Finite Group Databasebeta

← #156705

Subgroups — #156705

Order 512.

4184 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3C2×#1441C212yesmaximal, complemented
4C2×C22⋊#95C212yesmaximal
5#7557C214yesmaximal, complemented
6#7557C212yesmaximal, complemented
7#7555C212yesmaximal
8#7555C211yesmaximal, characteristic
9C25×C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C4⋊C4C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C2×#399—28no—
18C2×#399—24no—
19C2×#385—28no—
20C2×#385—24no—
21C22×#95C2212yes—
22C22×#95C2211yescharacteristic
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29#1441—28no—
30C22⋊#95—28no—
31C26C2311yescharacteristic
32C24×C4—24no—
33C24×C4—22no—
34C24×C4—24no—
35C24×C4—28no—
36C24×C4—44no—
37C24×C4—42no—
38C24×C4C2311yescharacteristic
39C24×C4C2312yes—
40C24×C4C2314yes—
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C24×C4C2312yes—
44C24×C4C2312yes—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—28no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—44no—
56C22×C4⋊C4—42no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4D418yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—28no—
70C22×C22⋊C4—44no—
71C22×C22⋊C4—42no—
72C22×C22⋊C4—44no—
73C22×C22⋊C4—44no—
74C22×C22⋊C4D418yes—
75C22×C22⋊C4D414yes—
76C22×C22⋊C4Q814yes—
77C22×C22⋊C4Q818yes—
78#399—232no—
79#385—232no—
80C2×#95—22no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—28no—
86C2×#95—28no—
87C2×#95—28no—
88C2×#95—28no—
89C2×#95—24no—
90C2×#95—28no—
91C2×#95—28no—
92C2×#95—28no—
93C2×#95—44no—
94C2×#95—44no—
95C2×#95—44no—
96C2×#95—44no—
97C2×#95—44no—
98C2×#95—44no—
99C2×#95—44no—
100C2×#95—42no—
101C25—22no—
102C25—22no—
103C25—24no—
104C25—24no—
105C25—24no—
106C25—22no—
107C25—24no—
108C25—22no—
109C25C2×D412yes—
110C25C2×D414yes—
111C25C2×Q812yes—
112C25C2×Q814yes—
113C25#4112yes—
114C25C2411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—216no—
142C23×C4—28no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
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—22no—
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—416no—
171C23×C4—48no—
172C23×C4—416no—
173C23×C4—416no—
174C23×C4—416no—
175C23×C4—48no—
176C23×C4—416no—
177C23×C4—44no—
178C23×C4—48no—
179C23×C4—48no—
180C23×C4—44no—
181C23×C4—48no—
182C23×C4C2×D412yes—
183C23×C4C2×D414yes—
184C23×C4C2×D414yes—
185C23×C4C2×D412yes—
186C23×C4C2×D414yes—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×Q818yes—
190C23×C4C2×Q814yes—
191C23×C4#4114yes—
192C23×C4#4118yes—
193C23×C4#4118yes—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—216no—
196C2×C4⋊C4—216no—
197C2×C4⋊C4—216no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—416no—
200C2×C4⋊C4—416no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—416no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—48no—
208C2×C22⋊C4—216no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—216no—
213C2×C22⋊C4—216no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—216no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—416no—
221C2×C22⋊C4—416no—
222C2×C22⋊C4—416no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—416no—
227#95—216no—
228#95—28no—
229#95—416no—
230#95—416no—
231#95—416no—
232#95—48no—
233#95—416no—
234#95—48no—
235C24—22no—
236C24—24no—
237C24—24no—
238C24—22no—
239C24—24no—
240C24—24no—
241C24—28no—
242C24—28no—
243C24—24no—
244C24—22no—
245C24—24no—
246C24—22no—
247C24—28no—
248C24—24no—
249C24—28no—
250C24—24no—
251C24—28no—
252C24—28no—
253C24—28no—
254C24—28no—
255C24—28no—
256C24—28no—
257C24—28no—
258C24—28no—
259C24—28no—
260C24—28no—
261C24—28no—
262C24—28no—
263C24—42no—
264C24—44no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—44no—
269C24—44no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—42no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—44no—
283C24#12014yes—
284C24C22⋊Q814yes—
285C24C22⋊Q818yes—
286C24C4⋊Q814yes—
287C24C4⋊Q818yes—
288C24C22×D411yescharacteristic
289C24C22×D412yes—
290C24C22×D412yes—
291C24C22×Q811yescharacteristic
292C24C22×Q812yes—
293C24C2×#4111yescharacteristic
294C24C2×#4112yes—
295C24C2×#4114yes—
296C24C2×#4114yes—
297C24H(5,𝔽2)14yes—
298C24H(5,𝔽2)14yes—
