Finite Group Databasebeta

← #4933515

Subgroups — #4933515

Order 512.

3556 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2731C212yesmaximal, complemented
3C2×#2436C211yesmaximal, characteristic
4C2×#2434C212yesmaximal, complemented
5#16511C214yesmaximal, complemented
6#16510C212yesmaximal, complemented
7#15856C212yesmaximal, complemented
8#15825C214yesmaximal, complemented
9#15340C212yesmaximal
10#15325C214yesmaximal, complemented
11#14985C214yesmaximal, complemented
12#14969C212yesmaximal, complemented
13#13834C211yesmaximal, characteristic, complemented
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2212yes—
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×#121C2211yescharacteristic
21#2734—28no—
22#2731—216no—
23#2436—28no—
24#2434—216no—
25C2×#399C2214yes—
26C2×#399C2212yes—
27C2×#398C2212yes—
28C2×#396C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#394C2214yes—
33C2×#393C2214yes—
34C2×#393C2214yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#386C2214yes—
46C2×#385C2214yes—
47C2×#385C2214yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1437C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69#1434C2212yes—
70#1434C2214yes—
71C23×Q8—28no—
72C23×Q8—24no—
73C23×Q8—42no—
74C23×Q8—42no—
75C23×Q8—44no—
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8—44no—
90C2×#124—24no—
91C2×#124—28no—
92C2×#124—24no—
93C2×#124—44no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×#121—24no—
98C2×#121—42no—
99C2×#121—42no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#399—44no—
127#399—48no—
128#398—28no—
129#396—216no—
130#395—216no—
131#395—216no—
132#395—216no—
133#394—216no—
134#393—216no—
135#393—216no—
136#391—28no—
137#391—24no—
138#391D418yes—
139#391D4116yescomplemented
140#390—44no—
141#389—28no—
142#388—216no—
143#388—216no—
144#387—216no—
145#387—216no—
146#386—216no—
147#385—216no—
148#385—216no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—416no—
195C22×Q8—432no—
196C22×Q8—416no—
197C22×Q8—432no—
198C22×Q8—416no—
199C22×Q8C2×D414yes—
200C22×Q8C2×D418yes—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—44no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C4⋊Q8—432no—
235C4⋊Q8—432no—
236#124—416no—
237#124—416no—
238C22⋊Q8—432no—
239#121—416no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
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—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D418yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42C2×D414yes—
312C2×C42C2×D418yes—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95C2×D4116yes—
330#95C2×D4116yes—
331#95C2×D4116yes—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24H(5,𝔽2)12yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—4128no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—48no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×Q8#127116yes—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
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—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D418yes—
455C22×C4C22×D418yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+418yes—
464C4⋊C4—216no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—416no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C22⋊C4—232no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—44no—
486C42—48no—
487C42—44no—
488C42—48no—
489C42—84no—
490C23—24no—
491C23—22no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—28no—
497C23—24no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23#56314yes—
527C23C23×D411yescharacteristic
528C23C23×D412yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+414yes—
534C23#58512yes—
535Q8—416no—
536Q8—416no—
537Q8—432no—
538Q8—816no—
539D4—88no—
540D4—816no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—416no—
571C2×C4—432no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4C2×#12718yes—
609C2×C4C2×#12714yes—
610C22—24no—
611C22—28no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—48no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—82no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22C22×#12711yescharacteristic
628C22#348411yescharacteristic
629C22#348412yes—
630C22#348411yescharacteristic
631C22#348412yes—
632C22#348612yes—
633C22#348614yes—
634C22#348912yes—
635C22#348914yes—
636C22C2×#56311yescharacteristic
637C22#354412yes—
638C22#354511yescharacteristic
639C22#354714yes—
640C22#355012yes—
641C22#355214yes—
642C22#356112yes—
643C4—24no—
644C4—28no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—48no—
650C4—44no—
651C4—48no—
652C4—84no—
653C4—82no—
654C2—44no—
655C2—162no—
656C2#3539011yescharacteristic
657C2#3592712yes—
658C2#4613611yescharacteristic
659C2#4632512yes—
660C2#4954912yes—
661C2#4955714yes—
662C2#5153112yes—
663C2#5207211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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
2. C2×#2731C2semidirect
4. C2×#2434C2semidirect
5. #16511C2semidirect
6. #16510C2semidirect
7. #15856C2semidirect
8. #15825C2semidirect
10. #15325C2semidirect
11. #14985C2semidirect
12. #14969C2semidirect
13. #13834C2semidirect
139. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#27312
C2×#24362
C2×#24342
#165112
#165102
#158562
#158252
#153402
#153252
#149852
#149692
#138342