Finite Group Databasebeta

← #4311777

Subgroups — #4311777

Order 512.

3302 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2432C212yesmaximal
3C2×#2311C211yesmaximal, characteristic
4#17011C212yesmaximal, complemented
5#16596C212yesmaximal, complemented
6#16264C214yesmaximal, complemented
7#15803C214yesmaximal, complemented
8#15311C214yesmaximal, complemented
9#14997C214yesmaximal, complemented
10#14907C212yesmaximal, complemented
11#14821C212yesmaximal, complemented
12#14501C211yesmaximal, characteristic, complemented
13#14450C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2734—28no—
21#2432—216no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#395C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29#2311—28no—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×C4×C22⋊C4C2212yes—
44#1446C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22×#41—24no—
71C22×#41—24no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—42no—
76C22×#41D418yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×#124—28no—
88C2×#124—44no—
89C2×#124—44no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C2×#117—24no—
94C2×#117—42no—
95C2×#117—42no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124#398—28no—
125#398—44no—
126#398—44no—
127#397—44no—
128#395—216no—
129#394—216no—
130#394—216no—
131#391—24no—
132#391—28no—
133#391D418yes—
134#390—28no—
135#390—24no—
136#390—24no—
137#389—28no—
138#387—216no—
139#386—216no—
140#385—216no—
141#385—216no—
142#380—216no—
143#380—44no—
144C4×C22⋊C4—28no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C2×#41—232no—
169C2×#41—416no—
170C2×#41—416no—
171C2×#41—416no—
172C2×#41—416no—
173C2×#41—416no—
174C2×#41—416no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8—41no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8C2×D414yes—
187C22×Q8C2×D414yes—
188C22×Q8#4114yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4C2×D418yes—
202C22×D4C2×D414yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
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—44no—
219C23×C4—44no—
220C23×C4C2×D414yes—
221C23×C4C2×D418yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C4⋊Q8—432no—
234#124—432no—
235C22⋊Q8—432no—
236#117—416no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
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—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—41no—
307C2×C42—41no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42C2×D414yes—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95C2×D4116yes—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24C22×D414yes—
349C24C22×D412yes—
350C24H(5,𝔽2)12yes—
351C24H(5,𝔽2)12yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8#121116yes—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—28no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
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—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—88no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+418yes—
470C4⋊C4—216no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—416no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—44no—
492C42—44no—
493C42—48no—
494C42—48no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23#53414yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×2−1+414yes—
543C23#58514yes—
544Q8—416no—
545Q8—816no—
546D4—432no—
547D4—416no—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—432no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
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—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
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—44no—
616C2×C4—816no—
617C2×C4C2×#12118yes—
618C2×C4C2×#12114yes—
619C2×C4#53414yes—
620C22—24no—
621C22—28no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—44no—
633C22—48no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22C22×#12111yescharacteristic
641C22C2×#53411yescharacteristic
642C22C2×#53411yescharacteristic
643C22#348411yescharacteristic
644C22#348411yescharacteristic
645C22#348612yes—
646C22#348914yes—
647C22#349011yescharacteristic
648C22#349011yescharacteristic
649C22#350012yes—
650C22#351614yes—
651C22#353312yes—
652C22#354412yes—
653C22#354714yes—
654C22#355012yes—
655C22#355112yes—
656C22#355914yes—
657C4—24no—
658C4—24no—
659C4—28no—
660C4—44no—
661C4—44no—
662C4—48no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—84no—
668C2—44no—
669C2—84no—
670C2#3498111yescharacteristic
671C2#3573712yes—
672C2#3737612yes—
673C2#4216612yes—
674C2#4642912yes—
675C2#4765712yes—
676C2#5197811yescharacteristic
677C2#5198812yes—
678C2#5207211yescharacteristic
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×#2734C2semidirect
4. #17011C2semidirect
5. #16596C2semidirect
6. #16264C2semidirect
7. #15803C2semidirect
8. #15311C2semidirect
9. #14997C2semidirect
10. #14907C2semidirect
11. #14821C2semidirect
12. #14501C2semidirect
13. #14450C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#24322
C2×#23112
#170112
#165962
#162642
#158032
#153112
#149972
#149072
#148212
#145012
#144502