299C24H(5,𝔽2)12yes—
300C242−1+412yes—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
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—432no—
359C22×C4—464no—
360C22×C4—416no—
361C22×C4—432no—
362C22×C4—464no—
363C22×C4—432no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
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—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
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—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4#12014yes—
453C22×C4C22⋊Q818yes—
454C22×C4#12318yes—
455C22×C4#12318yes—
456C4⋊C4—432no—
457C4⋊C4—416no—
458C4⋊C4—432no—
459C22⋊C4—232no—
460C22⋊C4—416no—
461C22⋊C4—432no—
462C22⋊C4—432no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23—216no—
475C23—216no—
476C23—216no—
477C23—216no—
478C23—28no—
479C23—216no—
480C23—216no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—44no—
485C23—48no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—48no—
495C23—48no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—44no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—48no—
512C23—48no—
513C23—44no—
514C23—48no—
515C23#39518yes—
516C23C2×#12011yescharacteristic
517C23C2×#12011yescharacteristic
518C23C2×C22⋊Q812yes—
519C23C2×C22⋊Q811yescharacteristic
520C23C2×C22⋊Q812yes—
521C23C2×#12312yes—
522C23C2×#12312yes—
523C23#52511yescharacteristic
524C23#52512yes—
525C23#52512yes—
526C23#52914yes—
527C23#52914yes—
528C23C2×C4⋊Q811yescharacteristic
529C23C2×C4⋊Q812yes—
530C23#53212yes—
531C23#53314yes—
532C23#53412yes—
533C23#53412yes—
534C23#53411yescharacteristic
535C23#53512yes—
536C23#53512yes—
537C23#53611yescharacteristic
538C23#53814yes—
539C23#53814yes—
540C23#53912yes—
541C23#53914yes—
542C23#54014yes—
543C23#54014yes—
544C23#54214yes—
545C23#54314yes—
546C23#54412yes—
547C23D4214yes—
548C23#54614yes—
549C23#54614yes—
550C23#54614yes—
551C23#54614yes—
552C23#54614yes—
553C23#54714yes—
554C23D4×Q814yes—
555C23#55014yes—
556C23#55114yes—
557C23#55112yes—
558C23#55214yes—
559C23#55314yes—
560C23#55414yes—
561C23#55414yes—
562C23#55414yes—
563C23#55414yes—
564C23#55414yes—
565C23#55912yes—
566C23#56012yes—
567C23#56012yes—
568C23#56214yes—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—464no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C22—216no—
636C22—28no—
637C22—44no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—416no—
643C22—44no—
644C22—416no—
645C22—416no—
646C22—416no—
647C22—48no—
648C22C2×#39511yescharacteristic
649C22#239512yes—
650C22#240311yescharacteristic
651C22#240412yes—
652C22#240512yes—
653C22#240512yes—
654C22#240611yescharacteristic
655C22#241814yes—
656C22#241914yes—
657C22#242314yes—
658C22#243112yes—
659C22#243411yescharacteristic
660C22#244014yes—
661C22#244514yes—
662C22#244914yes—
663C22#245014yes—
664C22#245014yes—
665C22#245414yes—
666C22#245714yes—
667C22#246714yes—
668C22#247014yes—
669C22#248712yes—
670C22#250212yes—
671C22#251712yes—
672C22#252214yes—
673C22#254914yes—
674C22#254914yes—
675C22#255614yes—
676C22#256412yes—
677C22#258312yes—
678C22#258311yescharacteristic
679C22#258914yes—
680C22#262612yes—
681C22#262711yescharacteristic
682C22#263512yes—
683C22#264914yes—
684C22#267314yes—
685C22#267814yes—
686C22#268714yes—
687C22#269214yes—
688C22#269712yes—
689C22#270114yes—
690C22#270614yes—
691C22#270912yes—
692C22#273214yes—
693C22#274212yes—
694C22#274814yes—
695C22#276814yes—
696C22#278014yes—
697C22#280712yes—
698C22#282612yes—
699C22#284811yescharacteristic
700C22#285512yes—
701C4—44no—
702C4—48no—
703C4—48no—
704C4—48no—
705C4—82no—
706C4—84no—
707C4—88no—
708C2—48no—
709C2#1472011yescharacteristic
710C2#1474214yes—
711C2#1482512yes—
712C2#1491412yes—
713C2#1526111yescharacteristic
714C2#1529214yes—
715C2#1546612yes—
716C2#1554514yes—
717C2#1573012yes—
718C2#1594811yescharacteristic
719C2#1598212yes—
720C2#1631914yes—
721C2#1647312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC25C24
prime radicalC26C23
LCSG ▷ C25 ▷ 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

Maximal subgroups

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

subgroupindex
C22×#3992
C22×#3852
C2×#14412
C2×C22⋊#952
#75572
#75552