Finite Group Databasebeta

← #371994

Subgroups — #371994

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1958C211yesmaximal, characteristic
2C2×#1933C211yesmaximal, characteristic, complemented
3C2×#1927C211yesmaximal, characteristic, complemented
4C2×#1864C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#14051C211yesmaximal, characteristic, complemented
7#14027C211yesmaximal, characteristic
8#11372C211yesmaximal, characteristic, complemented
9#11122C211yesmaximal, characteristic, complemented
10#11121C211yesmaximal, characteristic
11#11009C211yesmaximal, characteristic
12#10926C211yesmaximal, characteristic, complemented
13#10925C211yesmaximal, characteristic
14#10861C211yesmaximal, characteristic, complemented
15C8×#95C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1958C418yescomplemented
38#1933—24no—
39#1927—24no—
40#1864—24no—
41C4×#102—24no—
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51#1442—22no—
52#1440—22no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57#1434—22no—
58C22⋊#95—22no—
59C4×#95C2211yescharacteristic
60C23×Q8C2311yescharacteristic
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—41no—
71C2×#124—41no—
72C2×#124C2×C414yes—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—41no—
78C2×C4×D4—41no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—41no—
98C2×#106—41no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—41no—
107C2×#103—41no—
108C2×#103—41no—
109C2×#103C2×C414yescomplemented
110C2×#103C2×C414yescomplemented
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—41no—
121C2×#102C2×C414yes—
122C2×#102C2×C414yes—
123C2×C4×C8—22no—
124C2×C4×C8—41no—
125C2×C4×C8—41no—
126C2×C4×C8—41no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8—41no—
130C2×C4×C8C2×C414yescomplemented
131C2×C4×C8D414yes—
132#391—22no—
133#391—22no—
134#391—22no—
135#391—41no—
136#390—22no—
137#390—41no—
138#389—22no—
139#389—22no—
140#382—22no—
141#382C2×C414yescomplemented
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C2×#95—22no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—22no—
154C2×#95D412yes—
155C2×#95D412yes—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#340—22no—
162#336—22no—
163#336—22no—
164#336D414yes—
165C25—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8C22×C412yes—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C22×C412yes—
192C22×D4#4112yes—
193C22×D4#4112yes—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—82no—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C2×D412yes—
234C22×C8C2×D412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8#4112yes—
238C22×C8#4112yes—
239#124—28no—
240#124—44no—
241#124—44no—
242#124—44no—
243C4×D4—28no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4#4112yes—
284C2×C4⋊C4#4112yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—82no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C22×C412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42#4112yes—
314#106—44no—
315#106—44no—
316#103—28no—
317#103—28no—
318#103—44no—
319#103—44no—
320#102—28no—
321#102—28no—
322#102—28no—
323#102—44no—
324C4×C8—44no—
325C4×C8—44no—
326C4×C8—44no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—82no—
337#95—82no—
338#95—82no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24C22×D411yescharacteristic
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4#11714yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—48no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4C4×D414yes—
469C22×C4#12712yes—
470C22×C4#12712yes—
471C22×C4#13514yes—
472C22×C4#13514yes—
473C22×C4C23×C411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4C2×#4111yescharacteristic
483C22×C4H(5,𝔽2)11yescharacteristic
484C22×C4H(5,𝔽2)11yescharacteristic
485C22×C4H(5,𝔽2)11yescharacteristic
486C22×C42−1+411yescharacteristic
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8C4×D414yes—
514C2×C8C4×D414yes—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—84no—
540C22⋊C4—416no—
541C22⋊C4—48no—
542C22⋊C4—88no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—84no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—82no—
570C23—82no—
571C23—81no—
572C23—81no—
573C23—81no—
574C23—81no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23C2×C4×D411yescharacteristic
580C23C2×#12711yescharacteristic
581C23#53211yescharacteristic
582C23#54611yescharacteristic
583C23#54611yescharacteristic
584C23#54611yescharacteristic
585C23#54811yescharacteristic
586C23C2×#13512yes—
587C23C2×#13512yes—
588C23#57712yes—
589C23#57712yes—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—81no—
595Q8—81no—
596Q8—81no—
597Q8—81no—
598D4—24no—
599D4—24no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—816no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—82no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4#49514yes—
668C2×C4C2×#11711yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C4×#4111yescharacteristic
672C2×C4#51811yescharacteristic
673C2×C4#52011yescharacteristic
674C2×C4#52511yescharacteristic
675C2×C4#52511yescharacteristic
676C2×C4#52911yescharacteristic
677C2×C4#52911yescharacteristic
678C2×C4#53211yescharacteristic
679C2×C4#53211yescharacteristic
680C2×C4#53411yescharacteristic
681C2×C4#53711yescharacteristic
682C2×C4#53811yescharacteristic
683C2×C4#53911yescharacteristic
684C2×C4#54611yescharacteristic
685C2×C4#54611yescharacteristic
686C2×C4#54611yescharacteristic
687C2×C4#54811yescharacteristic
688C2×C4#55111yescharacteristic
689C2×C4#55111yescharacteristic
690C2×C4#55211yescharacteristic
691C2×C4#55311yescharacteristic
692C2×C4#55311yescharacteristic
693C2×C4#55311yescharacteristic
694C2×C4#55611yescharacteristic
695C2×C4#55911yescharacteristic
696C8—24no—
697C8—24no—
698C8—42no—
699C8—42no—
700C22—44no—
701C22—48no—
702C22—84no—
703C22—84no—
704C22—82no—
705C22—84no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22C4×#12711yescharacteristic
712C22#234211yescharacteristic
713C22#236711yescharacteristic
714C22#253911yescharacteristic
715C22#254011yescharacteristic
716C22#255811yescharacteristic
717C22#255911yescharacteristic
718C22C4×#13512yes—
719C22#295212yes—
720C22C2×#49511yescharacteristic
721C22#314711yescharacteristic
722C22#315411yescharacteristic
723C22#315411yescharacteristic
724C22#316812yes—
725C22#317312yes—
726C22#329612yes—
727C22#329612yes—
728C22#329612yes—
729C22#330112yes—
730C22#330312yes—
731C22#330312yes—
732C22#336012yes—
733C22#336512yes—
734C4—24no—
735C4—44no—
736C4—44no—
737C4—42no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—161no—
744C4—161no—
745C4#229011yescharacteristic
746C4#232511yescharacteristic
747C4#235211yescharacteristic
748C4#236411yescharacteristic
749C4#255711yescharacteristic
750C4#256011yescharacteristic
751C4#260111yescharacteristic
752C4#260211yescharacteristic
753C2—84no—
754C2#1408411yescharacteristic
755C2C4×#49511yescharacteristic
756C2#2169411yescharacteristic
757C2#2257212yes—
758C2#2321912yes—
759C2#2833011yescharacteristic
760C2#2833911yescharacteristic
761C2#2839811yescharacteristic
762C2#2840811yescharacteristic
763C2#2901212yes—
764C2#2904412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1933C2semidirect
3. C2×#1927C2semidirect
6. #14051C2semidirect
8. #11372C2semidirect
9. #11122C2semidirect
12. #10926C2semidirect
14. #10861C2semidirect
15. C8×#95C2semidirect
37. #1958C4semidirect
109. C2×#103C2×C4semidirect
110. C2×#103C2×C4semidirect
130. C2×C4×C8C2×C4semidirect
141. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19582
C2×#19332
C2×#19272
C2×#18642
C2×C4×#1022
#140512
#140272
#113722
#111222
#111212
#110092
#109262
#109252
#108612
C8×#952