Finite Group Databasebeta

← #624270

Subgroups — #624270

Order 512.

3058 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic, complemented
2C2×#2297C211yesmaximal, characteristic
3#15884C212yesmaximal, complemented
4#15854C212yesmaximal, complemented
5#15820C212yesmaximal, complemented
6#15662C212yesmaximal, complemented
7#15468C214yesmaximal, complemented
8#15399C214yesmaximal, complemented
9#15233C212yesmaximal, complemented
10#15125C212yesmaximal, complemented
11#14685C212yesmaximal, complemented
12#14337C212yesmaximal
13#13974C212yesmaximal, complemented
14#13973C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2628—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#393C2214yes—
26#2297—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#382C2211yescharacteristic
36C2×#381C2214yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2214yes—
48#1446C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1437C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C22×#41—24no—
78C22×#41—28no—
79C22×#41—28no—
80C22×#41—42no—
81C22×#41—42no—
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8—42no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C42—24no—
116C22×C42—28no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#399—28no—
126#395—28no—
127#393—28no—
128#393—48no—
129#391—24no—
130#391—24no—
131#390—24no—
132#390—24no—
133#389—28no—
134#388—48no—
135#386—28no—
136#386—28no—
137#382—24no—
138#381—48no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143#380—44no—
144#380—44no—
145C4×C4⋊C4—42no—
146C4×C4⋊C4—42no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C2×#41—232no—
177C2×#41—432no—
178C2×#41—432no—
179C2×#41—416no—
180C2×#41—416no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—42no—
202C22×D4—42no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4#4118yes—
221C23×C4#4118yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236#117—416no—
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—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
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⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—416no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24C22×D412yes—
362C24C2×#4114yes—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365#41—4128no—
366C2×Q8—28no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—88no—
373C2×Q8—88no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×D4—28no—
377C2×D4—216no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
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—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—24no—
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—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4#124116yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+414yes—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—88no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—28no—
507C42—28no—
508C42—44no—
509C42—44no—
510C42—48no—
511C42—84no—
512C42—84no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23C2×#12414yes—
569C23C23×D411yescharacteristic
570C23C22×#4112yes—
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+412yes—
576C23#58512yes—
577C23#58512yes—
578Q8—416no—
579Q8—816no—
580D4—416no—
581D4—432no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—232no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
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—432no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
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—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—432no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
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—44no—
657C2×C4—44no—
658C2×C4#52514yes—
659C2×C4C2×#12414yes—
660C2×C4C2×#12414yes—
661C2×C4#53414yes—
662C2×C4#55118yes—
663C2×C4#55118yes—
664C22—24no—
665C22—28no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C2×#52511yescharacteristic
689C22C22×#12411yescharacteristic
690C22C2×#53411yescharacteristic
691C22#346012yes—
692C22#346112yes—
693C22C2×#55112yes—
694C22C2×#55112yes—
695C22#348812yes—
696C22#348912yes—
697C22#349212yes—
698C22#349312yes—
699C22#349412yes—
700C22#349512yes—
701C22#351012yes—
702C22#351112yes—
703C22#354612yes—
704C22#354712yes—
705C22#354911yescharacteristic
706C22#355011yescharacteristic
707C22#355111yescharacteristic
708C22#355111yescharacteristic
709C4—24no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—84no—
718C4—84no—
719C4—84no—
720C4—84no—
721C4#256118yes—
722C2—44no—
723C2—84no—
724C2C2×#256111yescharacteristic
725C2#3525812yes—
726C2#3554212yes—
727C2#3727311yescharacteristic
728C2#3737611yescharacteristic
729C2#3796812yes—
730C2#3796912yes—
731C2#3797412yes—
732C2#3797512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2628C2semidirect
3. #15884C2semidirect
4. #15854C2semidirect
5. #15820C2semidirect
6. #15662C2semidirect
7. #15468C2semidirect
8. #15399C2semidirect
9. #15233C2semidirect
10. #15125C2semidirect
11. #14685C2semidirect
13. #13974C2semidirect
14. #13973C2semidirect
15. #13777C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#22972
#158842
#158542
#158202
#156622
#154682
#153992
#152332
#151252
#146852
#143372
#139742
#139732
#137772