Finite Group Databasebeta

← #627779

Subgroups — #627779

Order 512.

3658 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3#15960C214yesmaximal, complemented
4#15959C214yesmaximal, complemented
5#15957C214yesmaximal, complemented
6#15956C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14918C212yesmaximal
9#14680C211yesmaximal, characteristic
10#14680C211yesmaximal, characteristic
11#14339C211yesmaximal, characteristic, complemented
12#14275C211yesmaximal, characteristic, complemented
13#14275C211yesmaximal, characteristic, complemented
14#14039C212yesmaximal, complemented
15#13991C212yesmaximal
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2313—28no—
22C2×#388C2214yes—
23C2×#387C2212yescomplemented
24C2×#387C2212yescomplemented
25C2×#387C2212yescomplemented
26C2×#387C2212yescomplemented
27C2×#387C2214yescomplemented
28C2×#385—28no—
29C2×#385C2212yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2214yescomplemented
49#1444C2212yescomplemented
50#1444C2212yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1442C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#388—48no—
125#387—28no—
126#387—28no—
127#387—28no—
128#387—28no—
129#387—48no—
130#385—28no—
131#385—216no—
132#385—216no—
133#385—44no—
134#381—28no—
135#381—44no—
136#380—24no—
137#380—24no—
138#380—24no—
139#380—24no—
140#380—44no—
141#380—44no—
142#380—44no—
143#380—44no—
144#379—28no—
145#379—44no—
146C4×C22⋊C4—28no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—22no—
181C25—22no—
182C25—22no—
183C25—22no—
184C25—42no—
185C25—42no—
186C25#4114yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—416no—
230C23×C4—432no—
231C23×C4—432no—
232C23×C4—416no—
233C23×C4—416no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×C4⋊C4—232no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—232no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—416no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—44no—
320#95—24no—
321#95—24no—
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—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
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—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—28no—
352C24—28no—
353C24—216no—
354C24—28no—
355C24—216no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—28no—
369C24—28no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—44no—
380C24—48no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24C22×D412yes—
387C24C2×#4114yes—
388C24C2×#4112yes—
389C24C2×#4112yes—
390C24C2×#4112yes—
391C24C2×#4112yes—
392C24H(5,𝔽2)14yes—
393C24H(5,𝔽2)12yes—
394C242−1+414yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
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—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—44no—
491C22×C4—4128no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—84no—
506C22×C4#123116yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C2×#4114yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4H(5,𝔽2)14yes—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—88no—
528C22⋊C4—232no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C42—84no—
539C42—84no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—232no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—432no—
576C23—432no—
577C23—416no—
578C23—432no—
579C23—432no—
580C23—416no—
581C23—416no—
582C23—416no—
583C23—416no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—416no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23C2×#12314yes—
602C23#52514yes—
603C23#53714yes—
604C23#53918yes—
605C23#55214yes—
606C23#55314yes—
607C23#55514yes—
608C23#55614yes—
609C23C23×D411yescharacteristic
610C23C22×#4111yescharacteristic
611C23C22×#4111yescharacteristic
612C23C22×#4111yescharacteristic
613C23C22×#4111yescharacteristic
614C23C2×H(5,𝔽2)11yescharacteristic
615C23C2×H(5,𝔽2)12yes—
616C23C2×H(5,𝔽2)12yes—
617C23C2×2−1+411yescharacteristic
618C23C2×2−1+412yes—
619C23#58512yes—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—416no—
640C2×C4—432no—
641C2×C4—432no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—416no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—88no—
698C2×C4—88no—
699C2×C4C2×#12314yes—
700C2×C4C2×#12314yes—
701C22—28no—
702C22—24no—
703C22—28no—
704C22—28no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—28no—
711C22—28no—
712C22—28no—
713C22—28no—
714C22—464no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22#248518yes—
728C22C22×#12311yescharacteristic
729C22C2×#52511yescharacteristic
730C22C2×#53711yescharacteristic
731C22C2×#53912yes—
732C22#345912yes—
733C22C2×#55211yescharacteristic
734C22C2×#55311yescharacteristic
735C22C2×#55511yescharacteristic
736C22C2×#55611yescharacteristic
737C22#348411yescharacteristic
738C22#348411yescharacteristic
739C22#348611yescharacteristic
740C22#348711yescharacteristic
741C22#349012yes—
742C22#349012yes—
743C22#349212yes—
744C22#349512yes—
745C22#349512yes—
746C22#350512yes—
747C22#353412yes—
748C22#353512yes—
749C22#353612yes—
750C22#353812yes—
751C4—24no—
752C4—24no—
753C4—44no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—84no—
758C4—84no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4—82no—
764C2—28no—
765C2—48no—
766C2C2×#248511yescharacteristic
767C2#3498711yescharacteristic
768C2#3498911yescharacteristic
769C2#3535011yescharacteristic
770C2#3567211yescharacteristic
771C2#3671911yescharacteristic
772C2#3672211yescharacteristic
773C2#3788012yes—
774C2#3789012yes—
775C2#3794812yes—
776C2#3795612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2313C2semidirect
3. #15960C2semidirect
4. #15959C2semidirect
5. #15957C2semidirect
6. #15956C2semidirect
7. #14919C2semidirect
11. #14339C2semidirect
12. #14275C2semidirect
13. #14275C2semidirect
14. #14039C2semidirect
23. C2×#387C22semidirect
24. C2×#387C22semidirect
25. C2×#387C22semidirect
26. C2×#387C22semidirect
27. C2×#387C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
69. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
C22×#3852
#159602
#159592
#159572
#159562
#149192
#149182
#146802
#143392
#142752
#140392
#139912