Finite Group Databasebeta

← #366571

Subgroups — #366571

Order 512.

3118 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#102C211yesmaximal, characteristic
2C2×#2435C211yesmaximal, characteristic
3C2×#1958C212yesmaximal, complemented
4C2×#1892C211yesmaximal, characteristic, complemented
5C2×#1872C211yesmaximal, characteristic, complemented
6C2×#1786C211yesmaximal, characteristic
7C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
8#13881C211yesmaximal, characteristic, complemented
9#11051C212yesmaximal, complemented
10#11009C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C24×D4C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#124C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2211yescharacteristic, complemented
18C22×#103C2212yescomplemented
19C22×#103C2211yescharacteristic, complemented
20C22×#102—28no—
21C22×#102—24no—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×#102C2212yes—
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2212yescomplemented
28#2435—24no—
29#2435C418yescomplemented
30C2×#394—22no—
31C2×#390C2212yes—
32C2×#382C2212yescomplemented
33C2×#380C2211yescharacteristic
34C2×#379—22no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1958—28no—
37#1892—24no—
38#1872—24no—
39#1786—24no—
40C8×C22⋊C4—24no—
41C2×#340C2211yescharacteristic, complemented
42C2×#340C2212yescomplemented
43C2×#340C2211yescharacteristic, complemented
44C2×#336C2212yescomplemented
45#1442—22no—
46C4⋊#95C2212yescomplemented
47C22⋊#95—22no—
48C26—21no—
49C23×Q8C2311yescharacteristic
50C23×D4—24no—
51C23×D4—24no—
52C23×D4—22no—
53C23×D4—22no—
54C23×D4—21no—
55C23×D4—22no—
56C23×D4—22no—
57C23×D4—22no—
58C23×D4—42no—
59C23×D4—42no—
60C23×D4C2×C414yes—
61C23×D4C2312yes—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—42no—
68C23×C8D414yes—
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#124—24no—
73C2×#124—24no—
74C2×#124—24no—
75C2×#124—42no—
76C2×#124—42no—
77C2×#124C2×C418yescomplemented
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8C2×C414yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D412yes—
99C22×C42C2312yes—
100C2×#103—24no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#102—232no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—24no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—42no—
118C2×#102—42no—
119C2×#102D418yes—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8D414yes—
123C2×C4×C8—24no—
124C2×C4×C8—24no—
125C2×C4×C8—42no—
126#394—216no—
127#390—24no—
128#390C2×C418yescomplemented
129#382—24no—
130#382—42no—
131#380—22no—
132#380—22no—
133#379—216no—
134C4×C22⋊C4C2×C414yes—
135C4×C22⋊C4D414yes—
136C2×#95—24no—
137C2×#95—21no—
138C2×#95—24no—
139C2×#95D412yes—
140#340—24no—
141#340—22no—
142#340—22no—
143#336—24no—
144C25—22no—
145C25—24no—
146C25—44no—
147C25—44no—
148C25—42no—
149C25—42no—
150C25—44no—
151C25—44no—
152C25—82no—
153C25—82no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8C22×C412yes—
165C22×D4—28no—
166C22×D4—216no—
167C22×D4—28no—
168C22×D4—216no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4—48no—
187C22×D4—48no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—48no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C22×C414yes—
200C22×D4C2×D414yes—
201C22×D4#4114yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—82no—
215C23×C4C22×C412yes—
216C23×C4C2×D414yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—28no—
228C22×C8—21no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—21no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—28no—
239C22×C8—44no—
240C22×C8—48no—
241C22×C8—48no—
242C22×C8C2×D412yes—
243C22×C8C2×D414yes—
244C22×C8C2×D414yes—
245C22×C8C2×D412yes—
246#124—216no—
247#124—48no—
248#124—48no—
249#124—48no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—84no—
294C2×C22⋊C4—84no—
295C2×C22⋊C4C22⋊C418yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C22×C414yes—
301C2×C42C2×D414yes—
302#103—48no—
303#103—44no—
304#103—44no—
305#102—216no—
306#102—216no—
307#102—48no—
308#102—48no—
309C22⋊C8—28no—
310C4×C8—48no—
311#95—48no—
312#95—42no—
313#95—84no—
314#95C22⋊C418yes—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—416no—
323C24—48no—
324C24—416no—
325C24—48no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—48no—
335C24—416no—
336C24—88no—
337C24—816no—
338C24—816no—
339C24—816no—
340C24—816no—
341C24—88no—
342C24—84no—
343C24—84no—
344C24—88no—
345C24—84no—
346C24—84no—
347C24C2×C22⋊C412yes—
348C24#12412yes—
349C24#12712yes—
350C24#13514yes—
351C24#13514yes—
352C24C22×D411yescharacteristic
