Finite Group Databasebeta

← #4308623

Subgroups — #4308623

Order 512.

4013 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3#15205C214yesmaximal, complemented
4#15162C214yesmaximal, complemented
5#14919C212yesmaximal, complemented
6#14912C212yesmaximal, complemented
7#14727C212yesmaximal, complemented
8#14718C212yesmaximal
9#14706C211yesmaximal, characteristic, complemented
10#14706C211yesmaximal, characteristic, complemented
11#14705C211yesmaximal, characteristic, complemented
12#14705C211yesmaximal, characteristic, complemented
13#14681C211yesmaximal, characteristic, complemented
14#14304C214yesmaximal, complemented
15#14303C214yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2849—28no—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×#381C2212yes—
39C2×#379C2211yescharacteristic
40C2×#379C2212yes—
41C2×C4×C22⋊C4—28no—
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2214yes—
44C22×#95C2212yes—
45#1445C2214yescomplemented
46#1443C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yescomplemented
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yescomplemented
58#1438C2214yes—
59#1437C2212yescomplemented
60#1437C2212yescomplemented
61#1437C2212yescomplemented
62#1437C2212yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2212yescomplemented
66#1436C2212yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2212yescomplemented
70#1436C2212yescomplemented
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—44no—
83C24×C4—44no—
84C24×C4D418yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—44no—
96C2×#120—24no—
97C2×#120—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—216no—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42—44no—
125C22×C42—44no—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#397—216no—
131#394—216no—
132#386—28no—
133#386—28no—
134#386—44no—
135#386—44no—
136#385—28no—
137#385—28no—
138#385—44no—
139#385—44no—
140#381—24no—
141#381—24no—
142#381—24no—
143#381—24no—
144#381—44no—
145#379—24no—
146#379—44no—
147C4×C22⋊C4—216no—
148C4×C22⋊C4—216no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—28no—
175C25—22no—
176C25—22no—
177C25—24no—
178C25—24no—
179C25—44no—
180C25C2×D414yes—
181C25C2411yescharacteristic
182C25C2412yes—
183C25C2411yescharacteristic
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—44no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—232no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—416no—
214C23×C4—416no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—432no—
226C23×C4—432no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D418yes—
230C23×C4C2×D418yes—
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—416no—
246#123—416no—
247#120—416no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C22⋊C4—264no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4#41116yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—416no—
320C2×C42—416no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
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—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345C24—24no—
346C24—24no—
347C24—216no—
348C24—216no—
349C24—28no—
350C24—216no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—28no—
361C24—28no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—48no—
370C24—416no—
371C24—416no—
372C24—416no—
373C24—416no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24C22×D414yes—
379C24C22×D412yes—
380C24C2×#4118yes—
381C24C2×#4112yes—
382C24C2×#4112yes—
383C24H(5,𝔽2)14yes—
384C24H(5,𝔽2)14yes—
385C24C2511yescharacteristic
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—88no—
395C2×D4—88no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
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—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—4128no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4#123116yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4118yes—
494C22×C4C2×#4118yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C22⋊C4—232no—
505C22⋊C4—232no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—48no—
522C42—48no—
523C42—48no—
524C42—48no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—232no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—28no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—432no—
549C23—432no—
550C23—464no—
551C23—464no—
552C23—416no—
553C23—416no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23C2×#12314yes—
580C23#52518yes—
581C23#53414yes—
582C23#53414yes—
583C23#53818yes—
584C23#53818yes—
585C23C23×D411yescharacteristic
586C23C22×#4111yescharacteristic
587C23C22×#4111yescharacteristic
588C23C22×#4114yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)12yes—
591C23#58514yes—
592D4—816no—
593D4—88no—
594C2×C4—232no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—432no—
613C2×C4—432no—
614C2×C4—432no—
615C2×C4—432no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4C2×#12314yes—
667C2×C4C2×#12314yes—
668C22—216no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—28no—
673C22—28no—
674C22—28no—
675C22—28no—
676C22—28no—
677C22—464no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—84no—
687C22—84no—
688C22—81no—
689C22—81no—
690C22—82no—
691C22—82no—
692C22—81no—
693C22#262218yes—
694C22C22×#12311yescharacteristic
695C22C2×#52512yes—
696C22#344412yes—
697C22#344412yes—
698C22C2×#53411yescharacteristic
699C22C2×#53411yescharacteristic
700C22#344912yes—
701C22C2×#53812yes—
702C22C2×#53812yes—
703C22#345512yes—
704C22#345512yes—
705C22#351314yes—
706C22#351314yes—
707C22#351314yes—
708C22#351314yes—
709C4—24no—
710C4—24no—
711C4—48no—
712C4—48no—
713C4—44no—
714C4—84no—
715C4—84no—
716C4—81no—
717C4—81no—
718C4—82no—
719C4—82no—
720C4—81no—
721C2—28no—
722C2—48no—
723C2—161no—
724C2C2×#262211yescharacteristic
725C2#3463511yescharacteristic
726C2#3506212yes—
727C2#3506312yes—
728C2#3507414yes—
729C2#3507414yes—
730C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C22⋊C4C2semidirect
3. #15205C2semidirect
4. #15162C2semidirect
5. #14919C2semidirect
6. #14912C2semidirect
7. #14727C2semidirect
9. #14706C2semidirect
10. #14706C2semidirect
11. #14705C2semidirect
12. #14705C2semidirect
13. #14681C2semidirect
14. #14304C2semidirect
15. #14303C2semidirect
45. #1445C22semidirect
46. #1443C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
54. #1440C22semidirect
57. #1439C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C22×C4×C22⋊C42
#152052
#151622
#149192
#149122
#147272
#147182
#147062
#147052
#146812
#143042
#143032