Finite Group Databasebeta

← #155772

Subgroups — #155772

Order 512.

3454 conjugacy classes of subgroups.

The subgroup lattice has 791 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic, complemented
3C2×#1439C212yesmaximal
4C2×#1434C211yesmaximal, characteristic, complemented
5#7557C212yesmaximal, complemented
6#7557C212yesmaximal, complemented
7#7556C212yesmaximal, complemented
8#7555C212yesmaximal
9#7555C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2214yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#388—24no—
18C2×#388—24no—
19C2×#388C418yescomplemented
20C2×#382—24no—
21C2×#382—28no—
22C22×#95C2212yes—
23C22×#95C2212yes—
24C22×#95C2212yes—
25C22×#95C2212yes—
26C22×#95C2214yes—
27C22×#95C2211yescharacteristic
28C22×#95C2211yescharacteristic
29C22×#95C2212yes—
30C22×#95C2214yes—
31C22×#95C2214yes—
32#1439—28no—
33#1434—24no—
34C26C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C24×C4C2314yes—
38C24×C4C2314yes—
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×C4⋊C4—24no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—44no—
48C22×C4⋊C4—44no—
49C22×C4⋊C4—44no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D418yes—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—28no—
64C22×C22⋊C4—28no—
65C22×C22⋊C4—44no—
66C22×C22⋊C4—44no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2×C414yes—
72C22×C22⋊C4C2×C414yes—
73C22×C42—24no—
74C22×C42—24no—
75C22×C42C2×C414yes—
76C22×C42D414yes—
77C22×C42D414yes—
78#388—232no—
79#382—232no—
80C2×#95—24no—
81C2×#95—28no—
82C2×#95—28no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—28no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—28no—
93C2×#95—24no—
94C2×#95—28no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—44no—
98C2×#95—44no—
99C2×#95—44no—
100C2×#95—44no—
101C2×#95—44no—
102C2×#95—44no—
103C2×#95—44no—
104C2×#95C2×C414yes—
105C2×#95C2×C414yes—
106C25—21no—
107C25—21no—
108C25—22no—
109C25—22no—
110C25—22no—
111C25—24no—
112C25—24no—
113C25—22no—
114C25—22no—
115C25—22no—
116C25—22no—
117C25—22no—
118C25—22no—
119C25C22×C412yes—
120C25#4114yes—
121C25C2411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
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—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—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C2×D418yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D414yes—
197C23×C4#4118yes—
198C23×C4#4114yes—
199C23×C4#4114yes—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—416no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C22⋊C4—216no—
210C2×C22⋊C4—216no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—216no—
217C2×C22⋊C4—48no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—416no—
220C2×C22⋊C4—416no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—416no—
226C2×C22⋊C4—416no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—44no—
230C2×C42—216no—
231C2×C42—28no—
232C2×C42—216no—
233C2×C42—28no—
234C2×C42C22⋊C4116yescomplemented
235#95—216no—
236#95—216no—
237#95—216no—
238#95—28no—
239#95—28no—
240#95—416no—
241#95—48no—
242#95—416no—
243#95—416no—
244#95—48no—
245C24—22no—
246C24—22no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—28no—
256C24—28no—
257C24—24no—
258C24—24no—
259C24—28no—
260C24—28no—
261C24—24no—
262C24—24no—
263C24—41no—
264C24—42no—
265C24—41no—
266C24—42no—
267C24—42no—
268C24—44no—
269C24—42no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—44no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—42no—
291C24—42no—
292C24—44no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—42no—
297C24—44no—
298C24—42no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24#11714yes—
304C24C23×C411yescharacteristic
305C24C22×D411yescharacteristic
306C24C22×D411yescharacteristic
307C24C22×D412yes—
308C24C2×#4112yes—
309C24C2×#4112yes—
310C24C2×#4112yes—
311C24C2×#4114yes—
312C24H(5,𝔽2)14yes—
313C24H(5,𝔽2)12yes—
314C24H(5,𝔽2)12yes—
315C24H(5,𝔽2)12yes—
316C242−1+412yes—
317C242−1+412yes—
318C242−1+412yes—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
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—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
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—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
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×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×C4C2×C22⋊C414yes—
