Finite Group Databasebeta

← #4452743

Subgroups — #4452743

Order 512.

3256 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic, complemented
3#16838C214yesmaximal, complemented
4#16773C214yesmaximal, complemented
5#16526C214yesmaximal, complemented
6#16374C214yesmaximal, complemented
7#14921C212yesmaximal, complemented
8#14725C212yesmaximal, complemented
9#14720C211yesmaximal, characteristic, complemented
10#14719C211yesmaximal, characteristic, complemented
11#14717C211yesmaximal, characteristic, complemented
12#14716C211yesmaximal, characteristic
13#14020C212yesmaximal, complemented
14#13995C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2639—28no—
24C2×#400C2212yescomplemented
25C2×#397—28no—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#397C2212yescomplemented
29C2×#395C2212yescomplemented
30C2×#394C2214yes—
31C2×#394C2212yescomplemented
32C2×#394C2212yescomplemented
33C2×#393C2212yes—
34C2×#393C2212yescomplemented
35C2×#392C2212yes—
36C2×#390C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2214yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2212yescomplemented
55#1440C2214yes—
56#1440C2214yescomplemented
57#1439C2212yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1437C2214yes—
69#1437C2214yescomplemented
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1434C2214yescomplemented
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—44no—
90C2×#121—24no—
91C2×#121—44no—
92C2×#120—28no—
93C2×#120—44no—
94C2×#120—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2314yes—
129#400—28no—
130#397—28no—
131#397—216no—
132#397—216no—
133#395—44no—
134#394—28no—
135#394—48no—
136#394—44no—
137#393—28no—
138#393—44no—
139#392—44no—
140#390—48no—
141#386—28no—
142#386—28no—
143#386—28no—
144#386—44no—
145#386—44no—
146#385—44no—
147#380—28no—
148#379—44no—
149C4×C22⋊C4—44no—
150C2×#95—24no—
151C2×#95—24no—
152C2×#95D418yescomplemented
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25—22no—
176C25—24no—
177C25—24no—
178C25—22no—
179C25—24no—
180C25—22no—
181C25—42no—
182C25—42no—
183C25—42no—
184C25C2412yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—28no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—42no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
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×C4—42no—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2412yes—
247C22⋊Q8—416no—
248#121—416no—
249#120—432no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
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—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—44no—
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—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×C22⋊C4#4118yes—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—42no—
345#95—42no—
346#95C2×D418yes—
347#95C2×D418yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—28no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—416no—
375C24—48no—
376C24—48no—
377C24—416no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—48no—
385C24—48no—
386C24—88no—
387C24—84no—
388C24—84no—
389C24—88no—
390C24C22×D412yes—
391C24C2×#4114yes—
392C24H(5,𝔽2)14yes—
393C24H(5,𝔽2)14yes—
394C24H(5,𝔽2)12yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C2×Q8—44no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—88no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
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—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—88no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4—88no—
503C22×C4—88no—
504C22×C4—88no—
505C22×C4C22×D414yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4118yes—
510C22×C4H(5,𝔽2)14yes—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—216no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C42—88no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—24no—
550C23—28no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—832no—
599C23—832no—
600C23—88no—
601C23—88no—
602C23—88no—
603C23—88no—
604C23#53214yes—
605C23#53514yes—
606C23#53714yes—
607C23#53818yes—
608C23#54218yes—
609C23C23×D411yescharacteristic
610C23C22×#4112yes—
611C23C2×H(5,𝔽2)12yes—
612C23C2×H(5,𝔽2)11yescharacteristic
613C23C2×H(5,𝔽2)12yes—
614C23C2×H(5,𝔽2)12yes—
615C23C2×2−1+411yescharacteristic
616C23#58512yes—
617C23#58512yes—
618Q8—88no—
619D4—832no—
620D4—88no—
621D4—816no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—416no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—88no—
690C2×C4—88no—
691C2×C4—88no—
692C2×C4—88no—
693C2×C4—88no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—216no—
699C22—24no—
700C22—24no—
701C22—216no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—48no—
706C22—48no—
707C22—832no—
708C22—82no—
709C22—82no—
710C22—84no—
711C22—82no—
712C22—82no—
713C22—84no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—84no—
720C22—84no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—84no—
725C22#266218yes—
726C22C2×#53211yescharacteristic
727C22C2×#53511yescharacteristic
728C22C2×#53711yescharacteristic
729C22C2×#53812yes—
730C22C2×#54212yes—
731C22#348411yescharacteristic
732C22#348411yescharacteristic
733C22#348511yescharacteristic
734C22#348711yescharacteristic
735C22#351314yes—
736C22#351414yes—
737C22#353412yes—
738C22#353412yes—
739C22#353512yes—
740C22#353512yes—
741C22#353714yes—
742C22#356014yes—
743C4—44no—
744C4—42no—
745C4—42no—
746C4—44no—
747C4—44no—
748C4—48no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—84no—
755C2—28no—
756C2—84no—
757C2—162no—
758C2C2×#266211yescharacteristic
759C2#3539611yescharacteristic
760C2#3546611yescharacteristic
761C2#3670911yescharacteristic
762C2#3671711yescharacteristic
763C2#3672211yescharacteristic
764C2#3696111yescharacteristic
765C2#4081314yes—
766C2#4086614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14716C2
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×#2639C2semidirect
2. C22×#397C2semidirect
3. #16838C2semidirect
4. #16773C2semidirect
5. #16526C2semidirect
6. #16374C2semidirect
7. #14921C2semidirect
8. #14725C2semidirect
9. #14720C2semidirect
10. #14719C2semidirect
11. #14717C2semidirect
13. #14020C2semidirect
14. #13995C2semidirect
15. #13778C2semidirect
24. C2×#400C22semidirect
28. C2×#397C22semidirect
29. C2×#395C22semidirect
31. C2×#394C22semidirect
32. C2×#394C22semidirect
34. C2×#393C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
56. #1440C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1434C22semidirect
152. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C22×#3972
#168382
#167732
#165262
#163742
#149212
#147252
#147202
#147192
#147172
#147162
#140202
#139952
#137782