Finite Group Databasebeta

← #4452825

Subgroups — #4452825

Order 512.

2770 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic
3#16842C214yesmaximal, complemented
4#16538C214yesmaximal, complemented
5#16532C214yesmaximal, complemented
6#16314C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14920C212yesmaximal, complemented
10#14820C211yesmaximal, characteristic
11#14734C211yesmaximal, characteristic
12#14726C212yesmaximal
13#14043C212yesmaximal, complemented
14#13971C212yesmaximal
15#13778C211yesmaximal, characteristic, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2646—28no—
21C2×#400C2212yescomplemented
22C2×#400C2212yescomplemented
23C2×#399C2212yes—
24C2×#398C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#397—28no—
27C2×#397C2214yes—
28C2×#397C2214yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#389C2214yescomplemented
32C2×#388C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2214yescomplemented
47#1445C2212yescomplemented
48#1445C2212yescomplemented
49#1444C2214yescomplemented
50#1444C2212yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1437C2214yes—
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2314yes—
117#400—28no—
118#400—28no—
119#399—44no—
120#398—28no—
121#398—28no—
122#397—216no—
123#397—216no—
124#397—44no—
125#396—48no—
126#389—48no—
127#388—48no—
128#385—28no—
129#385—28no—
130#385—28no—
131#385—44no—
132#380—24no—
133#380—24no—
134#380—28no—
135#379—28no—
136#379—44no—
137C4×C22⋊C4—44no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—22no—
172C25—24no—
173C25—22no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25C2412yes—
178C25C2411yescharacteristic
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2414yes—
217C23×C4C2412yes—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—416no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4#4118yes—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95C2×D418yes—
315#95C2×D418yes—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—28no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—416no—
332C24—416no—
333C24—48no—
334C24—48no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—48no—
342C24—48no—
343C24—88no—
344C24—84no—
345C24—88no—
346C24—84no—
347C24C22×D412yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)14yes—
350C24H(5,𝔽2)12yes—
351C242−1+414yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
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—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—416no—
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—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C2×#4118yes—
446C22×C4H(5,𝔽2)14yes—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—88no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—416no—
490C23—416no—
491C23—416no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—832no—
511C23—832no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23#53214yes—
517C23#53614yes—
518C23#53714yes—
519C23#53918yes—
520C23#54118yes—
521C23C23×D411yescharacteristic
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×2−1+412yes—
527C23C2×2−1+411yescharacteristic
528C23#58512yes—
529C23#58512yes—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—88no—
603C22—216no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—216no—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—832no—
617C22—82no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22#266418yes—
624C22C2×#53211yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#53711yescharacteristic
627C22C2×#53912yes—
628C22C2×#54112yes—
629C22#348411yescharacteristic
630C22#348411yescharacteristic
631C22#348511yescharacteristic
632C22#348711yescharacteristic
633C22#351314yes—
634C22#351314yes—
635C22#353514yes—
636C22#353612yes—
637C22#353712yes—
638C22#356012yes—
639C22#356114yes—
640C22#356412yes—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—48no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—84no—
654C2—28no—
655C2—84no—
656C2C2×#266411yescharacteristic
657C2#3539411yescharacteristic
658C2#3539611yescharacteristic
659C2#3670611yescharacteristic
660C2#3671111yescharacteristic
661C2#3696611yescharacteristic
662C2#3697111yescharacteristic
663C2#4070014yes—
664C2#4076014yes—
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×#2646C2semidirect
3. #16842C2semidirect
4. #16538C2semidirect
5. #16532C2semidirect
6. #16314C2semidirect
7. #15015C2semidirect
8. #15014C2semidirect
9. #14920C2semidirect
13. #14043C2semidirect
15. #13778C2semidirect
21. C2×#400C22semidirect
22. C2×#400C22semidirect
24. C2×#398C22semidirect
25. C2×#398C22semidirect
31. C2×#389C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3972
#168422
#165382
#165322
#163142
#150152
#150142
#149202
#148202
#147342
#147262
#140432
#139712
#137782