Finite Group Databasebeta

← #559970

Subgroups — #559970

Order 512.

3762 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2#14680C212yesmaximal
3#14669C212yesmaximal, complemented
4#14409C212yesmaximal, complemented
5#14402C212yesmaximal, complemented
6#14398C214yesmaximal, complemented
7#14392C214yesmaximal, complemented
8#13907C212yesmaximal, complemented
9#13903C211yesmaximal, characteristic
10#13882C212yesmaximal, complemented
11#13878C212yesmaximal, complemented
12#13874C212yesmaximal, complemented
13#13873C212yesmaximal
14#13868C211yesmaximal, characteristic
15#13831C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#389C2214yescomplemented
22C2×#388C2214yes—
23C2×#387C2214yescomplemented
24C2×#385C4116yescomplemented
25C2×#385C2211yescharacteristic
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2211yescharacteristic
29C2×#385C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#381C2211yescharacteristic
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2214yes—
37C2×#380C2214yes—
38C2×#380C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1437C2212yes—
59#1437C2212yes—
60#1437C2212yes—
61#1437C2214yes—
62#1437C2212yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C26C2311yescharacteristic
77C24×C4—48no—
78C24×C4—42no—
79C24×C4—42no—
80C24×C4C2×C418yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#117—44no—
89C2×#117—42no—
90C22×C4⋊C4—48no—
91C22×C4⋊C4C2×C4116yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—42no—
103C22×C22⋊C4—42no—
104C22×C22⋊C4C2×C418yes—
105C22×C22⋊C4C2×C418yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#389—48no—
125#388—216no—
126#387—48no—
127#385C2×C418yes—
128#385C2×C4116yes—
129#385C2×C4116yes—
130#385C2×C418yes—
131#385C2×C418yes—
132#385C2×C418yes—
133#381—44no—
134#381—42no—
135#380—48no—
136#380—44no—
137#380—44no—
138#380—44no—
139#380—42no—
140#380—48no—
141#379—44no—
142#379—44no—
143C4×C22⋊C4—44no—
144C4×C22⋊C4—42no—
145C2×#95C2×C4116yes—
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×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25—24no—
171C25—22no—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—44no—
177C25C22×C414yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—432no—
196C23×C4—44no—
197C23×C4—416no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—432no—
209C23×C4—432no—
210C23×C4C22×C414yes—
211C23×C4C22×C418yes—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C22×C418yes—
216C23×C4C22×C414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229#117—416no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—232no—
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—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—416no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
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⋊C4C22×C414yes—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C414yes—
291C2×C22⋊C4C22×C414yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C414yes—
295C2×C22⋊C4#4118yes—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C418yes—
324#95C22×C418yes—
325#95C22×C418yes—
326#95C22×C418yes—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—216no—
331C24—216no—
332C24—28no—
333C24—28no—
334C24—28no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—216no—
348C24—48no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—44no—
356C24—48no—
357C24—44no—
358C24C23×C412yes—
359C24C23×C412yes—
360C24C23×C412yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)12yes—
363C24H(5,𝔽2)12yes—
364C24H(5,𝔽2)14yes—
365C242−1+414yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—4128no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C414yes—
464C22×C4C23×C414yes—
465C22×C4C2×#4118yes—
466C22×C4H(5,𝔽2)18yes—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C4⋊C4—84no—
476C22⋊C4—28no—
477C22⋊C4—216no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4#117116yescomplemented
491C42—48no—
492C42—88no—
493C42—84no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—232no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—432no—
523C23—432no—
524C23—416no—
525C23—432no—
526C23—432no—
527C23—464no—
528C23—416no—
529C23—44no—
530C23—44no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23C2×#11714yes—
544C23#51814yes—
545C23#51914yes—
546C23#52018yes—
547C23#56014yes—
548C23#56218yes—
549C23#56314yes—
550C23C24×C411yescharacteristic
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)14yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×2−1+412yes—
557C23C2×2−1+412yes—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—88no—
629C2×C4C2×#11718yes—
630C2×C4#51818yes—
631C22—24no—
632C22—28no—
633C22—24no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—216no—
638C22—28no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—464no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—48no—
655C22#237618yes—
656C22C22×#11711yescharacteristic
657C22C2×#51811yescharacteristic
658C22C2×#51812yes—
659C22C2×#51911yescharacteristic
660C22C2×#52012yes—
661C22#349212yes—
662C22#349212yes—
663C22#351914yes—
664C22#352014yes—
665C22C2×#56011yescharacteristic
666C22C2×#56212yes—
667C22C2×#56311yescharacteristic
668C22#353212yes—
669C22#353212yes—
670C22#353614yes—
671C22#353814yes—
672C4—28no—
673C4—28no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—84no—
681C4—84no—
682C4—84no—
683C4—84no—
684C2—28no—
685C2—44no—
686C2—44no—
687C2#3417811yescharacteristic
688C2#3419412yes—
689C2C2×#237611yescharacteristic
690C2#3435611yescharacteristic
691C2#3438012yes—
692C2#4574312yes—
693C2#4574412yes—
694C2#4574514yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14669C2semidirect
4. #14409C2semidirect
5. #14402C2semidirect
6. #14398C2semidirect
7. #14392C2semidirect
8. #13907C2semidirect
10. #13882C2semidirect
11. #13878C2semidirect
12. #13874C2semidirect
15. #13831C2semidirect
21. C2×#389C22semidirect
23. C2×#387C22semidirect
24. C2×#385C4semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
490. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
#146802
#146692
#144092
#144022
#143982
#143922
#139072
#139032
#138822
#138782
#138742
#138732
#138682
#138312