Finite Group Databasebeta

← #627445

Subgroups — #627445

Order 512.

3070 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#16085C214yesmaximal, complemented
4#16084C214yesmaximal, complemented
5#15425C214yesmaximal, complemented
6#15396C214yesmaximal, complemented
7#14920C212yesmaximal, complemented
8#14726C212yesmaximal
9#14695C212yesmaximal, complemented
10#14339C211yesmaximal, characteristic, complemented
11#14262C212yesmaximal
12#14128C211yesmaximal, characteristic, complemented
13#14128C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic
15#13868C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#396C2214yes—
24#2317—28no—
25C2×#389C2212yescomplemented
26C2×#389C2212yescomplemented
27C2×#388—28no—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yescomplemented
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384C2212yescomplemented
36C2×#382C2212yescomplemented
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#380C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2214yescomplemented
45#1444C2212yescomplemented
46#1444C2212yescomplemented
47#1442C2212yescomplemented
48#1442C2214yescomplemented
49#1442C2212yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71#1434C2214yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—28no—
87C2×#117—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4Q818yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
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×C42—24no—
123C22×C42—44no—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127#396—48no—
128#389—28no—
129#389—28no—
130#388—216no—
131#388—216no—
132#387—48no—
133#385—28no—
134#385—28no—
135#385—28no—
136#385—44no—
137#384—28no—
138#382—28no—
139#381—24no—
140#381—24no—
141#380—44no—
142#379—28no—
143#379—44no—
144C2×#95—24no—
145C2×#95—24no—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25—22no—
171C25—22no—
172C25—24no—
173C25—24no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25—42no—
178C25C2×Q814yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×Q814yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#117—232no—
221#117—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—232no—
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—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—416no—
279C2×C22⋊C4—416no—
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⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×Q818yes—
291C2×C22⋊C4C2×Q818yes—
292C2×C22⋊C4#4118yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—416no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—28no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—216no—
343C24—48no—
344C24—48no—
345C24—44no—
346C24—48no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—44no—
352C24—44no—
353C24—48no—
354C24—88no—
355C24—88no—
356C24—88no—
357C24—88no—
358C24C22×D412yes—
359C24C22×Q814yes—
360C24C22×Q812yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)14yes—
364C24C2511yescharacteristic
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
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—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—416no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4C22⋊Q8116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)14yes—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—216no—
486C42—416no—
487C42—48no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—416no—
506C23—416no—
507C23—416no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—832no—
527C23—88no—
528C23—88no—
529C23—832no—
530C23—88no—
531C23—88no—
532C23C2×C22⋊Q814yes—
533C23#52514yes—
534C23#53414yes—
535C23#55318yes—
536C23#55418yes—
537C23C23×D411yescharacteristic
538C23C23×Q811yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23#58512yes—
544C23#58512yes—
545C23#58512yes—
546C2×C4—232no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
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—88no—
618C2×C4—88no—
619C2×C4C2×C22⋊Q814yes—
620C2×C4C2×C22⋊Q814yes—
621C22—24no—
622C22—216no—
623C22—24no—
624C22—216no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—44no—
634C22—832no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22#257918yes—
642C22C22×C22⋊Q811yescharacteristic
643C22C2×#52511yescharacteristic
644C22C2×#53411yescharacteristic
645C22#345812yes—
646C22#346012yes—
647C22C2×#55312yes—
648C22C2×#55412yes—
649C22#348411yescharacteristic
650C22#348511yescharacteristic
651C22#348711yescharacteristic
652C22#348711yescharacteristic
653C22#349312yes—
654C22#350112yes—
655C22#351314yes—
656C22#352114yes—
657C22#354214yes—
658C22#354514yes—
659C4—24no—
660C4—24no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—84no—
671C4—82no—
672C2—28no—
673C2—84no—
674C2C2×#257911yescharacteristic
675C2#3515111yescharacteristic
676C2#3517411yescharacteristic
677C2#3535611yescharacteristic
678C2#3545411yescharacteristic
679C2#3670411yescharacteristic
680C2#3718711yescharacteristic
681C2#3821114yes—
682C2#3895914yes—
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×#2317C2semidirect
3. #16085C2semidirect
4. #16084C2semidirect
5. #15425C2semidirect
6. #15396C2semidirect
7. #14920C2semidirect
9. #14695C2semidirect
10. #14339C2semidirect
12. #14128C2semidirect
13. #14128C2semidirect
25. C2×#389C22semidirect
26. C2×#389C22semidirect
30. C2×#387C22semidirect
35. C2×#384C22semidirect
36. C2×#382C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C22×#3882
#160852
#160842
#154252
#153962
#149202
#147262
#146952
#143392
#142622
#141282
#138682