Finite Group Databasebeta

← #155752

Subgroups — #155752

Order 512.

4130 conjugacy classes of subgroups.

The subgroup lattice has 809 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3C2×#1441C212yesmaximal, complemented
4C2×C22⋊#95C211yesmaximal, characteristic
5#7557C212yesmaximal, complemented
6#7557C212yesmaximal, complemented
7#7556C212yesmaximal, complemented
8#7555C212yesmaximal
9#7555C212yesmaximal
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C2×#385—24no—
19C2×#385—24no—
20C2×#385C418yescomplemented
21C2×#380—24no—
22C2×#380—28no—
23C22×#95C2212yes—
24C22×#95C2212yes—
25C22×#95C2214yes—
26C22×#95C2211yescharacteristic
27C22×#95C2211yescharacteristic
28C22×#95C2214yes—
29C22×#95C2214yes—
30C22×#95C2212yes—
31C22×#95C2214yes—
32#1441—28no—
33C22⋊#95—24no—
34C26C2311yescharacteristic
35C24×C4—24no—
36C24×C4—24no—
37C24×C4—44no—
38C24×C4—44no—
39C24×C4C2×C414yes—
40C24×C4D414yes—
41C24×C4D414yes—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2314yes—
45C24×C4C2314yes—
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4D418yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—28no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—44no—
77C22×C22⋊C4—44no—
78C22×C22⋊C4—44no—
79C22×C22⋊C4—44no—
80C22×C22⋊C4C2×C414yes—
81C22×C22⋊C4C2×C414yes—
82C22×C22⋊C4Q818yes—
83#385—232no—
84#380—232no—
85C2×#95—28no—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—28no—
94C2×#95—28no—
95C2×#95—24no—
96C2×#95—28no—
97C2×#95—28no—
98C2×#95—24no—
99C2×#95—44no—
100C2×#95—44no—
101C2×#95—44no—
102C2×#95—44no—
103C2×#95—44no—
104C2×#95—44no—
105C2×#95—44no—
106C2×#95—44no—
107C2×#95—44no—
108C2×#95C2×C414yes—
109C2×#95C2×C414yes—
110C25—22no—
111C25—22no—
112C25—22no—
113C25—22no—
114C25—22no—
115C25—22no—
116C25—24no—
117C25—24no—
118C25—22no—
119C25—22no—
120C25C22×C412yes—
121C25C2×D412yes—
122C25C2×D412yes—
123C25C2×Q814yes—
124C25#4114yes—
125C25C2411yescharacteristic
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—216no—
146C23×C4—28no—
147C23×C4—216no—
148C23×C4—28no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—48no—
191C23×C4—48no—
192C23×C4—48no—
193C23×C4—48no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4—416no—
197C23×C4—416no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4C22⋊C4116yes—
201C23×C4C22×C412yes—
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C412yes—
207C23×C4C2×D418yes—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D414yes—
213C23×C4C2×Q818yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4#4114yes—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—216no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—416no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—416no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—416no—
249C2×C22⋊C4—48no—
250#95—216no—
251#95—28no—
252#95—28no—
253#95—416no—
254#95—48no—
255#95—416no—
256#95—416no—
257#95—416no—
258#95—48no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—22no—
267C24—24no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—28no—
282C24—28no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—28no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24C2×C22⋊C414yes—
314C24C4×Q818yes—
315C24C22⋊Q818yes—
316C24#12414yes—
317C24C4⋊Q814yes—
318C24C23×C411yescharacteristic
319C24C22×D411yescharacteristic
320C24C22×D411yescharacteristic
321C24C22×D412yes—
322C24C22×Q812yes—
323C24C2×#4112yes—
324C24C2×#4112yes—
325C24C2×#4112yes—
326C24C2×#4112yes—
327C24H(5,𝔽2)14yes—
328C24H(5,𝔽2)14yes—
329C24H(5,𝔽2)12yes—
330C24H(5,𝔽2)12yes—
331C242−1+412yes—
332C242−1+412yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—232no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—464no—
397C22×C4—432no—
398C22×C4—464no—
399C22×C4—432no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—44no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
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×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4C2×C22⋊C414yes—
501C22×C4C2×C22⋊C414yes—
502C22×C4#11714yes—
503C22×C4#11714yes—
504C22×C4C4×D418yes—
505C22×C4#12014yes—
506C22×C4#12118yes—
507C22×C4#12314yes—
508C4⋊C4—432no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C22⋊C4—232no—
512C22⋊C4—416no—
513C22⋊C4—432no—
514C22⋊C4—432no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—216no—
520C23—28no—
