Finite Group Databasebeta

← #155753

Subgroups — #155753

Order 512.

3842 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C2×C22⋊#95C211yesmaximal, characteristic
3C2×C22⋊#95C211yesmaximal, characteristic
4#7557C212yesmaximal, complemented
5#7557C212yesmaximal, complemented
6#7556C212yesmaximal, complemented
7#7556C212yesmaximal, complemented
8#7555C212yesmaximal
9#7555C212yesmaximal
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C2×#385—24no—
18C2×#385—24no—
19C2×#385C418yescomplemented
20C22×#95C2214yes—
21C22×#95C2211yescharacteristic
22C22×#95C2211yescharacteristic
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2212yes—
26C22×#95C2212yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29C22×#95C2214yes—
30C22⋊#95—24no—
31C22⋊#95—24no—
32C26C2311yescharacteristic
33C24×C4—44no—
34C24×C4—44no—
35C24×C4—44no—
36C24×C4C2×C414yes—
37C24×C4D414yes—
38C24×C4D414yes—
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2314yes—
42C24×C4C2314yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—44no—
52C22×C4⋊C4—44no—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C414yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—28no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4—44no—
71C22×C22⋊C4C2×C414yes—
72C22×C22⋊C4C2×C414yes—
73C22×C22⋊C4Q818yes—
74#385—232no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—28no—
84C2×#95—28no—
85C2×#95—28no—
86C2×#95—28no—
87C2×#95—44no—
88C2×#95—44no—
89C2×#95—44no—
90C2×#95—44no—
91C2×#95—44no—
92C2×#95—44no—
93C2×#95—44no—
94C2×#95—44no—
95C2×#95—44no—
96C2×#95—44no—
97C2×#95—44no—
98C2×#95—44no—
99C2×#95C2×C414yes—
100C2×#95C2×C414yes—
101C25—22no—
102C25—22no—
103C25—24no—
104C25—22no—
105C25—24no—
106C25—24no—
107C25—24no—
108C25—22no—
109C25C22×C412yes—
110C25C2×D412yes—
111C25C2×D412yes—
112C25C2×Q814yes—
113C25#4114yes—
114C25C2411yescharacteristic
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—48no—
172C23×C4—48no—
173C23×C4—48no—
174C23×C4—48no—
175C23×C4—48no—
176C23×C4—48no—
177C23×C4—48no—
178C23×C4—48no—
179C23×C4—48no—
180C23×C4—416no—
181C23×C4—416no—
182C23×C4—416no—
183C23×C4—416no—
184C23×C4C22⋊C4116yes—
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×Q818yes—
196C23×C4#4118yes—
197C23×C4#4114yes—
198C23×C4#4114yes—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C22⋊C4—216no—
208C2×C22⋊C4—216no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—216no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—216no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—416no—
217C2×C22⋊C4—416no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—48no—
226#95—28no—
227#95—28no—
228#95—416no—
229#95—48no—
230#95—48no—
231#95—48no—
232#95—48no—
233#95—416no—
234#95—416no—
235#95—416no—
236C24—22no—
237C24—22no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—28no—
257C24—28no—
258C24—28no—
259C24—28no—
260C24—28no—
261C24—28no—
262C24—28no—
263C24—28no—
264C24—28no—
265C24—28no—
266C24—28no—
267C24—28no—
268C24—44no—
269C24—44no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24C2×C22⋊C414yes—
285C24C4×Q818yes—
286C24C22⋊Q818yes—
287C24#12414yes—
288C24C4⋊Q814yes—
289C24C23×C411yescharacteristic
290C24C22×D411yescharacteristic
291C24C22×D411yescharacteristic
292C24C22×Q812yes—
293C24C2×#4112yes—
294C24C2×#4112yes—
295C24C2×#4112yes—
296C24C2×#4114yes—
297C24H(5,𝔽2)14yes—
298C24H(5,𝔽2)14yes—
299C24H(5,𝔽2)14yes—
300C24H(5,𝔽2)12yes—
301C242−1+412yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—416no—
359C22×C4—416no—
360C22×C4—416no—
361C22×C4—464no—
362C22×C4—464no—
363C22×C4—464no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4C2×C22⋊C414yes—
