Finite Group Databasebeta

← #4344939

Subgroups — #4344939

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2579C211yesmaximal, characteristic
3#16603C212yesmaximal, complemented
4#16521C214yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#15766C214yesmaximal, complemented
7#15730C212yesmaximal, complemented
8#15406C212yesmaximal, complemented
9#15101C212yesmaximal, complemented
10#14763C212yesmaximal, complemented
11#14758C212yesmaximal, complemented
12#14720C212yesmaximal, complemented
13#14513C212yesmaximal
14#14401C212yesmaximal
15#13784C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2579—28no—
22#2542—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#396C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#390C2212yes—
31C2×#389C2214yes—
32C2×#387C2212yes—
33C2×#386C2211yescharacteristic
34C2×#386C2211yescharacteristic
35C2×#386C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4D418yes—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125#399—28no—
126#399—28no—
127#398—28no—
128#398—48no—
129#396—28no—
130#395—44no—
131#395—44no—
132#390—28no—
133#389—48no—
134#387—28no—
135#386—24no—
136#386—24no—
137#386Q8116yescomplemented
138#385—28no—
139#385—24no—
140#385—24no—
141#385—48no—
142#384—28no—
143#380—24no—
144#380—48no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—42no—
147C2×#95—42no—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4C2×D414yes—
183C22×D4C2×Q818yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—42no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×Q818yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220C22⋊Q8—432no—
221C22⋊Q8—432no—
222C4×D4—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×Q818yes—
292C2×C22⋊C4C2×Q8116yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42C2×D414yes—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q8116yes—
325C24—24no—
326C24—24no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×Q814yes—
340C24C22×Q814yes—
341C24C2511yescharacteristic
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—88no—
346C2×Q8—88no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4C4⋊Q8116yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
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—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×Q818yes—
440C22×C4C22×Q818yes—
441C22×C4C22×Q814yes—
442C22×C4H(5,𝔽2)18yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+414yes—
446C4⋊C4—216no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C42—28no—
472C42—48no—
473C42—84no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23C2×C4⋊Q814yes—
505C23C2×C4⋊Q814yes—
506C23C23×D411yescharacteristic
507C23C23×Q812yes—
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)14yes—
511C23C2×2−1+412yes—
512C23#58512yes—
513C23#58512yes—
514Q8—816no—
515D4—416no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
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—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4—816no—
590C2×C4—88no—
591C2×C4C2×C4⋊Q814yes—
592C2×C4#53414yes—
593C22—28no—
594C22—28no—
595C22—24no—
596C22—24no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—48no—
602C22—48no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22C22×C4⋊Q811yescharacteristic
610C22C2×#53411yescharacteristic
611C22#344911yescharacteristic
612C22#348412yes—
613C22#348912yes—
614C22#350112yes—
615C22#350114yes—
616C22#350112yes—
617C22#350314yes—
618C22#353314yes—
619C22#353811yescharacteristic
620C22#354011yescharacteristic
621C22#354112yes—
622C22#354312yes—
623C22#354312yes—
624C22#355914yes—
625C4—24no—
626C4—24no—
627C4—48no—
628C4—48no—
629C4—42no—
630C4—42no—
631C4—48no—
632C4—42no—
633C4—44no—
634C4—82no—
635C4—84no—
636C4—82no—
637C4—82no—
638C2—44no—
639C2—44no—
640C2#3501111yescharacteristic
641C2#3592812yes—
642C2#3723211yescharacteristic
643C2#3737411yescharacteristic
644C2#4060012yes—
645C2#4643614yes—
646C2#4796212yes—
647C2#5070012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16603C2semidirect
4. #16521C2semidirect
5. #16365C2semidirect
6. #15766C2semidirect
7. #15730C2semidirect
8. #15406C2semidirect
9. #15101C2semidirect
10. #14763C2semidirect
11. #14758C2semidirect
12. #14720C2semidirect
137. #386Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#25792
#166032
#165212
#163652
#157662
#157302
#154062
#151012
#147632
#147582
#147202
#145132
#144012
#137842