Finite Group Databasebeta

← #4311776

Subgroups — #4311776

Order 512.

4114 conjugacy classes of subgroups.

The subgroup lattice has 803 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2431C212yesmaximal, complemented
3C2×#2311C211yesmaximal, characteristic, complemented
4#16810C212yesmaximal, complemented
5#16551C212yesmaximal, complemented
6#16260C214yesmaximal, complemented
7#15450C214yesmaximal, complemented
8#15310C214yesmaximal, complemented
9#15002C214yesmaximal, complemented
10#14886C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14501C211yesmaximal, characteristic, complemented
13#14492C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2212yes—
16C22×#124C2212yes—
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2733—28no—
22#2431—216no—
23C2×#397C2214yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#392C2214yes—
29C2×#392C2214yes—
30C2×#392C2214yes—
31C2×#392C2212yes—
32C2×#392C2214yes—
33C2×#392C2214yes—
34#2311—28no—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#388C2214yes—
43C2×#386C2214yes—
44C2×#386C2214yes—
45C2×#386C2214yes—
46C2×#386C2214yes—
47C2×#380C2214yes—
48C2×C4×C22⋊C4C2212yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2212yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1434C2212yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C22×#41—24no—
71C22×#41—24no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—42no—
76C22×#41D418yescomplemented
77C23×Q8C2311yescharacteristic
78C23×D4C2314yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2314yes—
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2312yes—
85C23×D4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C2×#127—28no—
90C2×#127—44no—
91C2×#127—44no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—28no—
95C2×#121—28no—
96C2×#121—44no—
97C2×#121—44no—
98C2×#121—48no—
99C2×#121—44no—
100C2×#117—24no—
101C2×#117—42no—
102C2×#117—42no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—42no—
105C22×C4⋊C4—42no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#397—48no—
133#394—28no—
134#394—44no—
135#394—44no—
136#394—48no—
137#392—216no—
138#392—216no—
139#392—216no—
140#392—48no—
141#392—48no—
142#392—44no—
143#391—24no—
144#391—28no—
145#391D418yescomplemented
146#390—28no—
147#390—24no—
148#390—24no—
149#390—28no—
150#388—216no—
151#386—216no—
152#386—216no—
153#386—216no—
154#386—48no—
155#380—216no—
156C4×C22⋊C4—28no—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2414yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C2×#41—232no—
176C2×#41—416no—
177C2×#41—416no—
178C2×#41—416no—
179C2×#41—416no—
180C2×#41—416no—
181C2×#41—416no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8C2×D414yes—
187C22×Q8C2×D414yes—
188C22×Q8#4114yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—28no—
206C22×D4—28no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—42no—
221C22×D4—42no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—42no—
237C22×D4—44no—
238C22×D4—42no—
239C22×D4C2×D418yes—
240C22×D4C2×D414yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D414yes—
259C23×C4C2×D418yes—
260C23×C4C2412yes—
261C23×C4C2414yes—
262C23×C4C2412yes—
263C23×C4C2414yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2414yes—
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270#127—432no—
271#124—432no—
272#121—432no—
273#121—432no—
274#117—416no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D414yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—41no—
346C2×C42—41no—
347C2×C42—41no—
348C2×C42—41no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42C2×D414yes—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—42no—
362#95—42no—
363#95—44no—
364#95—42no—
365#95—44no—
366#95C2×D4116yes—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—24no—
371C24—21no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24C22×D414yes—
399C24C22×D412yes—
400C24H(5,𝔽2)12yes—
401C24H(5,𝔽2)12yes—
402C24C2511yescharacteristic
403#41—4128no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8#121116yescomplemented
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—28no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—416no—
426C2×D4—416no—
427C2×D4—416no—
428C2×D4—416no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—416no—
433C2×D4—416no—
434C2×D4—48no—
435C2×D4—416no—
436C2×D4—48no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—88no—
445C2×D4—88no—
446C2×D4—88no—
447C2×D4—88no—
448C2×D4—88no—
449C2×D4—88no—
450C2×D4—88no—
451C2×D4—88no—
452C2×D4—88no—
453C2×D4—84no—
454C2×D4—88no—
455C2×D4—88no—
