Finite Group Databasebeta

← #4929369

Subgroups — #4929369

Order 512.

3118 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#15191C212yesmaximal, complemented
4#15167C212yesmaximal, complemented
5#15053C212yesmaximal, complemented
6#14932C212yesmaximal, complemented
7#14817C214yesmaximal, complemented
8#14743C214yesmaximal, complemented
9#14292C211yesmaximal, characteristic, complemented
10#14220C214yesmaximal, complemented
11#14183C214yesmaximal, complemented
12#14159C211yesmaximal, characteristic, complemented
13#13956C211yesmaximal, characteristic
14#13851C211yesmaximal, characteristic
15#13749C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20#2607—28no—
21#2432—28no—
22C2×#399C2212yes—
23C2×#397C2214yes—
24C2×#394C2214yes—
25C2×#393C2214yes—
26C2×#393C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2214yes—
31C2×#390C2211yescharacteristic
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×#381C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1440C2214yes—
57#1440C2212yes—
58#1440C2212yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2214yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—24no—
80C22×#41—44no—
81C22×#41—44no—
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#124—24no—
93C2×#124—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#399—28no—
123#397—216no—
124#394—216no—
125#393—28no—
126#393—216no—
127#391—28no—
128#391—24no—
129#391—28no—
130#391—48no—
131#390—24no—
132#388—48no—
133#387—28no—
134#387—28no—
135#386—44no—
136#386—44no—
137#385—216no—
138#381—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—216no—
142C4×C22⋊C4—44no—
143C4×C22⋊C4—44no—
144C4×C22⋊C4D418yes—
145C2×#95—24no—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C25C2412yes—
173C25C2411yescharacteristic
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—22no—
183C22×Q8—22no—
184C22×Q8—28no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—44no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—416no—
237#124—416no—
238C22⋊Q8—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C2×D418yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95C2×D418yes—
329#95#41116yescomplemented
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24C22×D412yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24H(5,𝔽2)14yes—
349C24C2511yescharacteristic
350#41—4128no—
351C2×Q8—28no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—48no—
362C2×Q8—84no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—88no—
366C2×D4—28no—
367C2×D4—216no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—44no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4#124116yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C2×#4118yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4118yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—88no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C42—44no—
485C42—44no—
486C42—82no—
487C42—82no—
488C42—84no—
489C42—88no—
490C42—88no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C2×#12414yes—
530C23C23×D411yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4114yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538Q8—416no—
539Q8—816no—
540Q8—88no—
541D4—416no—
542D4—432no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
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—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
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—816no—
614C2×C4C2×#12414yes—
615C2×C4C2×#12414yes—
616C2×C4#53218yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—44no—
626C22—48no—
627C22—44no—
628C22—44no—
629C22—48no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—84no—
636C22C22×#12411yescharacteristic
637C22C2×#53212yes—
638C22#344412yes—
639C22#344412yes—
640C22#348411yescharacteristic
641C22#348411yescharacteristic
642C22#348912yes—
643C22#349012yes—
644C22#349012yes—
645C22#350712yes—
646C22#350911yescharacteristic
647C22#351011yescharacteristic
648C22#351314yes—
649C22#351414yes—
650C22#352014yes—
651C22#352214yes—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—44no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—48no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—44no—
666C2—84no—
667C2#3536711yescharacteristic
668C2#3540812yes—
669C2#3543512yes—
670C2#3562811yescharacteristic
671C2#3571814yes—
672C2#3591212yes—
673C2#3594012yes—
674C2#3663911yescharacteristic
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×#2607C2semidirect
3. #15191C2semidirect
4. #15167C2semidirect
5. #15053C2semidirect
6. #14932C2semidirect
7. #14817C2semidirect
8. #14743C2semidirect
9. #14292C2semidirect
10. #14220C2semidirect
11. #14183C2semidirect
12. #14159C2semidirect
15. #13749C2semidirect
329. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
C2×#24322
#151912
#151672
#150532
#149322
#148172
#147432
#142922
#142202
#141832
#141592
#139562
#138512
#137492