Finite Group Databasebeta

← #4344940

Subgroups — #4344940

Order 512.

2402 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2579C211yesmaximal, characteristic
3#17057C212yesmaximal, complemented
4#16520C214yesmaximal, complemented
5#16507C212yesmaximal, complemented
6#15942C214yesmaximal, complemented
7#15622C212yesmaximal, complemented
8#15568C212yesmaximal, complemented
9#15101C212yesmaximal, complemented
10#14840C212yesmaximal, complemented
11#14821C212yesmaximal, complemented
12#14766C212yesmaximal, complemented
13#14513C212yesmaximal
14#14401C212yesmaximal
15#13784C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2579—28no—
20#2563—28no—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#396C2212yes—
25C2×#395C2214yes—
26C2×#393C2212yes—
27C2×#393C2212yes—
28C2×#390C2212yes—
29C2×#387C2212yes—
30C2×#386C2211yescharacteristic
31C2×#386C2211yescharacteristic
32C2×#386C2212yes—
33C2×#385C2211yescharacteristic
34C2×#385C2212yes—
35C2×#385C2211yescharacteristic
36C2×#385C2214yes—
37C2×#382C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2214yes—
46#1445C2214yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
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×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119#398—28no—
120#398—44no—
121#398—44no—
122#396—28no—
123#395—48no—
124#393—28no—
125#393—28no—
126#390—28no—
127#387—28no—
128#386—24no—
129#386—24no—
130#386Q8116yescomplemented
131#385—28no—
132#385—24no—
133#385—24no—
134#385—48no—
135#382—28no—
136#380—28no—
137#380—28no—
138#380—24no—
139#380—48no—
140C4×C4⋊C4—24no—
141C4×C22⋊C4—42no—
142C2×#95—42no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—42no—
179C22×D4C2×D414yes—
180C22×D4C2×Q818yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—42no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×Q818yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217C4×D4—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×Q818yes—
281C2×C22⋊C4C2×Q8116yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D414yes—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C2×Q8116yes—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C22×Q814yes—
331C24C22×Q814yes—
332C24C2511yescharacteristic
333C2×Q8—48no—
334C2×Q8—88no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8—88no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4C4⋊Q8116yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
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—22no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×Q818yes—
432C22×C4C22×Q818yes—
433C22×C4C22×Q814yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)12yes—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—84no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—28no—
461C42—48no—
462C42—84no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—24no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C2×C4⋊Q814yes—
494C23C2×C4⋊Q814yes—
495C23C23×D411yescharacteristic
496C23C23×Q812yes—
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23#58512yes—
501C23#58514yes—
502C23#58512yes—
503Q8—816no—
504D4—416no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—816no—
579C2×C4—88no—
580C2×C4C2×C4⋊Q814yes—
581C2×C4#53414yes—
582C22—28no—
583C22—28no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—42no—
597C22—42no—
598C22C22×C4⋊Q811yescharacteristic
599C22C2×#53411yescharacteristic
600C22#344911yescharacteristic
601C22#348512yes—
602C22#348812yes—
603C22#350114yes—
604C22#350112yes—
605C22#350112yes—
606C22#350414yes—
607C22#353711yescharacteristic
608C22#354011yescharacteristic
609C22#354212yes—
610C22#354512yes—
611C22#354512yes—
612C22#355114yes—
613C22#356614yes—
614C4—24no—
615C4—24no—
616C4—44no—
617C4—42no—
618C4—48no—
619C4—48no—
620C4—42no—
621C4—42no—
622C4—84no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4—82no—
627C2—44no—
628C2—44no—
629C2#3501111yescharacteristic
630C2#3595012yes—
631C2#3711911yescharacteristic
632C2#3726911yescharacteristic
633C2#4141912yes—
634C2#4684414yes—
635C2#4883912yes—
636C2#5090212yes—
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. #17057C2semidirect
4. #16520C2semidirect
5. #16507C2semidirect
6. #15942C2semidirect
7. #15622C2semidirect
8. #15568C2semidirect
9. #15101C2semidirect
10. #14840C2semidirect
11. #14821C2semidirect
12. #14766C2semidirect
130. #386Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25792
#170572
#165202
#165072
#159422
#156222
#155682
#151012
#148402
#148212
#147662
#145132
#144012
#137842