Finite Group Databasebeta

← #5955430

Subgroups — #5955430

Order 512.

3782 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3#15019C211yesmaximal, characteristic, complemented
4#14934C211yesmaximal, characteristic, complemented
5#14918C212yesmaximal
6#14843C212yesmaximal, complemented
7#14820C211yesmaximal, characteristic
8#14819C214yesmaximal, complemented
9#14810C214yesmaximal, complemented
10#14803C214yesmaximal, complemented
11#14797C214yesmaximal, complemented
12#14717C211yesmaximal, characteristic
13#13796C211yesmaximal, characteristic, complemented
14#13776C212yesmaximal, complemented
15C4×#379C212yesmaximal
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C2×#400C2214yescomplemented
23C2×#397C2212yes—
24C2×#393C2212yes—
25#2329—28no—
26C2×#391C2212yescomplemented
27C2×#389C2212yescomplemented
28C2×#387C2212yescomplemented
29C2×#387C2212yescomplemented
30C2×#385—28no—
31C2×#385C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#380C2211yescharacteristic
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2214yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yescomplemented
46#1445C2212yescomplemented
47#1445C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2212yescomplemented
51#1442C2212yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1434C2214yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123#400—48no—
124#397—44no—
125#393—44no—
126#391—28no—
127#389—28no—
128#387—28no—
129#387—28no—
130#385—28no—
131#385—216no—
132#385—216no—
133#381—28no—
134#381—28no—
135#380—24no—
136#379—28no—
137#379—28no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—48no—
141C4×C22⋊C4—48no—
142C2×#95—24no—
143C2×#95D418yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25—22no—
173C25—22no—
174C25—24no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—42no—
179C25—42no—
180C25#4114yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—42no—
186C22×Q8—42no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—416no—
216C23×C4—464no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4#4118yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C22⋊Q8—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C22⋊C4—232no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4#4118yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—42no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328#95C2×D418yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24—28no—
335C24—216no—
336C24—28no—
337C24—216no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—216no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—44no—
356C24—44no—
357C24—48no—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24C2×#4112yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)14yes—
365C242−1+414yes—
366C24C2511yescharacteristic
367C2×Q8—28no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—88no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—232no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—4128no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4#124116yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)14yes—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C22⋊C4—232no—
481C22⋊C4—216no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C42—84no—
492C42—84no—
493C42—88no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—232no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—464no—
520C23—416no—
521C23—416no—
522C23—416no—
523C23—464no—
524C23—416no—
525C23—416no—
526C23—416no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23#52514yes—
538C23#52514yes—
539C23C2×#12414yes—
540C23#52918yes—
541C23#52918yes—
542C23#53918yes—
543C23C23×D411yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552Q8—88no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—416no—
575C2×C4—464no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4C2×#12418yes—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—216no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—216no—
635C22—28no—
636C22—216no—
637C22—464no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22#239818yes—
648C22C2×#52511yescharacteristic
649C22C2×#52511yescharacteristic
650C22C22×#12411yescharacteristic
651C22C2×#52912yes—
652C22C2×#52912yes—
653C22C2×#53912yes—
654C22#345912yes—
655C22#348411yescharacteristic
656C22#348411yescharacteristic
657C22#348511yescharacteristic
658C22#348611yescharacteristic
659C22#349014yes—
660C22#349212yes—
661C22#349514yes—
662C22#350612yes—
663C22#351314yes—
664C22#352014yes—
665C4—28no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—84no—
678C2—28no—
679C2—48no—
680C2C2×#239811yescharacteristic
681C2#3498611yescharacteristic
682C2#3498911yescharacteristic
683C2#3499914yes—
684C2#3535411yescharacteristic
685C2#3535711yescharacteristic
686C2#3536711yescharacteristic
687C2#3537214yes—
688C2#3564411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2329C2semidirect
3. #15019C2semidirect
4. #14934C2semidirect
6. #14843C2semidirect
8. #14819C2semidirect
9. #14810C2semidirect
10. #14803C2semidirect
11. #14797C2semidirect
13. #13796C2semidirect
14. #13776C2semidirect
22. C2×#400C22semidirect
26. C2×#391C22semidirect
27. C2×#389C22semidirect
28. C2×#387C22semidirect
29. C2×#387C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3852
#150192
#149342
#149182
#148432
#148202
#148192
#148102
#148032
#147972
#147172
#137962
#137762
C4×#3792