Finite Group Databasebeta

← #1494349

Subgroups — #1494349

Order 512.

3776 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2717C212yesmaximal, complemented
3C2×#2406C212yesmaximal
4C2×#2432C211yesmaximal, characteristic
5#15856C212yesmaximal, complemented
6#15539C214yesmaximal, complemented
7#15430C212yesmaximal, complemented
8#15384C214yesmaximal, complemented
9#15143C212yesmaximal, complemented
10#14839C214yesmaximal, complemented
11#14230C212yesmaximal, complemented
12#13880C214yesmaximal, complemented
13#13824C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2724—28no—
22#2717—216no—
23#2432—28no—
24#2406—216no—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#398C2212yes—
28C2×#397C2214yes—
29C2×#397C2214yes—
30C2×#397C2214yes—
31C2×#395C2214yes—
32C2×#395C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yes—
35C2×#391C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#384C2214yes—
47C2×#384C2212yes—
48C2×#381C2212yes—
49C2×#380C2214yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2212yes—
60#1444C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2212yes—
66#1436C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4×#95C2212yes—
71C22×#41—28no—
72C22×#41—24no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—44no—
86C2×#124—24no—
87C2×#124—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#123—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—42no—
96C2×C22⋊Q8—42no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#399—28no—
128#399—44no—
129#398—28no—
130#397—216no—
131#397—216no—
132#397—216no—
133#395—216no—
134#395—216no—
135#394—216no—
136#394—216no—
137#391—24no—
138#390D418yes—
139#389—28no—
140#387—216no—
141#386—28no—
142#386D4116yes—
143#385—216no—
144#385—216no—
145#385—216no—
146#385—216no—
147#385—216no—
148#384—44no—
149#384—48no—
150#381—28no—
151#380—216no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—216no—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C25C2411yescharacteristic
173C25C2412yes—
174C2×#41—432no—
175C2×#41—416no—
176C2×#41—432no—
177C2×#41—416no—
178C2×#41—416no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—42no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C2×D418yes—
200C22×D4C2×D414yes—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×D418yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2414yes—
236C4⋊Q8—416no—
237#124—416no—
238#123—432no—
239#123—432no—
240C22⋊Q8—416no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—432no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4C2×D414yes—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D4116yes—
308C2×C22⋊C4C2×D4116yes—
309C2×C22⋊C4C2×D418yes—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42C2×D414yes—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—42no—
335#95C2×D4116yes—
336#95#4118yescomplemented
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24C22×D414yes—
353C24C22×D414yes—
354C24C22×D414yes—
355C24H(5,𝔽2)12yes—
356C24C2511yescharacteristic
357#41—4128no—
358C2×Q8—28no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×D4—216no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4#127116yes—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—232no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C22×D418yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D418yes—
462C22×C4C22×D414yes—
463C22×C4C2×#4118yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C42−1+414yes—
468C22×C42−1+418yes—
469C4⋊C4—44no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—84no—
482C22⋊C4—232no—
483C22⋊C4—232no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C42—44no—
493C42—48no—
494C42—416no—
495C42—44no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—41no—
523C23—44no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—44no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—41no—
533C23—44no—
534C23C2×#12718yes—
535C23C23×D412yes—
536C23C23×D411yescharacteristic
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+414yes—
542C23C2×2−1+412yes—
543Q8—416no—
544Q8—88no—
545D4—416no—
546D4—432no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—432no—
572C2×C4—416no—
573C2×C4—432no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
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—42no—
611C2×C4—44no—
612C2×C4—816no—
613C2×C4C2×#12714yes—
614C2×C4#54718yes—
615C2×C4#547116yes—
616C22—28no—
617C22—216no—
618C22—48no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—42no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C22×#12711yescharacteristic
636C22C2×#54714yes—
637C22C2×#54712yes—
638C22#348111yescharacteristic
639C22#348412yes—
640C22#348411yescharacteristic
641C22#348412yes—
642C22#348411yescharacteristic
643C22#349914yes—
644C22#349912yes—
645C22#352014yes—
646C22#352212yes—
647C22#353812yes—
648C22#354412yes—
649C22#355011yescharacteristic
650C22#355214yes—
651C4—24no—
652C4—216no—
653C4—28no—
654C4—42no—
655C4—44no—
656C4—48no—
657C4—44no—
658C4—84no—
659C4—82no—
660C4—82no—
661C4—84no—
662C2—48no—
663C2—82no—
664C2#3539011yescharacteristic
665C2#3617912yes—
666C2#4300412yes—
667C2#4487514yes—
668C2#5132612yes—
669C2#5154211yescharacteristic
670C2#5907011yescharacteristic
671C2#5910712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2724C2semidirect
2. C2×#2717C2semidirect
5. #15856C2semidirect
6. #15539C2semidirect
7. #15430C2semidirect
8. #15384C2semidirect
9. #15143C2semidirect
10. #14839C2semidirect
11. #14230C2semidirect
12. #13880C2semidirect
13. #13824C2semidirect
336. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#27172
C2×#24062
C2×#24322
#158562
#155392
#154302
#153842
#151432
#148392
#142302
#138802
#138242