes—
637C2×C4#54618yes—
638C2×C4#55114yes—
639C22—24no—
640C22—28no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22C22×C4×D412yes—
658C22C2×#51811yescharacteristic
659C22C2×#51911yescharacteristic
660C22#342811yescharacteristic
661C22C4×2−1+412yes—
662C22C2×D4211yescharacteristic
663C22C2×#54612yes—
664C22C2×#55111yescharacteristic
665C22#348612yes—
666C22#348912yes—
667C22#349712yes—
668C22#350012yes—
669C22#351612yes—
670C22#351612yes—
671C22#352312yes—
672C22#352412yes—
673C22#353811yescharacteristic
674C22#354011yescharacteristic
675C22#354112yes—
676C22#354112yes—
677C22#354712yes—
678C4—216no—
679C4—48no—
680C4—48no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—82no—
689C4—84no—
690C4#235818yes—
691C2—44no—
692C2—82no—
693C2C2×#235811yescharacteristic
694C2#3429112yes—
695C2#3431212yes—
696C2#3437011yescharacteristic
697C2#3441211yescharacteristic
698C2#4524512yes—
699C2#4524912yes—
700C2#4531612yes—
701C2#4532012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C22⋊C4C2semidirect
3. C2×#2323C2semidirect
4. #14537C2semidirect
5. #14524C2semidirect
7. #14232C2semidirect
9. #14011C2semidirect
10. #13978C2semidirect
11. #13973C2semidirect
12. #13969C2semidirect
15. #13817C2semidirect
22. #2409C4semidirect
137. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×Q8×C22⋊C42
C2×#23232
#145372
#145242
#144102
#142322
#141452
#140112
#139782
#139732
#139692
#139052
#138392
#138172