463C22×C4C2×C22⋊C414yes—
464C22×C4#11714yes—
465C22×C4#11714yes—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C22⋊C4—232no—
469C22⋊C4—432no—
470C22⋊C4—416no—
471C22⋊C4—432no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—216no—
477C23—24no—
478C23—216no—
479C23—28no—
480C23—28no—
481C23—216no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—216no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—44no—
493C23—44no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—44no—
498C23—44no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23#39118yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×#11711yescharacteristic
527C23C2×#11711yescharacteristic
528C23C2×C4×Q812yes—
529C23C4×#4112yes—
530C23#51812yes—
531C23#51812yes—
532C23#51812yes—
533C23#52012yes—
534C23C2×C22⋊Q812yes—
535C23#52512yes—
536C23#52512yes—
537C23#52512yes—
538C23C2×#12411yescharacteristic
539C23C2×C4⋊Q811yescharacteristic
540C23#53212yes—
541C23#53412yes—
542C23#53412yes—
543C23#53412yes—
544C23#53412yes—
545C23#53412yes—
546C23#53412yes—
547C23#53511yescharacteristic
548C23#53611yescharacteristic
549C23#53712yes—
550C23#53814yes—
551C23#53814yes—
552C23#53814yes—
553C23#53914yes—
554C23#53914yes—
555C23#53914yes—
556C23#54014yes—
557C23#54014yes—
558C23#54214yes—
559C23#54214yes—
560C23#54314yes—
561C23#54414yes—
562C23#55114yes—
563C23#55114yes—
564C23#55114yes—
565C23#55314yes—
566C23#55314yes—
567C23#55314yes—
568C23#55314yes—
569C23#55414yes—
570C23#55414yes—
571C23#55414yes—
572C23#55414yes—
573C23#55914yes—
574C23#56014yes—
575C23#56014yes—
576C23#56014yes—
577C23H(3,𝔽4)14yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—464no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4—82no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C22—28no—
650C22—28no—
651C22—28no—
652C22—44no—
653C22—44no—
654C22—416no—
655C22—44no—
656C22—416no—
657C22—416no—
658C22—44no—
659C22—416no—
660C22—416no—
661C22—416no—
662C22#228011yescharacteristic
663C22C2×#39111yescharacteristic
664C22#229711yescharacteristic
665C22#230811yescharacteristic
666C22#230811yescharacteristic
667C22#230811yescharacteristic
668C22#231111yescharacteristic
669C22#231812yes—
670C22#231912yes—
671C22#232914yes—
672C22#233514yes—
673C22#234512yes—
674C22#234512yes—
675C22#235312yes—
676C22#235712yes—
677C22#236814yes—
678C22#237514yes—
679C22#256112yes—
680C22#256414yes—
681C22#257914yes—
682C22#258312yes—
683C22#258312yes—
684C22#259112yes—
685C22#260014yes—
686C22#260214yes—
687C22#262612yes—
688C22#262712yes—
689C22#263014yes—
690C22#263014yes—
691C22#263512yes—
692C22#263812yes—
693C22#263912yes—
694C22#265912yes—
695C22#266112yes—
696C22#266312yes—
697C22#274614yes—
698C22#274814yes—
699C22#274914yes—
700C22#275314yes—
701C22#275714yes—
702C22#276314yes—
703C22#277814yes—
704C22#278014yes—
705C22#278014yes—
706C22#278214yes—
707C22#279014yes—
708C22#279014yes—
709C22#279614yes—
710C22#279714yes—
711C22#280414yes—
712C22#280714yes—
713C22#281214yes—
714C22#281214yes—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—82no—
720C4—82no—
721C4—88no—
722C4—88no—
723C2—48no—
724C2#1392311yescharacteristic
725C2#1394011yescharacteristic
726C2#1415911yescharacteristic
727C2#1428111yescharacteristic
728C2#1428711yescharacteristic
729C2#1434411yescharacteristic
730C2#1445614yes—
731C2#1445714yes—
732C2#1465711yescharacteristic
733C2#1688514yes—
734C2#1689314yes—
735C2#1689414yes—
736C2#1698714yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7557C2semidirect
5. #7557C2semidirect
6. #7556C2semidirect
7. #7556C2semidirect
19. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C2×C22⋊#952
#75572
#75562
#75552