482C22×C4C2×C22⋊C414yes—
483C22×C4C2×C22⋊C414yes—
484C22×C4#11714yes—
485C22×C4C4×D418yes—
486C22×C4#12014yes—
487C22×C4#12118yes—
488C22×C4#12314yes—
489C4⋊C4—432no—
490C4⋊C4—416no—
491C22⋊C4—232no—
492C22⋊C4—416no—
493C22⋊C4—464no—
494C22⋊C4—416no—
495C42—232no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—81no—
536C23—81no—
537C23—84no—
538C23—81no—
539C23—81no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—84no—
545C23—84no—
546C23—82no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—82no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23C22×C22⋊C411yescharacteristic
556C23C2×#11711yescharacteristic
557C23C2×#11711yescharacteristic
558C23C2×C4×D412yes—
559C23C4×#4112yes—
560C23#51812yes—
561C23#52012yes—
562C23#52012yes—
563C23#52012yes—
564C23C2×#12011yescharacteristic
565C23C2×#12112yes—
566C23C2×#12311yescharacteristic
567C23#52512yes—
568C23#52512yes—
569C23#52512yes—
570C23#53212yes—
571C23#53414yes—
572C23#53511yescharacteristic
573C23#53511yescharacteristic
574C23#53512yes—
575C23#53512yes—
576C23#53511yescharacteristic
577C23#53611yescharacteristic
578C23#53611yescharacteristic
579C23#53611yescharacteristic
580C23#53614yes—
581C23#53712yes—
582C23#53712yes—
583C23#53712yes—
584C23#53814yes—
585C23#53814yes—
586C23#53814yes—
587C23#53914yes—
588C23#53914yes—
589C23#53914yes—
590C23#53914yes—
591C23#53914yes—
592C23#54114yes—
593C23#54214yes—
594C23#54612yes—
595C23#54614yes—
596C23#54712yes—
597C23#54812yes—
598C23#55012yes—
599C23#55014yes—
600C23#55114yes—
601C23#55214yes—
602C23#55214yes—
603C23#55212yes—
604C23#55312yes—
605C23#55314yes—
606C23#55514yes—
607C23#55512yes—
608C23#55514yes—
609C23#55612yes—
610C23H(3,𝔽4)14yes—
611C23#56214yes—
612C23#56214yes—
613C23#56314yes—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—216no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—216no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—48no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—82no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—84no—
685C2×C4—84no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—82no—
689C2×C4—84no—
690C2×C4—84no—
691C2×C4#38618yes—
692C22—44no—
693C22—44no—
694C22—416no—
695C22—44no—
696C22—84no—
697C22—84no—
698C22—88no—
699C22—88no—
700C22—82no—
701C22—88no—
702C22—82no—
703C22—84no—
704C22—84no—
705C22—82no—
706C22—82no—
707C22#228011yescharacteristic
708C22C2×#38611yescharacteristic
709C22#229711yescharacteristic
710C22#230911yescharacteristic
711C22#231111yescharacteristic
712C22#231111yescharacteristic
713C22#231111yescharacteristic
714C22#231812yes—
715C22#231812yes—
716C22#232214yes—
717C22#233414yes—
718C22#234312yes—
719C22#234612yes—
720C22#235212yes—
721C22#235412yes—
722C22#236314yes—
723C22#237914yes—
724C22#240712yes—
725C22#241412yes—
726C22#241414yes—
727C22#241512yes—
728C22#243112yes—
729C22#248512yes—
730C22#250012yes—
731C22#260914yes—
732C22#261012yes—
733C22#261112yes—
734C22#261311yescharacteristic
735C22#261411yescharacteristic
736C22#261814yes—
737C22#262311yescharacteristic
738C22#262311yescharacteristic
739C22#263112yes—
740C22#263312yes—
741C22#263614yes—
742C22#263612yes—
743C22#263712yes—
744C22#266212yes—
745C22#266412yes—
746C22#268414yes—
747C22#268614yes—
748C22#269014yes—
749C22#269414yes—
750C22#269414yes—
751C22#272114yes—
752C22#273014yes—
753C22#273314yes—
754C22#274414yes—
755C22#276514yes—
756C22#277514yes—
757C22#278114yes—
758C22#278114yes—
759C22#278214yes—
760C22#278314yes—
761C22#278514yes—
762C22#281912yes—
763C22#281912yes—
764C22#282012yes—
765C22#282012yes—
766C4—28no—
767C4—44no—
768C4—44no—
769C4—44no—
770C4—88no—
771C4—88no—
772C4—82no—
773C4—82no—
774C2—84no—
775C2#1387111yescharacteristic
776C2#1395711yescharacteristic
777C2#1413611yescharacteristic
778C2#1425611yescharacteristic
779C2#1429111yescharacteristic
780C2#1434511yescharacteristic
781C2#1439314yes—
782C2#1439914yes—
783C2#1466211yescharacteristic
784C2#1668012yes—
785C2#1668212yes—
786C2#1668414yes—
787C2#1668514yes—
788C2#1702912yes—
789C2#1704612yes—
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
2. C22×#382C2semidirect
4. C2×#1434C2semidirect
5. #7557C2semidirect
6. #7557C2semidirect
7. #7556C2semidirect
19. C2×#388C4semidirect
234. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C22×#3822
C2×#14392
C2×#14342
#75572
#75562
#75552