Finite Group Databasebeta

← #576606

Subgroups — #576606

Order 512.

4090 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic, complemented
3C2×#2309C211yesmaximal, characteristic
4#14673C211yesmaximal, characteristic, complemented
5#14573C214yesmaximal, complemented
6#14374C214yesmaximal, complemented
7#14366C214yesmaximal, complemented
8#14201C214yesmaximal, complemented
9#14095C214yesmaximal, complemented
10#14063C214yesmaximal, complemented
11#13869C211yesmaximal, characteristic, complemented
12#13867C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C22×#117C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2405C4116yescomplemented
20C2×#394C2214yes—
21#2312—28no—
22#2309—28no—
23C2×#390C2214yes—
24C2×#390C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yescomplemented
27C2×#388C2214yescomplemented
28C2×#387C2214yescomplemented
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#381C2212yescomplemented
34C2×#380C2212yescomplemented
35C2×#379C2212yes—
36C2×#379C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×#379C2212yes—
39C2×#379C2211yescharacteristic
40C2×#379C2212yes—
41C2×#379C2211yescharacteristic
42C2×C4×C4⋊C4C2214yescomplemented
43C2×C4×C22⋊C4C2212yescomplemented
44C2×C4×C22⋊C4C2212yescomplemented
45#1444C2214yescomplemented
46#1441C2214yescomplemented
47#1440C2214yes—
48#1440C2214yescomplemented
49#1439C2214yes—
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yescomplemented
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1437C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2212yes—
66C22⋊#95C2212yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C22⋊#95C2212yescomplemented
70C26C2311yescharacteristic
71C26C2311yescharacteristic
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—44no—
75C23×D4—44no—
76C23×D4C2×C418yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—44no—
85C2×#123—44no—
86C2×#123C2×C4116yescomplemented
87C2×#117—28no—
88C2×#117—44no—
89C2×#117—44no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4C2×C418yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124#394C2×C4132yes—
125#390—216no—
126#390—216no—
127#388—216no—
128#388—216no—
129#388—48no—
130#387—48no—
131#386—44no—
132#386—44no—
133#386C2×C4116yescomplemented
134#385C2×C4116yescomplemented
135#381—44no—
136#380—28no—
137#379—24no—
138#379—24no—
139#379—28no—
140#379—44no—
141#379C2×C418yes—
142#379C2×C4116yescomplemented
143#379D418yes—
144C4×C4⋊C4—48no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—44no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C25—22no—
163C25—22no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25C22×C414yes—
177C25C22×C414yes—
178C25C2×D414yes—
179C25#4114yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×D4—232no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—432no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—432no—
199C22×D4—416no—
200C22×D4—416no—
201C22×D4C22×C418yes—
202C22×D4C22×C418yes—
203C23×C4—22no—
204C23×C4—22no—
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—22no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235#123—432no—
236#117—432no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
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⋊C4C22×C4116yes—
249C2×C22⋊C4—232no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
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—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C22⋊C4C22×C4116yes—
311C2×C22⋊C4C22×C418yes—
312C2×C22⋊C4C22×C418yes—
313C2×C22⋊C4C22×C4116yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C4116yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—28no—
370C24—44no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—44no—
375C24—44no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—48no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—48no—
384C24—48no—
385C24—48no—
386C24—44no—
387C24—44no—
388C24—84no—
389C24—82no—
390C24—82no—
391C24—82no—
392C24—82no—
393C24—84no—
394C24—88no—
395C24—88no—
396C24—88no—
397C24—88no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—88no—
402C24C4×D4116yes—
