Finite Group Databasebeta

← #154229

Subgroups — #154229

Order 512.

4514 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#381C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4C2×#1440C212yesmaximal
5C2×#1436C212yesmaximal, complemented
6#7557C212yesmaximal, complemented
7#7556C212yesmaximal, complemented
8#7555C212yesmaximal
9#7554C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C23×C42C2212yes—
18C2×#385—24no—
19C2×#385—24no—
20C2×#385C418yescomplemented
21C2×#381—28no—
22C2×#381—24no—
23C2×C4×C22⋊C4—24no—
24C2×C4×C22⋊C4—28no—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29C22×#95C2214yes—
30C22×#95C2214yes—
31C22×#95C2211yescharacteristic
32C22×#95C2211yescharacteristic
33C22×#95C2212yes—
34#1440—28no—
35#1436—28no—
36C26C2311yescharacteristic
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—44no—
42C24×C4C2×C414yes—
43C24×C4D414yes—
44C24×C4D414yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2314yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2314yes—
52C24×C4C2311yescharacteristic
53C22×C4⋊C4—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4C2×C414yes—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—28no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—44no—
79C22×C22⋊C4C2×C414yes—
80C22×C22⋊C4C2×C414yes—
81C22×C22⋊C4D418yes—
82C22×C42—28no—
83C22×C42—24no—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42—44no—
87#385—232no—
88#381—232no—
89C4×C22⋊C4—232no—
90C2×#95—28no—
91C2×#95—24no—
92C2×#95—28no—
93C2×#95—28no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—28no—
105C2×#95—28no—
106C2×#95—28no—
107C2×#95—28no—
108C2×#95—28no—
109C2×#95—28no—
110C2×#95—44no—
111C2×#95—44no—
112C2×#95C2×C414yes—
113C2×#95C2×C414yes—
114C2×#95D418yes—
115C25—22no—
116C25—22no—
117C25—22no—
118C25—24no—
119C25—24no—
120C25—22no—
121C25—24no—
122C25—24no—
123C25C22×C412yes—
124C25C2×D412yes—
125C25C2×D412yes—
126C25C2×D414yes—
127C25#4114yes—
128C25C2411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—216no—
133C23×C4—28no—
134C23×C4—216no—
135C23×C4—28no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—216no—
165C23×C4—28no—
166C23×C4—216no—
167C23×C4—28no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4—48no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4—416no—
202C23×C4C22⋊C4116yes—
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C412yes—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C2×D418yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D418yes—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4#4118yes—
219C23×C4#4118yes—
220C23×C4#4114yes—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—44no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—232no—
233C2×C22⋊C4—216no—
234C2×C22⋊C4—216no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—232no—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4—416no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—44no—
250C2×C42—216no—
251C2×C42—416no—
252C2×C42—416no—
253C2×C42—48no—
254C2×C42—48no—
255#95—232no—
256#95—216no—
257#95—216no—
258#95—28no—
259#95—28no—
260#95—216no—
261#95—416no—
262#95—416no—
263#95—416no—
264C24—22no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—28no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—28no—
278C24—24no—
279C24—24no—
280C24—28no—
281C24—28no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—28no—
294C24—28no—
295C24—28no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—44no—
308C24—44no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24C2×C22⋊C414yes—
313C24C4×D418yes—
314C24#12014yes—
315C24#12118yes—
316C24#12314yes—
317C24C23×C411yescharacteristic
318C24C22×D411yescharacteristic
319C24C22×D411yescharacteristic
320C24C22×D412yes—
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24C2×#4112yes—
325C24C2×#4112yes—
326C24C2×#4114yes—
327C24C2×#4114yes—
328C24H(5,𝔽2)14yes—
329C242−1+414yes—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—232no—
387C22×C4—232no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—416no—
404C22×C4—464no—
405C22×C4—432no—
406C22×C4—432no—
407C22×C4—432no—
408C22×C4—432no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
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—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
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—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
