Finite Group Databasebeta

← #4344499

Subgroups — #4344499

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2322C211yesmaximal, characteristic
3#16666C212yesmaximal, complemented
4#16560C212yesmaximal, complemented
5#16481C212yesmaximal, complemented
6#16400C212yesmaximal, complemented
7#16060C214yesmaximal, complemented
8#15586C214yesmaximal, complemented
9#15102C212yesmaximal, complemented
10#15063C212yesmaximal, complemented
11#14826C212yesmaximal, complemented
12#14822C212yesmaximal, complemented
13#14513C212yesmaximal
14#14505C212yesmaximal
15#13960C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2563—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#393C2212yes—
24C2×#393C2212yes—
25#2322—28no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#388C2212yes—
29C2×#387C2212yes—
30C2×#386C2211yescharacteristic
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#385C2211yescharacteristic
34C2×#382C2212yes—
35C2×#381C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1445C2212yes—
45#1445C2214yes—
46#1445C2212yes—
47#1444C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2212yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1434C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
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×#123—24no—
88C2×#123—24no—
89C2×#123Q818yescomplemented
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119#400—44no—
120#400—44no—
121#393—28no—
122#393—28no—
123#390—28no—
124#389—28no—
125#388—28no—
126#387—28no—
127#386—24no—
128#386—28no—
129#385—28no—
130#385—24no—
131#382—28no—
132#381—24no—
133#380—28no—
134#380—28no—
135#380—42no—
136C4×C4⋊C4—24no—
137C4×C22⋊C4—24no—
138C2×#95—42no—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4C2×D414yes—
180C22×D4C2×Q814yes—
181C22×D4#4114yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
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×Q814yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220#123—232no—
221C4×D4—416no—
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—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—42no—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×Q818yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×Q814yes—
278C2×C22⋊C4C2×Q818yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
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#95—44no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—24no—
322C24—21no—
323C24—24no—
324C24—24no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C22×Q812yes—
331C24C22×Q812yes—
332C24C2×#4112yes—
333C24C2×#4112yes—
334C24C2511yescharacteristic
335C2×Q8—48no—
336C2×Q8—88no—
337C2×Q8—88no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4C22⋊Q8116yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
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—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q812yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—88no—
454C22⋊C4—28no—
455C22⋊C4—216no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—84no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—84no—
463C22⋊C4—88no—
464C42—28no—
465C42—48no—
466C42—84no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—28no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C2×C22⋊Q814yes—
500C23C2×C22⋊Q814yes—
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23#58511yescharacteristic
507C23#58511yescharacteristic
508C23#58514yes—
509C23#58514yes—
510Q8—816no—
511D4—416no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—416no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
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—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4—88no—
586C2×C4—816no—
587C2×C4—88no—
588C2×C4C2×C22⋊Q814yes—
589C2×C4#53414yes—
590C22—28no—
591C22—28no—
592C22—24no—
593C22—24no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22C22×C22⋊Q811yescharacteristic
609C22C2×#53411yescharacteristic
610C22#344911yescharacteristic
611C22#348512yes—
612C22#350012yes—
613C22#350111yescharacteristic
614C22#350111yescharacteristic
615C22#350414yes—
616C22#350511yescharacteristic
617C22#350511yescharacteristic
618C22#352114yes—
619C22#354012yes—
620C22#354512yes—
621C22#354512yes—
622C22#355112yes—
623C22#356614yes—
624C22#356614yes—
625C4—24no—
626C4—24no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—84no—
633C4—84no—
634C4—84no—
635C4—84no—
636C4—82no—
637C4—82no—
638C2—44no—
639C2—44no—
640C2#3501011yescharacteristic
641C2#3603112yes—
642C2#3711911yescharacteristic
643C2#3741911yescharacteristic
644C2#4138512yes—
645C2#4684912yes—
646C2#4693512yes—
647C2#4905912yes—
648C2#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. #16666C2semidirect
4. #16560C2semidirect
5. #16481C2semidirect
6. #16400C2semidirect
7. #16060C2semidirect
8. #15586C2semidirect
9. #15102C2semidirect
10. #15063C2semidirect
11. #14826C2semidirect
12. #14822C2semidirect
89. C2×#123Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23222
#166662
#165602
#164812
#164002
#160602
#155862
#151022
#150632
#148262
#148222
#145132
#145052
#139602