Finite Group Databasebeta

← #581618

Subgroups — #581618

Order 512.

3826 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2309C211yesmaximal, characteristic
2#14673C211yesmaximal, characteristic, complemented
3#14431C214yesmaximal, complemented
4#14429C214yesmaximal, complemented
5#14365C214yesmaximal, complemented
6#14127C211yesmaximal, characteristic, complemented
7#14084C214yesmaximal, complemented
8#14034C214yesmaximal, complemented
9#13944C214yesmaximal, complemented
10#13867C211yesmaximal, characteristic
11#13867C211yesmaximal, characteristic
12#13775C211yesmaximal, characteristic, complemented
13#13775C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#117C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2309C4116yescomplemented
19C2×#390C2214yes—
20C2×#390C2214yes—
21C2×#388C2214yescomplemented
22C2×#388C2214yescomplemented
23C2×#387C2214yescomplemented
24C2×#387C2214yescomplemented
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#385C2212yescomplemented
30C2×#385C2212yescomplemented
31C2×#379C2211yescharacteristic
32C2×#379C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2212yes—
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yescomplemented
39C2×C4×C22⋊C4C2212yescomplemented
40#1445C2214yescomplemented
41#1445C2214yescomplemented
42#1440C2214yescomplemented
43#1440C2214yescomplemented
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1438C2214yes—
51#1438C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yescomplemented
55#1437C2214yescomplemented
56#1437C2214yescomplemented
57#1437C2214yescomplemented
58C4⋊#95C2214yescomplemented
59#1434C2214yescomplemented
60#1434C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C4×#95C2214yescomplemented
68C26C2311yescharacteristic
69C26C2311yescharacteristic
70C23×D4—24no—
71C23×D4—24no—
72C23×D4—44no—
73C23×D4—44no—
74C23×D4C2×C418yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#117—44no—
83C2×#117—44no—
84C2×#117C2×C4116yescomplemented
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117#390C2×C4132yescomplemented
118#390C2×C4132yescomplemented
119#388—48no—
120#388—48no—
121#387—48no—
122#387—48no—
123#386—44no—
124#386—44no—
125#386—44no—
126#386—44no—
127#385—44no—
128#385—44no—
129#379—24no—
130#379—24no—
131#379—44no—
132#379—44no—
133#379C2×C418yes—
134#379C2×C418yes—
135#379C2×C4116yescomplemented
136C4×C22⋊C4—44no—
137C4×C22⋊C4—44no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25—22no—
155C25—22no—
156C25—42no—
157C25—42no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25C22×C414yes—
169C25C22×C414yes—
170C25#4114yes—
171C25#4114yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×D4—232no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—432no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—432no—
191C22×D4—416no—
192C22×D4—416no—
193C22×D4C22×C418yes—
194C22×D4C22×C418yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C22×C414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225#117—432no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C4116yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—416no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C418yes—
297C2×C22⋊C4C22×C418yes—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C4116yes—
302C2×C22⋊C4C22×C4116yes—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C22×C4116yes—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
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—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—28no—
357C24—44no—
358C24—44no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—44no—
363C24—44no—
364C24—48no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—44no—
373C24—44no—
374C24—48no—
375C24—82no—
376C24—82no—
377C24—82no—
378C24—82no—
379C24—84no—
380C24—84no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24#117116yes—
390C24C23×C414yes—
391C24C23×C412yes—
392C24C23×C412yes—
393C24C23×C412yes—
394C24C2×#4112yes—
395C24C2×#4112yes—
396C24C2×#4112yes—
397C24C2×#4112yes—
398C24C2×#4112yes—
399C24C2×#4112yes—
400C24C2511yescharacteristic
401C2×D4—216no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—4128no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4C23×C418yes—
486C22×C4C23×C418yes—
487C22×C4C23×C414yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—232no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—416no—
518C42—88no—
519C42—88no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—28no—
560C23—28no—
561C23—416no—
562C23—416no—
563C23—416no—
564C23—416no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—88no—
596C23—88no—
597C23—88no—
598C23—88no—
599C23—88no—
600C23—88no—
601C23—832no—
602C23—832no—
603C23—832no—
604C23—88no—
605C23—88no—
606C23—88no—
607C23—88no—
608C23—832no—
609C23C2×#11714yes—
610C23C2×#11714yes—
611C23C2×#11714yes—
612C23C24×C411yescharacteristic
613C23C22×#4111yescharacteristic
614C23C22×#4111yescharacteristic
615C23C2×H(5,𝔽2)12yes—
616C23C2×H(5,𝔽2)12yes—
617C23C2×H(5,𝔽2)12yes—
618C23C2×H(5,𝔽2)12yes—
619C23#58512yes—
620C23#58512yes—
621D4—432no—
622D4—432no—
623C2×C4—232no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—88no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4#51818yes—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—48no—
705C22—48no—
706C22—48no—
707C22—48no—
708C22—48no—
709C22—48no—
710C22—44no—
711C22—48no—
712C22—48no—
713C22—44no—
714C22—44no—
715C22—832no—
716C22—832no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—84no—
722C22—84no—
723C22C22×#11711yescharacteristic
724C22C2×#51812yes—
725C22C4×H(5,𝔽2)12yes—
726C22C4×H(5,𝔽2)12yes—
727C22#349011yescharacteristic
728C22#349011yescharacteristic
729C22#349011yescharacteristic
730C22#349011yescharacteristic
731C22#349012yes—
732C22#349111yescharacteristic
733C22#349111yescharacteristic
734C22#350512yes—
735C22#350512yes—
736C22#354314yes—
737C22#354614yes—
738C22#355814yes—
739C22#356214yes—
740C4—28no—
741C4—44no—
742C4—44no—
743C4—48no—
744C4—48no—
745C4—82no—
746C4—82no—
747C4—84no—
748C4—84no—
749C2—44no—
750C2—44no—
751C2—44no—
752C2—84no—
753C2#3417611yescharacteristic
754C2#3425111yescharacteristic
755C2#3425111yescharacteristic
756C2#3443814yes—
757C2#4638511yescharacteristic
758C2#4638711yescharacteristic
759C2#4638911yescharacteristic
760C2#4639011yescharacteristic
761C2#4644714yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14673C2semidirect
3. #14431C2semidirect
4. #14429C2semidirect
5. #14365C2semidirect
6. #14127C2semidirect
7. #14084C2semidirect
8. #14034C2semidirect
9. #13944C2semidirect
12. #13775C2semidirect
13. #13775C2semidirect
18. #2309C4semidirect
21. C2×#388C22semidirect
22. C2×#388C22semidirect
23. C2×#387C22semidirect
24. C2×#387C22semidirect
29. C2×#385C22semidirect
30. C2×#385C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C2×C4×C22⋊C4C22semidirect
40. #1445C22semidirect
41. #1445C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
54. #1437C22semidirect
55. #1437C22semidirect
56. #1437C22semidirect
57. #1437C22semidirect
58. C4⋊#95C22semidirect
59. #1434C22semidirect
60. #1434C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C4×#95C22semidirect
84. C2×#117C2×C4semidirect
117. #390C2×C4semidirect
118. #390C2×C4semidirect
135. #379C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23092
#146732
#144312
#144292
#143652
#141272
#140842
#140342
#139442
#138672
#137752