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×C4C2×C22⋊C414yes—
494C22×C4C2×C22⋊C414yes—
495C22×C4#11714yes—
496C22×C4#11714yes—
497C22×C4C4×D418yes—
498C22×C4#12014yes—
499C22×C4#12118yes—
500C22×C4#123116yes—
501C22×C4#12314yes—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C22⋊C4—264no—
505C22⋊C4—232no—
506C22⋊C4—432no—
507C22⋊C4—416no—
508C42—432no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—216no—
514C23—28no—
515C23—28no—
516C23—216no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—216no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—216no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—44no—
539C23—48no—
540C23—44no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—44no—
555C23—48no—
556C23—48no—
557C23—44no—
558C23—44no—
559C23—48no—
560C23—48no—
561C23#38618yes—
562C23C22×C22⋊C411yescharacteristic
563C23C2×#11711yescharacteristic
564C23C2×#11711yescharacteristic
565C23C2×C4×D412yes—
566C23C2×C4×D412yes—
567C23C4×#4112yes—
568C23C4×#4112yes—
569C23#51812yes—
570C23#51912yes—
571C23C2×#12011yescharacteristic
572C23C2×#12011yescharacteristic
573C23C2×#12112yes—
574C23C2×#12112yes—
575C23C2×#12314yes—
576C23C2×#12311yescharacteristic
577C23C2×#12311yescharacteristic
578C23#52512yes—
579C23#52512yes—
580C23#52514yes—
581C23#52512yes—
582C23#52512yes—
583C23#52514yes—
584C23#52914yes—
585C23#52914yes—
586C23#52914yes—
587C23#52914yes—
588C23#53212yes—
589C23#53212yes—
590C23#53214yes—
591C23#53412yes—
592C23#53412yes—
593C23#53612yes—
594C23#53612yes—
595C23#53914yes—
596C23#53914yes—
597C23#54214yes—
598C23#54214yes—
599C23D4214yes—
600C23#54614yes—
601C23#54614yes—
602C23#54714yes—
603C23#54714yes—
604C23#54714yes—
605C23#54814yes—
606C23#55114yes—
607C23#55114yes—
608C23#55214yes—
609C23#55214yes—
610C23#55214yes—
611C23#55314yes—
612C23#55314yes—
613C23#55614yes—
614C23#55914yes—
615C2×C4—24no—
616C2×C4—216no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—216no—
624C2×C4—28no—
625C2×C4—216no—
626C2×C4—216no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—464no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—44no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—48no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—48no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—48no—
675C2×C4—48no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—82no—
685C2×C4—82no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—84no—
689C2×C4—84no—
690C2×C4—84no—
691C2×C4#38618yes—
692C22—28no—
693C22—28no—
694C22—28no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—416no—
699C22—416no—
700C22—416no—
701C22—416no—
702C22—416no—
703C22—44no—
704C22—416no—
705C22#228011yescharacteristic
706C22C2×#38611yescharacteristic
707C22C2×#38611yescharacteristic
708C22#229012yes—
709C22#229312yes—
710C22#229711yescharacteristic
711C22#229711yescharacteristic
712C22C4×#12314yes—
713C22#230811yescharacteristic
714C22#231011yescharacteristic
715C22#233214yes—
716C22#234412yes—
717C22#234412yes—
718C22#235312yes—
719C22#235712yes—
720C22#235814yes—
721C22#236214yes—
722C22#238812yes—
723C22#239214yes—
724C22#239512yes—
725C22#240612yes—
726C22#241012yes—
727C22#241012yes—
728C22#241212yes—
729C22#241212yes—
730C22#241314yes—
731C22#241514yes—
732C22#241914yes—
733C22#242214yes—
734C22#243212yes—
735C22#244314yes—
736C22#244714yes—
737C22#245514yes—
738C22#245814yes—
739C22#246114yes—
740C22#246614yes—
741C22#246814yes—
742C22#247014yes—
743C22#248412yes—
744C22#249912yes—
745C22#251514yes—
746C22#252714yes—
747C22#255314yes—
748C22#255814yes—
749C22#257314yes—
750C22#257514yes—
751C22#260014yes—
752C22#260214yes—
753C22#263814yes—
754C22#266314yes—
755C22#284912yes—
756C22#285312yes—
757C22#285414yes—
758C4—28no—
759C4—44no—
760C4—44no—
761C4—44no—
762C4—416no—
763C4—82no—
764C4—82no—
765C4—88no—
766C2—48no—
767C2#1383911yescharacteristic
768C2#1387011yescharacteristic
769C2#1394011yescharacteristic
770C2#1398314yes—
771C2#1400814yes—
772C2#1413411yescharacteristic
773C2#1413811yescharacteristic
774C2#1429211yescharacteristic
775C2#1470611yescharacteristic
776C2#1473114yes—
777C2#1476114yes—
778C2#1483714yes—
779C2#1496514yes—
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
5. C2×#1436C2semidirect
6. #7557C2semidirect
7. #7556C2semidirect
9. #7554C2semidirect
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×#3812
C22×C4×C22⋊C42
C2×#14402
C2×#14362
#75572
#75562
#75552
#75542