403C24C23×C414yes—
404C24C23×C412yes—
405C24C23×C412yes—
406C24C23×C412yes—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C2×#4112yes—
411C24C2×#4112yes—
412C24C2×#4112yes—
413C24C2511yescharacteristic
414C2×D4—216no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—4128no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—216no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
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—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—88no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—84no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—88no—
499C22×C4C23×C414yes—
500C22×C4C23×C418yes—
501C22×C4C23×C414yes—
502C22×C4C23×C414yes—
503C22×C4C22×D418yes—
504C22×C4H(5,𝔽2)14yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)14yes—
507C4⋊C4—232no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—88no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—88no—
533C42—416no—
534C42—416no—
535C42—88no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—28no—
576C23—28no—
577C23—416no—
578C23—416no—
579C23—416no—
580C23—416no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—88no—
612C23—88no—
613C23—832no—
614C23—88no—
615C23—88no—
616C23—88no—
617C23—88no—
618C23—832no—
619C23—832no—
620C23—832no—
621C23—88no—
622C23—88no—
623C23—88no—
624C23—88no—
625C23C2×C4×D414yes—
626C23C2×C4×D414yes—
627C23C2×C4×D414yes—
628C23C24×C411yescharacteristic
629C23C23×D411yescharacteristic
630C23C22×#4111yescharacteristic
631C23C2×H(5,𝔽2)12yes—
632C23C2×H(5,𝔽2)12yes—
633C23C2×H(5,𝔽2)12yes—
634C23#58512yes—
635C23#58512yes—
636C23#58512yes—
637D4—432no—
638D4—432no—
639C2×C4—232no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—24no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
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—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
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—88no—
695C2×C4—88no—
696C2×C4—88no—
697C2×C4—88no—
698C2×C4#51818yes—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—48no—
709C22—48no—
710C22—48no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—48no—
724C22—48no—
725C22—48no—
726C22—44no—
727C22—44no—
728C22—48no—
729C22—48no—
730C22—44no—
731C22—832no—
732C22—832no—
733C22—84no—
734C22—84no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22C22×C4×D411yescharacteristic
740C22C2×#51812yes—
741C22#342812yes—
742C22C4×H(5,𝔽2)12yes—
743C22#348412yes—
744C22#348512yes—
745C22#349011yescharacteristic
746C22#349011yescharacteristic
747C22#349011yescharacteristic
748C22#349111yescharacteristic
749C22#349111yescharacteristic
750C22#349111yescharacteristic
751C22#349612yes—
752C22#354014yes—
753C22#354214yes—
754C22#354614yes—
755C22#356214yes—
756C4—28no—
757C4—48no—
758C4—44no—
759C4—44no—
760C4—48no—
761C4—44no—
762C4—84no—
763C4—84no—
764C4—82no—
765C2—44no—
766C2—44no—
767C2—44no—
768C2—84no—
769C2#3419811yescharacteristic
770C2#3419911yescharacteristic
771C2#3427311yescharacteristic
772C2#3447814yes—
773C2#4635314yes—
774C2#4638911yescharacteristic
775C2#4639011yescharacteristic
776C2#4699711yescharacteristic
777C2#4699811yescharacteristic
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. C2×#2312C2semidirect
4. #14673C2semidirect
5. #14573C2semidirect
6. #14374C2semidirect
7. #14366C2semidirect
8. #14201C2semidirect
9. #14095C2semidirect
10. #14063C2semidirect
11. #13869C2semidirect
13. #13775C2semidirect
19. #2405C4semidirect
26. C2×#388C22semidirect
27. C2×#388C22semidirect
28. C2×#387C22semidirect
33. C2×#381C22semidirect
34. C2×#380C22semidirect
42. C2×C4×C4⋊C4C22semidirect
43. C2×C4×C22⋊C4C22semidirect
44. C2×C4×C22⋊C4C22semidirect
45. #1444C22semidirect
46. #1441C22semidirect
48. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
57. #1437C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
69. C22⋊#95C22semidirect
86. C2×#123C2×C4semidirect
133. #386C2×C4semidirect
134. #385C2×C4semidirect
142. #379C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
C2×#23122
C2×#23092
#146732
#145732
#143742
#143662
#142012
#140952
#140632
#138692
#138672
#137752