456C2×D4—88no—
457C2×D4—88no—
458C2×D4—88no—
459C2×D4—88no—
460C2×D4—88no—
461C2×D4—88no—
462C2×D4—88no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—216no—
497C22×C4—216no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—24no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—24no—
522C22×C4—48no—
523C22×C4—48no—
524C22×C4—48no—
525C22×C4—48no—
526C22×C4—48no—
527C22×C4—48no—
528C22×C4—48no—
529C22×C4—88no—
530C22×C4—84no—
531C22×C4—88no—
532C22×C4C22×D414yes—
533C22×C4C22×D414yes—
534C22×C4C22×D412yes—
535C22×C4C22×D414yes—
536C22×C4C22×D414yes—
537C22×C4C22×D412yes—
538C22×C4C22×D412yes—
539C22×C4C2×#4114yes—
540C22×C4C2×#4112yes—
541C22×C4H(5,𝔽2)18yes—
542C22×C4H(5,𝔽2)12yes—
543C22×C4H(5,𝔽2)12yes—
544C4⋊C4—44no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—44no—
549C4⋊C4—48no—
550C4⋊C4—84no—
551C22⋊C4—416no—
552C22⋊C4—48no—
553C22⋊C4—416no—
554C22⋊C4—416no—
555C22⋊C4—416no—
556C22⋊C4—416no—
557C22⋊C4—416no—
558C22⋊C4—88no—
559C22⋊C4—88no—
560C22⋊C4—88no—
561C22⋊C4—88no—
562C22⋊C4—88no—
563C22⋊C4—88no—
564C22⋊C4—88no—
565C22⋊C4—88no—
566C42—44no—
567C42—44no—
568C42—48no—
569C42—48no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—28no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—28no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—28no—
584C23—28no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—44no—
626C23—44no—
627C23—44no—
628C23—44no—
629C23—44no—
630C23—44no—
631C23—44no—
632C23—44no—
633C23—44no—
634C23—44no—
635C23—44no—
636C23—44no—
637C23—44no—
638C23—44no—
639C23—44no—
640C23—44no—
641C23—44no—
642C23—44no—
643C23—44no—
644C23#53414yes—
645C23C23×D411yescharacteristic
646C23C23×D411yescharacteristic
647C23C22×#4111yescharacteristic
648C23C2×H(5,𝔽2)11yescharacteristic
649C23C2×H(5,𝔽2)11yescharacteristic
650C23C2×H(5,𝔽2)11yescharacteristic
651C23C2×H(5,𝔽2)11yescharacteristic
652C23C2×H(5,𝔽2)14yes—
653C23#58514yes—
654Q8—416no—
655D4—432no—
656D4—416no—
657D4—816no—
658D4—832no—
659D4—832no—
660D4—816no—
661D4—816no—
662D4—832no—
663C2×C4—232no—
664C2×C4—22no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—24no—
675C2×C4—24no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—22no—
679C2×C4—22no—
680C2×C4—24no—
681C2×C4—24no—
682C2×C4—24no—
683C2×C4—24no—
684C2×C4—24no—
685C2×C4—28no—
686C2×C4—24no—
687C2×C4—28no—
688C2×C4—28no—
689C2×C4—28no—
690C2×C4—432no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—44no—
705C2×C4—42no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—42no—
711C2×C4—44no—
712C2×C4—44no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—42no—
716C2×C4—42no—
717C2×C4—44no—
718C2×C4—44no—
719C2×C4—44no—
720C2×C4—44no—
721C2×C4—44no—
722C2×C4—44no—
723C2×C4—44no—
724C2×C4—816no—
725C2×C4C2×#12118yes—
726C2×C4C2×#12114yes—
727C2×C4#53414yes—
728C22—24no—
729C22—28no—
730C22—44no—
731C22—44no—
732C22—42no—
733C22—44no—
734C22—44no—
735C22—42no—
736C22—44no—
737C22—44no—
738C22—44no—
739C22—44no—
740C22—48no—
741C22—48no—
742C22—84no—
743C22—84no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22—82no—
748C22—84no—
749C22—84no—
750C22—84no—
751C22—84no—
752C22—84no—
753C22—84no—
754C22—84no—
755C22—84no—
756C22—84no—
757C22—84no—
758C22—84no—
759C22—84no—
760C22—84no—
761C22—84no—
762C22—84no—
763C22C22×#12111yescharacteristic
764C22C2×#53411yescharacteristic
765C22C2×#53411yescharacteristic
766C22#348411yescharacteristic
767C22#348411yescharacteristic
768C22#348712yes—
769C22#348814yes—
770C22#349011yescharacteristic
771C22#349011yescharacteristic
772C22#349812yes—
773C22#351714yes—
774C22#353212yes—
775C22#353414yes—
776C22#354512yes—
777C22#354614yes—
778C22#354912yes—
779C22#355112yes—
780C4—24no—
781C4—24no—
782C4—28no—
783C4—44no—
784C4—44no—
785C4—48no—
786C4—82no—
787C4—84no—
788C4—84no—
789C4—82no—
790C2—44no—
791C2—84no—
792C2—164no—
793C2#3498111yescharacteristic
794C2#3602912yes—
795C2#3727312yes—
796C2#4198112yes—
797C2#4642512yes—
798C2#4765912yes—
799C2#5190211yescharacteristic
800C2#5193012yes—
801C2#5208311yescharacteristic
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×#2431C2semidirect
3. C2×#2311C2semidirect
4. #16810C2semidirect
5. #16551C2semidirect
6. #16260C2semidirect
7. #15450C2semidirect
8. #15310C2semidirect
9. #15002C2semidirect
10. #14886C2semidirect
11. #14719C2semidirect
12. #14501C2semidirect
13. #14492C2semidirect
76. C22×#41D4semidirect
145. #391D4semidirect
410. C2×Q8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#24312
C2×#23112
#168102
#165512
#162602
#154502
#153102
#150022
#148862
#147192
#145012
#144922