Finite Group Databasebeta

← #4462511

Subgroups — #4462511

Order 512.

3150 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic, complemented
2C22×#393C211yesmaximal, characteristic
3#17152C214yesmaximal, complemented
4#17127C214yesmaximal, complemented
5#17104C214yesmaximal, complemented
6#17100C214yesmaximal, complemented
7#15019C211yesmaximal, characteristic, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14935C212yesmaximal, complemented
10#14918C212yesmaximal
11#14874C212yesmaximal, complemented
12#14820C211yesmaximal, characteristic
13#14734C211yesmaximal, characteristic
14#14726C212yesmaximal
15#14347C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2646—28no—
21C2×#400C2212yescomplemented
22C2×#400C2212yescomplemented
23C2×#399C2212yes—
24C2×#398C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2214yescomplemented
29C2×#395C2214yescomplemented
30C2×#393—28no—
31C2×#393C2214yes—
32C2×#393C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2214yescomplemented
48#1446C2214yescomplemented
49#1445C2212yescomplemented
50#1445C2214yescomplemented
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1443C2214yescomplemented
54#1443C2214yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2212yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118#400—28no—
119#400—28no—
120#399—44no—
121#398—28no—
122#398—28no—
123#397—44no—
124#396—48no—
125#395—48no—
126#395—48no—
127#393—216no—
128#393—216no—
129#385—28no—
130#385—28no—
131#385—44no—
132#385—44no—
133#381—44no—
134#381—44no—
135#380—24no—
136#380—24no—
137#380—44no—
138#379—44no—
139#379—44no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95D418yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25—22no—
172C25—24no—
173C25—24no—
174C25—22no—
175C25—24no—
176C25—22no—
177C25—42no—
178C25—42no—
179C25—42no—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—416no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8—464no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—416no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95C2×D418yes—
323#95C2×D418yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—28no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—416no—
340C24—48no—
341C24—416no—
342C24—48no—
343C24—44no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—48no—
350C24—48no—
351C24—88no—
352C24—84no—
353C24—84no—
354C24—88no—
355C24C22×D412yes—
356C24H(5,𝔽2)14yes—
357C24H(5,𝔽2)12yes—
358C242−1+414yes—
359C242−1+414yes—
360C242−1+412yes—
361C24C2511yescharacteristic
362C2×Q8—416no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×Q8—416no—
366C2×Q8—416no—
367C2×Q8—416no—
368C2×Q8—4128no—
369C2×Q8—416no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)14yes—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—416no—
504C23—416no—
505C23—416no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—832no—
525C23—832no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23#53514yes—
531C23#53614yes—
532C23#53714yes—
533C23#56218yes—
534C23#56318yes—
535C23C23×D411yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58512yes—
544Q8—464no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4—88no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—216no—
624C22—216no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—832no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—84no—
638C22#282018yes—
639C22C2×#53511yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#53711yescharacteristic
642C22#348411yescharacteristic
643C22#348411yescharacteristic
644C22#348511yescharacteristic
645C22#348611yescharacteristic
646C22C2×#56212yes—
647C22C2×#56312yes—
648C22#353214yes—
649C22#353514yes—
650C22#353612yes—
651C22#353814yes—
652C22#353812yes—
653C22#355812yes—
654C22#356112yes—
655C22#356114yes—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—84no—
668C4—82no—
669C2—28no—
670C2—84no—
671C2C2×#282011yescharacteristic
672C2#3670311yescharacteristic
673C2#3670811yescharacteristic
674C2#3671911yescharacteristic
675C2#3695711yescharacteristic
676C2#3696611yescharacteristic
677C2#3711811yescharacteristic
678C2#5239614yes—
679C2#5241214yes—
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. #17152C2semidirect
4. #17127C2semidirect
5. #17104C2semidirect
6. #17100C2semidirect
7. #15019C2semidirect
8. #14936C2semidirect
9. #14935C2semidirect
11. #14874C2semidirect
15. #14347C2semidirect
21. C2×#400C22semidirect
22. C2×#400C22semidirect
24. C2×#398C22semidirect
25. C2×#398C22semidirect
28. C2×#395C22semidirect
29. C2×#395C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1443C22semidirect
54. #1443C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3932
#171522
#171272
#171042
#171002
#150192
#149362
#149352
#149182
#148742
#148202
#147342
#147262
#143472