Finite Group Databasebeta

← #156758

Subgroups — #156758

Order 512.

4032 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×#381C211yesmaximal, characteristic
3C2×#1442C212yesmaximal, complemented
4C2×C22⋊#95C212yesmaximal
5#7557C214yesmaximal, complemented
6#7556C212yesmaximal, complemented
7#7555C211yesmaximal, characteristic
8#7555C212yesmaximal
9C25×C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C2×#399—28no—
17C2×#399—24no—
18C2×#381—28no—
19C2×#381—24no—
20C22×#95C2212yes—
21C22×#95C2211yescharacteristic
22C22×#95C2212yes—
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2212yes—
27C22×#95C2214yes—
28C22×#95C2212yes—
29#1442—28no—
30C22⋊#95—28no—
31C26C2311yescharacteristic
32C24×C4—22no—
33C24×C4—22no—
34C24×C4—24no—
35C24×C4—24no—
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—42no—
39C24×C4—42no—
40C24×C4Q814yes—
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C24×C4C2312yes—
44C24×C4C2314yes—
45C24×C4C2311yescharacteristic
46C24×C4C2312yes—
47C24×C4C2312yes—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—42no—
56C22×C4⋊C4—44no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—28no—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4—42no—
71C22×C22⋊C4—44no—
72C22×C22⋊C4—42no—
73C22×C22⋊C4D418yes—
74C22×C22⋊C4D414yes—
75#399—232no—
76#381—232no—
77C2×#95—22no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—24no—
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—28no—
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—42no—
100C2×#95D418yes—
101C25—22no—
102C25—24no—
103C25—24no—
104C25—22no—
105C25—24no—
106C25—22no—
107C25—22no—
108C25—24no—
109C25C2×D412yes—
110C25C2×D414yes—
111C25C2×Q812yes—
112C25#4112yes—
113C25#4114yes—
114C25C2411yescharacteristic
115C23×C4—216no—
116C23×C4—28no—
117C23×C4—28no—
118C23×C4—28no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—22no—
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—216no—
139C23×C4—216no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
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—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
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—48no—
176C23×C4—416no—
177C23×C4—416no—
178C23×C4—416no—
179C23×C4—416no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—48no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—48no—
186C23×C4C2×D414yes—
187C23×C4C2×D412yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4C2×D412yes—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4C2×Q814yes—
194C23×C4#4118yes—
195C23×C4#4114yes—
196C23×C4#4118yes—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—216no—
199C2×C4⋊C4—416no—
200C2×C4⋊C4—416no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C22⋊C4—216no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—216no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—216no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—416no—
215C2×C22⋊C4—416no—
216C2×C22⋊C4—416no—
217C2×C22⋊C4—416no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—416no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—44no—
224#95—216no—
225#95—216no—
226#95—28no—
227#95—216no—
228#95—416no—
229#95—416no—
230#95—416no—
231#95—416no—
232#95—48no—
233C24—22no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24—22no—
238C24—24no—
239C24—28no—
240C24—28no—
241C24—24no—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—24no—
246C24—28no—
247C24—24no—
248C24—28no—
249C24—28no—
250C24—28no—
251C24—28no—
252C24—28no—
253C24—28no—
254C24—28no—
255C24—28no—
256C24—28no—
257C24—28no—
258C24—28no—
259C24—28no—
260C24—28no—
261C24—42no—
262C24—44no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—44no—
267C24—44no—
268C24—44no—
269C24—44no—
270C24—44no—
271C24—44no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—44no—
280C24—42no—
281C24#12014yes—
282C24C22⋊Q814yes—
283C24C22⋊Q818yes—
284C24#12418yes—
285C24#12414yes—
286C24C22×D411yescharacteristic
287C24C22×D412yes—
288C24C22×D412yes—
289C24C22×Q811yescharacteristic
290C24C2×#4111yescharacteristic
291C24C2×#4112yes—
