Finite Group Databasebeta

← #4453540

Subgroups — #4453540

Order 512.

3326 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic, complemented
3#16381C214yesmaximal, complemented
4#16357C214yesmaximal, complemented
5#16284C214yesmaximal, complemented
6#16270C214yesmaximal, complemented
7#15233C212yesmaximal, complemented
8#15157C212yesmaximal, complemented
9#15124C212yesmaximal, complemented
10#15123C212yesmaximal, complemented
11#14350C211yesmaximal, characteristic, complemented
12#13970C212yesmaximal, complemented
13#13970C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20#2398—28no—
21C2×#400C2212yes—
22C2×#397C2212yes—
23C2×#394C2212yes—
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28C2×#392C2214yes—
29#2328—28no—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×#376C2211yescharacteristic
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2314yes—
74C23×D4C2314yes—
75C24×C4C2312yes—
76C24×C4C2314yes—
77C2×#127—28no—
78C2×#127—24no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—44no—
82C2×#121—44no—
83C2×#121—44no—
84C2×#121—44no—
85C2×C4×Q8—24no—
86C2×C4×Q8—42no—
87C2×C4×Q8—42no—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—44no—
91C2×C4×D4—44no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#400—28no—
121#397—28no—
122#394—28no—
123#393—28no—
124#393—48no—
125#392—48no—
126#392—48no—
127#392—48no—
128#390—28no—
129#390—44no—
130#390—44no—
131#386—216no—
132#386—216no—
133#386—48no—
134#386—48no—
135#382—24no—
136#380—44no—
137#380—44no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140#376D418yescomplemented
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2414yes—
161C25C2412yes—
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4#4118yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225#127—232no—
226#121—432no—
227#121—432no—
228C4×Q8—416no—
229C4×D4—432no—
230C4×D4—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325#95C2×D418yes—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24C2×#4118yes—
355C24H(5,𝔽2)14yes—
356C24C2511yescharacteristic
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×D4—232no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)14yes—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—48no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23C23×D411yescharacteristic
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)12yes—
583C23#58512yes—
584C23#58512yes—
585Q8—416no—
586D4—416no—
587D4—432no—
588D4—832no—
589D4—832no—
590D4—832no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—832no—
658C2×C4—816no—
659C2×C4#52514yes—
660C2×C4#53514yes—
661C2×C4#53514yes—
662C2×C4#53818yes—
663C2×C4#53818yes—
664C22—48no—
665C22—44no—
666C22—48no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—48no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—48no—
675C22—48no—
676C22—44no—
677C22—48no—
678C22—82no—
679C22—84no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—84no—
684C22—82no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22C2×#52511yescharacteristic
695C22C2×#53511yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#53812yes—
698C22C2×#53812yes—
699C22#348812yes—
700C22#348812yes—
701C22#351312yes—
702C22#351412yes—
703C22#351912yes—
704C22#352112yes—
705C22#353711yescharacteristic
706C22#353711yescharacteristic
707C22#353911yescharacteristic
708C22#353911yescharacteristic
709C22#354212yes—
710C22#354212yes—
711C22#354312yes—
712C22#354612yes—
713C22#355812yes—
714C22#356212yes—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—84no—
722C4—84no—
723C4—82no—
724C4—84no—
725C4#263118yes—
726C2—88no—
727C2—84no—
728C2C2×#263111yescharacteristic
729C2#3552212yes—
730C2#3727011yescharacteristic
731C2#3727011yescharacteristic
732C2#3727812yes—
733C2#4256612yes—
734C2#4258112yes—
735C2#4259812yes—
736C2#4273212yes—
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×#2398C2semidirect
2. C2×#2328C2semidirect
3. #16381C2semidirect
4. #16357C2semidirect
5. #16284C2semidirect
6. #16270C2semidirect
7. #15233C2semidirect
8. #15157C2semidirect
9. #15124C2semidirect
10. #15123C2semidirect
11. #14350C2semidirect
12. #13970C2semidirect
13. #13970C2semidirect
140. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
C2×#23282
#163812
#163572
#162842
#162702
#152332
#151572
#151242
#151232
#143502
#139702