Finite Group Databasebeta

← #559944

Subgroups — #559944

Order 512.

3634 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2#14681C212yesmaximal
3#14669C212yesmaximal, complemented
4#14515C212yesmaximal, complemented
5#14514C212yesmaximal, complemented
6#14457C214yesmaximal, complemented
7#14456C214yesmaximal, complemented
8#13904C212yesmaximal, complemented
9#13903C211yesmaximal, characteristic
10#13874C212yesmaximal, complemented
11#13874C212yesmaximal, complemented
12#13873C212yesmaximal
13#13870C212yesmaximal, complemented
14#13868C211yesmaximal, characteristic
15#13831C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#388C2214yes—
22C2×#385C4116yescomplemented
23C2×#385C2211yescharacteristic
24C2×#385C2211yescharacteristic
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2211yescharacteristic
28C2×#385C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2212yes—
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1443C2214yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1437C2212yes—
57#1437C2212yes—
58#1437C2212yes—
59#1437C2214yes—
60#1437C2212yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C4⋊#95C2214yescomplemented
67#1434C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C26C2311yescharacteristic
77C24×C4—44no—
78C24×C4—44no—
79C24×C4—42no—
80C24×C4—42no—
81C24×C4C2×C418yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#117—44no—
90C2×#117—42no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2×C4116yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
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#388—216no—
125#385C2×C418yes—
126#385C2×C4116yes—
127#385C2×C4116yes—
128#385C2×C418yes—
129#385C2×C418yes—
130#385C2×C418yes—
131#381—44no—
132#381—44no—
133#381—44no—
134#381—44no—
135#381—44no—
136#381—42no—
137#380—44no—
138#380—42no—
139#379—44no—
140#379—44no—
141C4×C22⋊C4—44no—
142C4×C22⋊C4—42no—
143C2×#95C2×C4116yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
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—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—432no—
197C23×C4—416no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—42no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—416no—
215C23×C4—432no—
216C23×C4—416no—
217C23×C4C22×C414yes—
218C23×C4C22×C418yes—
219C23×C4C22×C418yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C22×C418yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236#117—416no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C418yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—232no—
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—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
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⋊C4C22×C414yes—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C414yes—
296C2×C22⋊C4C22×C414yes—
297C2×C22⋊C4C22×C414yes—
298C2×C22⋊C4C22×C414yes—
299C2×C22⋊C4C22×C414yes—
300C2×C22⋊C4#4118yes—
301C2×C42—28no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C22×C418yes—
331#95C22×C418yes—
332#95C22×C418yes—
333#95C22×C418yes—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—216no—
338C24—216no—
339C24—28no—
340C24—28no—
341C24—28no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—216no—
355C24—48no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—44no—
363C24—48no—
364C24—44no—
365C24C23×C412yes—
366C24C23×C412yes—
367C24C23×C412yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)12yes—
370C24H(5,𝔽2)14yes—
371C24H(5,𝔽2)12yes—
372C24H(5,𝔽2)14yes—
373C242−1+414yes—
374C24C2511yescharacteristic
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—216no—
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—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—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—4128no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C23×C414yes—
478C22×C4C23×C414yes—
479C22×C4C23×C414yes—
480C22×C4C23×C412yes—
481C22×C4C23×C412yes—
482C22×C4C23×C414yes—
483C22×C4C2×#4118yes—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—88no—
490C4⋊C4—84no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4#117116yescomplemented
506C42—48no—
507C42—88no—
508C42—84no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—232no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—416no—
541C23—464no—
542C23—432no—
543C23—416no—
544C23—44no—
545C23—44no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23C2×#11714yes—
559C23#51814yes—
560C23#51818yes—
561C23#51914yes—
562C23#56014yes—
563C23#56018yes—
564C23H(3,𝔽4)14yes—
565C23C24×C411yescharacteristic
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×2−1+412yes—
572C23#58514yes—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—432no—
595C2×C4—432no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—432no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
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—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
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—88no—
649C2×C4C2×#11718yes—
650C22—28no—
651C22—24no—
652C22—28no—
653C22—24no—
654C22—28no—
655C22—216no—
656C22—28no—
657C22—28no—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—464no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—44no—
673C22—44no—
674C22#237518yes—
675C22C22×#11711yescharacteristic
676C22C2×#51811yescharacteristic
677C22C2×#51812yes—
678C22C2×#51911yescharacteristic
679C22#342812yes—
680C22#349212yes—
681C22#349212yes—
682C22#351314yes—
683C22#351314yes—
684C22C2×#56011yescharacteristic
685C22C2×#56012yes—
686C22C2×H(3,𝔽4)11yescharacteristic
687C22#353312yes—
688C22#353312yes—
689C22#353514yes—
690C22#353514yes—
691C4—28no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—84no—
700C4—84no—
701C4—84no—
702C4—84no—
703C2—28no—
704C2—44no—
705C2—44no—
706C2#3417811yescharacteristic
707C2#3419012yes—
708C2C2×#237511yescharacteristic
709C2#3435711yescharacteristic
710C2#3435912yes—
711C2#4553814yes—
712C2#4553912yes—
713C2#4556912yes—
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. #14515C2semidirect
5. #14514C2semidirect
6. #14457C2semidirect
7. #14456C2semidirect
8. #13904C2semidirect
10. #13874C2semidirect
11. #13874C2semidirect
13. #13870C2semidirect
15. #13831C2semidirect
22. C2×#385C4semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
505. 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
#146812
#146692
#145152
#145142
#144572
#144562
#139042
#139032
#138742
#138732
#138702
#138682
#138312