521C23—216no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—216no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—216no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—44no—
538C23—44no—
539C23—48no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—44no—
545C23—48no—
546C23—48no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23#39118yes—
574C23C22×C22⋊C411yescharacteristic
575C23C2×#11711yescharacteristic
576C23C2×#11711yescharacteristic
577C23C2×C4×D412yes—
578C23C2×C4×Q812yes—
579C23#51812yes—
580C23#51812yes—
581C23#52012yes—
582C23#52012yes—
583C23C2×#12011yescharacteristic
584C23C2×#12112yes—
585C23C2×C22⋊Q812yes—
586C23C2×#12311yescharacteristic
587C23#52512yes—
588C23#52512yes—
589C23C2×#12411yescharacteristic
590C23C2×C4⋊Q811yescharacteristic
591C23#53412yes—
592C23#53412yes—
593C23#53412yes—
594C23#53412yes—
595C23#53414yes—
596C23#53511yescharacteristic
597C23#53511yescharacteristic
598C23#53514yes—
599C23#53611yescharacteristic
600C23#53611yescharacteristic
601C23#53712yes—
602C23#53712yes—
603C23#53814yes—
604C23#53814yes—
605C23#53814yes—
606C23#53814yes—
607C23#53914yes—
608C23#53914yes—
609C23#54014yes—
610C23#54214yes—
611C23#54314yes—
612C23#54414yes—
613C23#54612yes—
614C23#54614yes—
615C23#54712yes—
616C23#54812yes—
617C23D4×Q814yes—
618C23#55012yes—
619C23#55212yes—
620C23#55214yes—
621C23#55314yes—
622C23#55314yes—
623C23#55312yes—
624C23#55414yes—
625C23#55414yes—
626C23#55414yes—
627C23#55512yes—
628C23#55612yes—
629C23#56014yes—
630C23#56214yes—
631C23#56214yes—
632C23#56314yes—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—216no—
636C2×C4—28no—
637C2×C4—216no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—464no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—48no—
670C2×C4—48no—
671C2×C4—48no—
672C2×C4—48no—
673C2×C4—48no—
674C2×C4—48no—
675C2×C4—48no—
676C2×C4—48no—
677C2×C4—48no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4—48no—
681C2×C4—48no—
682C2×C4—48no—
683C2×C4—48no—
684C2×C4—48no—
685C2×C4—42no—
686C2×C4—84no—
687C2×C4—82no—
688C2×C4—84no—
689C2×C4—84no—
690C2×C4—82no—
691C2×C4—84no—
692C2×C4—84no—
693C2×C4—84no—
694C2×C4—84no—
695C2×C4—84no—
696C2×C4—84no—
697C2×C4—84no—
698C2×C4—84no—
699C2×C4—84no—
700C2×C4—84no—
701C2×C4—84no—
702C2×C4—84no—
703C2×C4—82no—
704C2×C4—82no—
705C2×C4—84no—
706C2×C4—84no—
707C2×C4—82no—
708C2×C4—82no—
709C2×C4#38618yes—
710C22—28no—
711C22—28no—
712C22—28no—
713C22—44no—
714C22—44no—
715C22—416no—
716C22—416no—
717C22—44no—
718C22—48no—
719C22—48no—
720C22—48no—
721C22—44no—
722C22—416no—
723C22—416no—
724C22—48no—
725C22#228011yescharacteristic
726C22C2×#38611yescharacteristic
727C22C2×#39111yescharacteristic
728C22#230811yescharacteristic
729C22#230811yescharacteristic
730C22#231111yescharacteristic
731C22#231111yescharacteristic
732C22#231812yes—
733C22#231912yes—
734C22#232114yes—
735C22#232914yes—
736C22#234512yes—
737C22#234512yes—
738C22#235212yes—
739C22#235412yes—
740C22#236014yes—
741C22#237614yes—
742C22#240712yes—
743C22#241412yes—
744C22#241512yes—
745C22#241714yes—
746C22#243411yescharacteristic
747C22#243511yescharacteristic
748C22#248512yes—
749C22#250211yescharacteristic
750C22#250311yescharacteristic
751C22#257914yes—
752C22#258312yes—
753C22#259112yes—
754C22#261312yes—
755C22#261714yes—
756C22#262212yes—
757C22#262612yes—
758C22#262712yes—
759C22#263014yes—
760C22#263512yes—
761C22#263812yes—
762C22#263912yes—
763C22#265912yes—
764C22#268814yes—
765C22#269214yes—
766C22#270814yes—
767C22#271114yes—
768C22#271414yes—
769C22#272014yes—
770C22#272914yes—
771C22#273214yes—
772C22#276814yes—
773C22#277014yes—
774C22#277914yes—
775C22#278014yes—
776C22#278114yes—
777C22#278414yes—
778C22#278714yes—
779C22#278814yes—
780C22#281312yes—
781C22#281312yes—
782C22#281412yes—
783C22#281412yes—
784C4—28no—
785C4—44no—
786C4—44no—
787C4—44no—
788C4—82no—
789C4—88no—
790C4—82no—
791C4—88no—
792C2—48no—
793C2#1387111yescharacteristic
794C2#1392311yescharacteristic
795C2#1413911yescharacteristic
796C2#1425511yescharacteristic
797C2#1428111yescharacteristic
798C2#1434411yescharacteristic
799C2#1439214yes—
800C2#1439814yes—
801C2#1465811yescharacteristic
802C2#1655712yes—
803C2#1656114yes—
804C2#1656214yes—
805C2#1660612yes—
806C2#1698612yes—
807C2#1700512yes—
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
3. C2×#1441C2semidirect
5. #7557C2semidirect
6. #7557C2semidirect
7. #7556C2semidirect
20. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C22×#3802
C2×#14412
C2×C22⋊#952
#75572
#75562
#75552