292C24C2×#4112yes—
293C24C2×#4114yes—
294C24C2×#4114yes—
295C24H(5,𝔽2)14yes—
296C24H(5,𝔽2)12yes—
297C24H(5,𝔽2)12yes—
298C242−1+414yes—
299C22×C4—232no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
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—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—464no—
351C22×C4—416no—
352C22×C4—416no—
353C22×C4—416no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—464no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—416no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
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—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4#12014yes—
455C22×C4C22⋊Q818yes—
456C22×C4#12318yes—
457C4⋊C4—416no—
458C4⋊C4—432no—
459C22⋊C4—416no—
460C22⋊C4—432no—
461C22⋊C4—432no—
462C22⋊C4—416no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—24no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—216no—
475C23—216no—
476C23—216no—
477C23—216no—
478C23—216no—
479C23—216no—
480C23—28no—
481C23—48no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23#39318yes—
516C23C2×#12011yescharacteristic
517C23C2×#12011yescharacteristic
518C23C2×C22⋊Q811yescharacteristic
519C23C2×C22⋊Q812yes—
520C23C2×C22⋊Q812yes—
521C23C2×#12312yes—
522C23#52511yescharacteristic
523C23#52512yes—
524C23#52512yes—
525C23#52512yes—
526C23C2×#12411yescharacteristic
527C23C2×#12412yes—
528C23#52914yes—
529C23#52914yes—
530C23#53212yes—
531C23#53314yes—
532C23#53411yescharacteristic
533C23#53412yes—
534C23#53512yes—
535C23#53511yescharacteristic
536C23#53612yes—
537C23#53612yes—
538C23#53812yes—
539C23#53914yes—
540C23#53914yes—
541C23#53914yes—
542C23#54014yes—
543C23#54014yes—
544C23#54214yes—
545C23#54312yes—
546C23#54414yes—
547C23D4214yes—
548C23#54614yes—
549C23#54614yes—
550C23#54614yes—
551C23#54614yes—
552C23#54714yes—
553C23#54714yes—
554C23#54714yes—
555C23#55014yes—
556C23#55114yes—
557C23#55114yes—
558C23#55114yes—
559C23#55114yes—
560C23#55214yes—
561C23#55214yes—
562C23#55214yes—
563C23#55314yes—
564C23#55912yes—
565C23#55912yes—
566C23#56012yes—
567C23#56214yes—
568C23#56312yes—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—216no—
574C2×C4—216no—
575C2×C4—464no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
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—48no—
638C22—416no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—416no—
644C22—48no—
645C22—44no—
646C22—416no—
647C22—416no—
648C22C2×#39311yescharacteristic
649C22#239612yes—
650C22#240311yescharacteristic
651C22#240411yescharacteristic
652C22#240512yes—
653C22#240612yes—
654C22#240612yes—
655C22#241914yes—
656C22#243112yes—
657C22#243511yescharacteristic
658C22#243714yes—
659C22#243814yes—
660C22#243814yes—
661C22#244114yes—
662C22#244514yes—
663C22#245514yes—
664C22#245714yes—
665C22#247014yes—
666C22#247114yes—
667C22#247214yes—
668C22#247914yes—
669C22#248814yes—
670C22#248814yes—
671C22#250312yes—
672C22#251614yes—
673C22#255314yes—
674C22#256112yes—
675C22#256111yescharacteristic
676C22#257514yes—
677C22#259812yes—
678C22#260712yes—
679C22#261112yes—
680C22#262611yescharacteristic
681C22#262712yes—
682C22#262812yes—
683C22#266714yes—
684C22#267314yes—
685C22#268014yes—
686C22#268714yes—
687C22#269414yes—
688C22#270114yes—
689C22#270614yes—
690C22#270614yes—
691C22#270912yes—
692C22#272412yes—
693C22#273814yes—
694C22#274314yes—
695C22#274614yes—
696C22#275212yes—
697C22#277112yes—
698C22#284112yes—
699C22#284811yescharacteristic
700C22#285512yes—
701C4—44no—
702C4—48no—
703C4—48no—
704C4—88no—
705C4—84no—
706C4—82no—
707C4—84no—
708C2—48no—
709C2#1471711yescharacteristic
710C2#1477214yes—
711C2#1489312yes—
712C2#1493412yes—
713C2#1526111yescharacteristic
714C2#1539914yes—
715C2#1546914yes—
716C2#1566912yes—
717C2#1591612yes—
718C2#1594911yescharacteristic
719C2#1598212yes—
720C2#1630014yes—
721C2#1666812yes—
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×#1442C2semidirect
5. #7557C2semidirect
6. #7556C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C22×#3812
C2×#14422
C2×C22⋊#952
#75572
#75562
#75552