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×D4—232no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—432no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—432no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—48no—
409C2×D4C4×D418yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
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—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—48no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—816no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C2×C22⋊C418yes—
460C22×C4C2×C22⋊C412yes—
461C22×C4C4×D414yes—
462C22×C4C4×D414yes—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4#12412yes—
466C22×C4#12712yes—
467C22×C4C23×C411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—416no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8#12014yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—48no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C22⋊C4—216no—
512C22⋊C4—432no—
513C22⋊C4—48no—
514C22⋊C4—816no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—22no—
532C23—28no—
533C23—432no—
534C23—44no—
535C23—48no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—44no—
545C23—44no—
546C23—864no—
547C23—864no—
548C23—816no—
549C23—816no—
550C23—816no—
551C23—816no—
552C23—816no—
553C23—84no—
554C23—84no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23—816no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23#39014yes—
565C23#41618yes—
566C23#48818yes—
567C23#49514yes—
568C23#49514yes—
569C23C22×C22⋊C411yescharacteristic
570C23C2×#12411yescharacteristic
571C23C2×#12711yescharacteristic
572C23D4212yes—
573C23#54612yes—
574C23C2×#13512yes—
575C23C2×#13512yes—
576C23#57612yes—
577C23#57712yes—
578Q8—42no—
579Q8—44no—
580Q8—42no—
581Q8—81no—
582Q8—81no—
583Q8—82no—
584D4—28no—
585D4—28no—
586D4—432no—
587D4—44no—
588D4—44no—
589D4—44no—
590D4—44no—
591D4—44no—
592D4—44no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—816no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—82no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—84no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—81no—
634C2×C4#39014yes—
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×D412yes—
638C2×C4#51812yes—
639C2×C4C2×#12011yescharacteristic
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4#52511yescharacteristic
643C2×C4#52512yes—
644C2×C4#53411yescharacteristic
645C2×C4#53511yescharacteristic
646C2×C4#53812yes—
647C2×C4D4212yes—
648C2×C4D4212yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#55112yes—
653C8—24no—
654C8—42no—
655C8—44no—
656C22—216no—
657C22—216no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—416no—
665C22—48no—
666C22—864no—
667C22—88no—
668C22—88no—
669C22—88no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22#195818yescomplemented
677C22C2×#39011yescharacteristic
678C22D4×C22⋊C412yes—
679C22#243311yescharacteristic
680C22#247212yes—
681C22#253911yescharacteristic
682C22C2×#41612yes—
683C22#289712yes—
684C22C2×#48812yes—
685C22#313612yes—
686C22C2×#49511yescharacteristic
687C22C2×#49511yescharacteristic
688C22#315411yescharacteristic
689C22#315511yescharacteristic
690C22#328014yes—
691C22#328214yes—
692C22#329614yes—
693C22#329714yes—
694C4—24no—
695C4—48no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—161no—
700C4—161no—
701C4C4×#12011yescharacteristic
702C4#232111yescharacteristic
703C4#235812yes—
704C4#241011yescharacteristic
705C4#241111yescharacteristic
706C4#242412yes—
707C2—28no—
708C2—88no—
709C2#1399511yescharacteristic
710C2C2×#195811yescharacteristic
711C2#2120611yescharacteristic
712C2#2241414yes—
713C2#2577611yescharacteristic
714C2#2578111yescharacteristic
715C2#2725614yes—
716C2#2833011yescharacteristic
717C2#2833111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1958C2semidirect
4. C2×#1892C2semidirect
5. C2×#1872C2semidirect
7. C2×C8×C22⋊C4C2semidirect
8. #13881C2semidirect
9. #11051C2semidirect
10. #11009C2semidirect
11. #10925C2semidirect
17. C22×#103C22semidirect
18. C22×#103C22semidirect
19. C22×#103C22semidirect
27. C22×C4×C8C22semidirect
29. #2435C4semidirect
32. C2×#382C22semidirect
41. C2×#340C22semidirect
42. C2×#340C22semidirect
43. C2×#340C22semidirect
44. C2×#336C22semidirect
46. C4⋊#95C22semidirect
77. C2×#124C2×C4semidirect
128. #390C2×C4semidirect
676. C22#1958semidirect

Maximal subgroups

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

subgroupindex
C23×#1022
C2×#24352
C2×#19582
C2×#18922
C2×#18722
C2×#17862
C2×C8×C22⋊C42
#138812
#110512
#110092
#109252