Finite Group Databasebeta

← #848809

Subgroups — #848809

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×C4×#124C211yesmaximal, characteristic
3#16686C212yesmaximal, complemented
4#16630C212yesmaximal, complemented
5#16039C214yesmaximal, complemented
6#15586C214yesmaximal, complemented
7#15201C212yesmaximal, complemented
8#15183C212yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#14703C212yesmaximal, complemented
11#14513C212yesmaximal
12#14473C212yesmaximal
13#14139C212yesmaximal, complemented
14#13922C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2409—28no—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C4×#124—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C43C2211yescharacteristic
46#1446C2214yes—
47#1446C2214yes—
48#1446C2214yes—
49#1445C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124Q818yescomplemented
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×C4×Q8—24no—
97C2×C4×Q8—42no—
98C2×C4×Q8—42no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—44no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#395—28no—
130#393—28no—
131#391—28no—
132#391—24no—
133#391—28no—
134#391—44no—
135#391—44no—
136#391—44no—
137#390—24no—
138#389—44no—
139#388—28no—
140#388—28no—
141#386—28no—
142#386—28no—
143#385—28no—
144#382—24no—
145#380—24no—
146#380—44no—
147#380—42no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C43—24no—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—28no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—41no—
195C22×Q8—41no—
196C22×Q8—432no—
197C22×Q8—416no—
198C22×Q8—416no—
199C22×Q8—432no—
200C22×Q8—416no—
201C22×Q8C2×Q814yescomplemented
202C22×Q8#4118yes—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4C2×D414yes—
208C22×D4C2×Q814yes—
209C22×D4#4114yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243#124—232no—
244C22⋊Q8—432no—
245C4×Q8—416no—
246C4×D4—416no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×Q818yes—
303C2×C22⋊C4C2×Q818yescomplemented
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—41no—
319C2×C42—41no—
320C2×C42C2×D414yes—
321C2×C42C2×Q814yescomplemented
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×Q812yes—
367C24C22×Q812yes—
368C24C2×#4112yes—
369C24C2×#4112yes—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—416no—
385C2×Q8—4128no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4C22⋊Q8116yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
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—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
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—24no—
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—48no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—84no—
483C22×C4—88no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×Q814yes—
492C22×C4C22×Q814yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4118yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C42−1+414yes—
501C4⋊C4—44no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—88no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—88no—
520C42—28no—
521C42—28no—
522C42—48no—
523C42—44no—
524C42—48no—
525C42—48no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×C22⋊Q814yes—
571C23C2×C22⋊Q814yes—
572C23C23×D411yescharacteristic
573C23C23×Q811yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+412yes—
580C23#58511yescharacteristic
581C23#58511yescharacteristic
582C23#58512yes—
583Q8—416no—
584Q8—432no—
585Q8—432no—
586D4—416no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
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—432no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—416no—
638C2×C4—416no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—816no—
668C2×C4C2×C22⋊Q814yes—
669C2×C4#53414yes—
670C2×C4#53818yes—
671C2×C4#54618yes—
672C2×C4#55114yes—
673C2×C4H(3,𝔽4)14yes—
674C22—28no—
675C22—28no—
676C22—24no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22C22×C22⋊Q811yescharacteristic
697C22C2×#53411yescharacteristic
698C22#344911yescharacteristic
699C22C2×#53812yes—
700C22C2×#54612yes—
701C22C2×#55111yescharacteristic
702C22Q8×#4112yes—
703C22#348212yes—
704C22#348912yes—
705C22#350111yescharacteristic
706C22#350111yescharacteristic
707C22#350312yes—
708C22#350511yescharacteristic
709C22#350511yescharacteristic
710C22#351412yes—
711C22#351612yes—
712C22#352412yes—
713C22C2×H(3,𝔽4)11yescharacteristic
714C22#354712yes—
715C22#355011yescharacteristic
716C22#355111yescharacteristic
717C22#356112yes—
718C22#356212yes—
719C4—24no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—48no—
724C4—44no—
725C4—44no—
726C4—44no—
727C4—82no—
728C4—84no—
729C4—84no—
730C4—84no—
731C4#268418yes—
732C2—44no—
733C2—44no—
734C2#3501011yescharacteristic
735C2#3627412yes—
736C2#3737611yescharacteristic
737C2#3890312yes—
738C2#3965412yes—
739C2#4010412yes—
740C2#4600411yescharacteristic
741C2#4607311yescharacteristic
742C2C2×#268411yescharacteristic
743C2#5993311yescharacteristic
744C2#5994611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16686C2semidirect
4. #16630C2semidirect
5. #16039C2semidirect
6. #15586C2semidirect
7. #15201C2semidirect
8. #15183C2semidirect
9. #15179C2semidirect
10. #14703C2semidirect
13. #14139C2semidirect
14. #13922C2semidirect
92. C2×#124Q8semidirect
201. C22×Q8C2×Q8semidirect
303. C2×C22⋊C4C2×Q8semidirect
321. C2×C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×C4×#1242
#166862
#166302
#160392
#155862
#152012
#151832
#151792
#147032
#145132
#144732
#141392
#139222
#138722