Finite Group Databasebeta

← #4381121

Subgroups — #4381121

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2616C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#16661C214yesmaximal, complemented
6#15512C214yesmaximal, complemented
7#15360C214yesmaximal, complemented
8#15354C214yesmaximal, complemented
9#15346C214yesmaximal, complemented
10#14761C214yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13937C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19#2616—28no—
20#2583—28no—
21#2406—28no—
22C2×#398C2212yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#394C2214yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29#2313—28no—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#381C2212yes—
42C2×#381C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2212yes—
49#1445C2214yes—
50#1445C2212yes—
51#1444C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C22⋊#95C2214yes—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—44no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—44no—
86C2×#123—44no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120#398—28no—
121#395—28no—
122#395—28no—
123#395—28no—
124#394—28no—
125#394—216no—
126#394—28no—
127#391—44no—
128#390—44no—
129#389—216no—
130#389—48no—
131#388—48no—
132#387—216no—
133#387—216no—
134#386—28no—
135#385—28no—
136#385—28no—
137#385—216no—
138#381—24no—
139#381—28no—
140#380—24no—
141#380—28no—
142#380D418yes—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—28no—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C2×#41—432no—
169C2×#41—432no—
170C2×#41—416no—
171C2×#41—416no—
172C2×#41—416no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#123—432no—
229C22⋊Q8—432no—
230C22⋊Q8—416no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
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—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D418yes—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C2×#4112yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344#41—4128no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—84no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×D4—216no—
356C2×D4—28no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4#121116yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+418yes—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—88no—
467C22⋊C4—216no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—84no—
479C42—88no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×#12114yes—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+414yes—
525C23#58512yes—
526C23#58512yes—
527Q8—416no—
528Q8—88no—
529D4—416no—
530D4—432no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—432no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
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—44no—
599C2×C4—88no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—84no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22C22×#12111yescharacteristic
622C22#344912yes—
623C22#348412yes—
624C22#348412yes—
625C22#348511yescharacteristic
626C22#348612yes—
627C22#348711yescharacteristic
628C22#349011yescharacteristic
629C22#349011yescharacteristic
630C22#349312yes—
631C22#349912yes—
632C22#350912yes—
633C22#353514yes—
634C22#354114yes—
635C22#355414yes—
636C22#355914yes—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—48no—
644C4—82no—
645C4—84no—
646C4—82no—
647C4—84no—
648C2—44no—
649C2—84no—
650C2#3515011yescharacteristic
651C2#3534811yescharacteristic
652C2#3566711yescharacteristic
653C2#4059914yes—
654C2#4635812yes—
655C2#4636112yes—
656C2#4788512yes—
657C2#4788812yes—
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×#2583C2semidirect
4. C2×#2313C2semidirect
5. #16661C2semidirect
6. #15512C2semidirect
7. #15360C2semidirect
8. #15354C2semidirect
9. #15346C2semidirect
10. #14761C2semidirect
11. #14660C2semidirect
12. #14297C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24062
C2×#26162
C2×#23132
#166612
#155122
#153602
#153542
#153462
#147612
#146602
#142972
#139372